\magnification=\magstep1 \hsize=16.0true cm \hoffset=.0true cm \vsize=22.true cm \voffset=.4true cm \baselineskip=15pt \hfuzz=2pt \parindent=0truept \parskip=0pt \def\Bigskip{\vskip4em} \def\lemma#1{\underbar{\bf{Lemma.}}\ {#1}\vskip 3em} \def\reals#1{{\bf R}^{#1}} \def\norm#1{|\!|{#1}|\!|} \def\lnorm#1{\left|\!\left|{#1}\right|\!\right|} \def\parv{\par\vskip 1em} \def\lapl{\Delta} \def\repulsion{\oh\sum_{i\ne j}{1\over {|x_i-x_j|}}} \def\dzdr{\>{{dz\,dR}\over {R^5}}} \def\dr{{{dR}\over {R^5}}} \def\2int{\int\!\!\!\int} \font\BBigrm=cmr10 scaled \magstep4 \font\Bbigrm=cmr10 scaled \magstep3 \font\Bigrm=cmr10 scaled \magstep2 \def\sss{\scriptscriptstyle} \def\scapro#1,#2{\left<#1,#2\right>} \def\littlescapro#1,#2{\left<\right. #1,#2\left. \right>} \def\smallc{\scriptstyle c} \def\frac#1,#2{{}^{#1}\!\!\kern1pt \lower-1.0pt\hbox{$\scriptscriptstyle /$} \!{}_{#2}} \def\fra#1,#2{\hbox{$ #1\over #2$}} \def\ffra#1,#2{\hbox{$ \scriptscriptstyle{{ {\lower-0.5pt\hbox{$\scriptscriptstyle #1$}}\over #2}}$}} \def\BBracket#1{\Biggl(#1\Biggr)} \def\Bracket#1{\Bigl(#1\Bigr)} \def\bracket#1{\bigl(#1\bigr)} \def\bbracket#1{\biggl(#1\biggr)} \def\bra#1{\left(#1\right)} \def\braces#1{\left\{#1\right\}} \def\vs#1,#2{#1_1,\ldots,#1_{#2}} \def\vsz#1,#2{#1_0,\ldots,#1_{#2}} \def\abs#1{\left|#1\right|} \def\muchbigger{>\!\!\!>} \def\muchsmaller{<\!\!\!<} \def\bover#1,#2{{\displaystyle{#1}\over{\displaystyle{#2}}}} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def\Vol#1{{\rm\ Vol\,}\bra{#1}} \def\sign{{\rm sign\ }} \def\for{\qquad{\rm\ for\ }\quad} \def\and{\qquad{\rm\ and\ }\qquad} \def\bigo#1{{\it O}\!\bra{#1}} \def\littleo#1{{\it o}\!\bra{#1}} \def\eqbydef{\quad{\buildrel {\rm def} \over \equiv}\quad} \def\oh{\hbox{$ 1\over 2$}} \def\dxz{\,dx_1\,\cdots\,dx_Z} \def\brac#1{\left\{{#1}\right\}} \def\dist{{\rm dist}\,} \def\restr{\>\lceil\>} \def\domain{{\cal D}} \def\lipnorm#1{\lnorm{#1}_{\rm Lip}} \def\feffsec#1{ \centerline{\Bigrmb #1} \vskip3em \centerline{\bigrmb Charles L. Fefferman\footnote{*} {\rm Partially supported by a NSF grant at Princeton University}} \vskip1em \centerline{\it Department of Mathematics, Princeton University} \vskip2em \centerline{\bigrmb Luis A. Seco} \vskip1em \centerline{\it Department of Mathematics, California Institute of Technology} \vskip10em } \def\feffsectwo#1#2{ \centerline{\Bigrmb #1} \vskip1em \centerline{\Bigrmb #2} \vskip3em \centerline{\bigrmb Charles L. Fefferman\footnote{$ ^\ast $} {\medtype\rm Partially supported by an NSF grant at Princeton University}} \vskip1em \centerline{\it Department of Mathematics, Princeton University} \vskip2em \centerline{\bigrmb Luis A. Seco} \vskip1em \centerline{\it Department of Mathematics, California Institute of Technology} \vskip5em } \def\myaddress{\bigskip \centerline{Luis A. Seco} \centerline{ Department of Mathematics} \centerline{ Princeton University} \centerline{ Princeton NJ 08544} \centerline{ U.S.A} \bigskip} \def\mycopenhagenaddress{\bigskip \centerline{Luis A. Seco} \centerline{Matematisk Institut} \centerline{K\o benhavns Universitet} \centerline{ Universitetsparken 5} \centerline{DK--2100 K\o benhavn \O} \centerline{Denmark} \centerline{Tel. (Gerd Grubb) 45--3135 3133 Ext. 443} \centerline{Fax. 45--3135 4254} \centerline{e--mail (P. Solovej): philip@math.ku.dk} \bigskip} \def\myaarhusaddress{\bigskip \centerline{Luis A. Seco} \centerline{c/o E. Balslev} \centerline{Matematisk Institut} \centerline{ Ny Munkegade} \centerline{ Universitetsparken, 8000 \AA rhus} \centerline{Denmark} \centerline{Fax. 45--86--131769} \centerline{e-mail (E. Balslev): balslev@mi.aau.dk} \bigskip} \def\mymadridaddress{\bigskip \centerline{Luis A. Seco} \centerline{c/o Antonio C\'ordoba} \centerline{ Division de Matem\'aticas C--XVI} \centerline{ Facultad de Ciencias} \centerline{ Universidad Aut\'onoma de Madrid} \centerline{ Cantoblanco, 28049 Madrid} \centerline{ Spain} \centerline{Tel. (A. C\'ordoba): 34--1--3974986} \centerline{Fax. 34--1--3974889} \centerline{e-mail (A. Sanchez--Calle): sanca@emduam11.bitnet} \bigskip} \def\ourspainaddress{\bigskip \centerline{Luis A. Seco} \centerline{Maria T. Quintanilla} \centerline{ Chacoli Ategorri 4-6A} \centerline{ 48004 Bilbao} \centerline{ Spain} \centerline{Tel. 34--4--4117088} \bigskip} \def\luiswoodlawn{\bigskip \centerline{Luis A. Seco} \centerline{37 Woodlawn Av. West} \centerline{ Toronto, ONTARIO M4V 1G6} \centerline{ Canada} \centerline{Tel. 416--978 2033} \bigskip} \def\maitewoodlawn{\bigskip \centerline{Maria T. Quintanilla} \centerline{37 Woodlawn Av. West} \centerline{ Toronto, ONTARIO M4V 1G6} \centerline{ Canada} \centerline{Tel. 416--978 2033} \bigskip} \def\myspainaddress{\bigskip \centerline{Luis A. Seco} \centerline{ Chacoli Ategorri 4-6A} \centerline{ 48004 Bilbao} \centerline{ Spain} \centerline{Tel. 34--4--4117088} \bigskip} \def\mycaladdress{\bigskip \centerline{Luis A. Seco} \centerline{253--37 Caltech} \centerline{ Pasadena CA 91125} \bigskip} \def\supp{{\rm supp\ }} \font\BBigrm=cmr10 scaled \magstep4 \font\Bbigrm=cmr10 scaled \magstep3 \font\Bigrm=cmr10 scaled \magstep2 \font\bigrm=cmr10 scaled \magstep1 \font\smarm=cmr10 scaled \magstep0 \font\smfpt=cmr5 scaled \magstep0 \font\bfpt=cmr5 scaled \magstep1 \font\Bfpt=cmr5 scaled \magstep2 \font\bbfpt=cmr5 scaled \magstep3 \font\smept=cmr8 scaled \magstep0 \font\bept=cmr8 scaled \magstep1 \font\Bept=cmr8 scaled \magstep2 \font\bbept=cmr8 scaled \magstep3 \font\BBigrmb=cmbx10 scaled \magstep4 \font\Bbigrmb=cmbx10 scaled \magstep3 \font\Bigrmb=cmbx10 scaled \magstep2 \font\bigrmb=cmbx10 scaled \magstep1 \font\sigrmb=cmbx10 scaled \magstep0 \font\eightit=cmti8 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\eightsl=cmsl8 \font\eighttt=cmtt8 \def\thmstyleb{\vskip2em\goodbreak\noindent\bf } \def\thmstylea{\it} \def\intsend{\hskip1pt} \def\intsmid{\hskip1pt{,}\hskip1pt} \def\theorem#1{{\thmstyleb Theorem #1:} \thmstylea} \def\algorithm#1{{\thmstyleb Algorithm #1:} \thmstylea} \def\endt{\vskip2em\goodbreak\rm} \def\proof{\endt{\bf{Proof: }}} \def\cproof{\endt{\bf{Proof (Computer--Assisted): }}} \def\description{\endt{\bf{Description: }}} \def\corol#1{{\thmstyleb Corollary #1:} \thmstylea} \def\lem#1{{\thmstyleb Lemma #1:} \thmstylea} \def\propo#1{{\thmstyleb Proposition #1:} \thmstylea} \def\definition{\thmstyleb{\bf Definition: }\sl} \def\itemize{\parindent=40pt\parskip=8pt} \def\smallitemize{\parindent=15pt\parskip=8pt} \def\ttitle#1{\goodbreak\vskip4em\centerline{\BBigrmb{#1}}\vskip4em} \def\Title#1{\goodbreak\vskip4em{\Bigrmb{#1}}\vskip2em} \def\title#1{\goodbreak\vskip8pt\noindent{\bf{#1}}\enspace } \def\chapter#1{\vfill\eject\hbox{\ }\vskip60pt{\BBigrmb #1}\vskip6em} \def\senia#1{\hskip200pt\vtop{\hfuzz=5pt\baselineskip=4pt\hsize1.9in\raggedright \smept{#1}}} \def\mathendpf{\hbox{ \it Q\kern-2.5pt\lower2.5pt\hbox{\rm E}\lower-2pt\hbox{\kern-4.5pt D}} } \def\endpf{\hfill{ \it Q\kern-2.5pt\lower2.5pt\hbox{\rm E}\lower-2pt\hbox{\kern-4.5pt D}} \goodbreak \vskip2em} \def\coint#1,#2{[\intsend #1\intsmid #2\intsend)} \def\ocint#1,#2{(\intsend #1\intsmid #2\intsend]} \def\ooint#1,#2{(\intsend #1\intsmid #2\intsend)} \def\ccint#1,#2{[\intsend #1\intsmid #2\intsend]} \def\bibliography{ \vskip3em \Title{References}\vskip1em\parindent13pt} \def\medtype{\let\rm=\eightrm \let\bf=\eightbf \let \mus=\eightmus \let\tt=\eighttt \let\it=\eightit \let\sl\eightsl \baselineskip=3pt minus .75pt\rm} \def\Calogero{[Ca]} \def\BBcalogeroBB{\item{\calogero} Calogero, F. ``{\sl Variable Phase Approach to the Potential Scattering\/}'' Academic Press, NY 1967.\vskip1em} \def\Hughes{[Hu]} \def\BBHughesBB{ \item{\Hughes} Hughes, W. To appear in {\it Advances in Mathematics}. \vskip1em} \def\BBLiebtfBB{ \item{\Liebtf}Lieb, E. (1981) ``{\sl Thomas--Fermi and Related Theories of Atoms and Mol\-ecules}'' Reviews of Modern Physics Vol 53 no. 4. \vskip1em} \def\BBLiebSimonBB{ \item{\LiebSimon}Lieb, E. and Simon, B. (1977) ``{\sl Thomas--Fermi Theory of Atoms, Molecules and Solids}'' Adv. Math. 23, pp 22---116. \vskip1em} \def\SiedentopWeikard{[SW1]} \def\BBSiedentopWeikardBB{ \item{\SiedentopWeikard} Siedentop, H., Weikard, R. (1987) ``{\sl On the Leading Energy Correction for the Statistical Model of the Atom: Interacting Case}'' Communications in Mathematical Physics {\bf 112} 471-490 \vskip1em} \def\SiedentopWeikardLBII{[SW3]} \def\BBSiedentopWeikardLBIIBB{ \item{\SiedentopWeikardLBII} Siedentop, H., Weikard, R. (1990) ``{\sl A Lower Bound of Scott Type by a New Microlocalization Technique }'' To appear. \vskip1em} \def\SiedentopWeikardLB{[SW2]} \def\BBSiedentopWeikardLBBB{ \item{\SiedentopWeikardLB} Siedentop, H., Weikard, R. ``{\sl On the Leading Correction of the Thomas--Fermi Model: Lower Bound}'' and an appendix by A.M.K. M\"uller. Inv. Math., Vol., 97, pp 159---193, 1989. \vskip1em} \def\Schwinger{[Sc]} \def\BBSchwingerBB{ \item{\Schwinger} Schwinger, J. (1981) ``{\sl Thomas--Fermi Model: The Second Correction}'' Physical Review A24, {\bf 5}, 2353---2361. \goodbreak\vskip1em} \def\Dirac{[Di]} \def\BBDiracBB{ \item{\Dirac} Dirac, P. (1930) ``{\sl Note on Exchange Fenomena in the Thomas--Fermi Atom}'' Proc. Cambridge Philos. Soc. {\bf 26}, 376---385. \vskip1em} \def\Thomas{[Th]} \def\BBThomasBB{ \item{\Thomas} Thomas, L. H. (1927) ``{\sl The Calculation of Atomic Fields}'' Proc. Cambridge Philos. Soc. {\bf 23} 542---548. \vskip1em} \def\BBFermiBB{ \item{\Fermi} Fermi, E. (1927) ``{\sl Un Metodo Statistico per la Determinazione di alcune Priorieta dell'Atome}'' Rend. Accad. Naz. Lincei {\bf 6}, 602---607. \vskip1em} \def\Scott{[Sc]} \def\BBScottBB{ \item{\Scott} Scott, J. M. C. (1952) ``{\sl The Binding Energy of the Thomas--Fermi Atom }'' Phil. Mag. {\bf 43} 859---867. \vskip1em} \def\BBLiebBB{ \item{\Lieb} Lieb, E. H. (1979) ``{\sl A Lower Bound for Coulomb Energies }'' Phys. Lett. {\bf 70A} 444---446. \vskip1em} \def\GilbarTrudinger{[GT]} \def\BBGilbarTrudingerBB{ \item{\GilbarTrudinger} Gilbar, and Trudinger, (1979) ``{\sl Elliptic Partial Differential Equations }'' Springer-Verlag \vskip1em} \def\thesis{[Se1]} \def\BBthesisBB{ \item{\thesis}Seco, L. ``{\sl Lower Bounds for the Ground State Energy of Atoms }'' Thesis, Princeton University, 1989. \vskip1em} \def\BBFeffermanSeconIBB{ \item{\FeffermanSeconI} Fefferman, C., Seco, L. (1989) ``{\sl An Upper Bound for the Number of Electrons in a Large Ion}'' Proceedings of the Nat. Acad. Sci., USA Vol. {\bf 86}, no. 10, May 1989, pp 3464--3465.\vskip1em} \def\BBFeffermanSeconIIBB{ \item{\FeffermanSeconII} Fefferman, C., Seco, L. (1990) ``{\sl Asymptotic Neutrality of Large Ions }'' Comm. Math. Phys. Vol. 128, pp 109---130. \vskip1em} \def\BBSecoSigalSolovejBB{ \item{\SecoSigalSolovej} Seco, L., Sigal, I. M., Solovej, J. P., ``{\sl Bound on the Ionization Energy of Large Atoms }'' To appear in {\it Comm. Math. Phys.} \vskip1em} \def\Hille{[Hi]} \def\BBHilleBB{ \item{\Hille} Hille, E., ``{\sl On the Thomas--Fermi Equation }'' Proc. Nat. Acad. Sci, USA, 62, 7---10. \vskip1em} \def\FeffSeca{[FS1]} \def\BBFeffSecaBB{ \item{\FeffSeca} Fefferman, C. and Seco, L. ``{\sl The Ground--State Energy of a Large Atom }'' Bull. A.M.S., Vol {\bf 23} no. 2, 525---530, 1990 \vskip1em} \def\FeffSecb{[FS2]} \def\BBFeffSecbBB{ \item{\FeffSecb} Fefferman, C. and Seco, L. ``{\sl Eigenvalues and Eigenfunctions of Ordinary Differential Operators }'' To appear in {\sl Adv. Math.} \vskip1em} \def\FeffSecc{[FS3]} \def\BBFeffSeccBB{ \item{\FeffSecc} Fefferman, C. and Seco, L. ``{\sl The Eigenvalue Sum for a One--Dimensional Potential }'' To appear in {\sl Adv. Math.} \vskip1em} \def\FeffSecd{[FS4]} \def\BBFeffSecdBB{ \item{\FeffSecd} Fefferman, C. and Seco, L. ``{\sl The Density in a One-Dimensional Potential }'' To appear in {\sl Adv. Math.} \vskip1em} \def\FeffSece{[FS5]} \def\BBFeffSeceBB{ \item{\FeffSece} Fefferman, C. and Seco, L. ``{\sl The Eigenvalue Sum for a Three--Dimensional Radial Potential }'' To appear in {\sl Adv. Math.} \vskip1em} \def\FeffSecf{[FS6]} \def\BBFeffSecfBB{ \item{\FeffSecf} Fefferman, C. and Seco, L. ``{\sl The Density in a Three--Dimensional Radial Potential }'' To appear in {\sl Adv. Math.} \vskip1em} \def\FeffSecg{[FS7]} \def\BBFeffSecgBB{ \item{\FeffSecg} Fefferman, C. and Seco, L. ``{\sl On the Dirac and Schwinger Corrections to the Ground--State Energy of an Atom }'' To appear in {\sl Adv. Math.} \vskip1em} \def\Arnold{[Ar]} \def\BBArnoldBB{ \item{\Arnold} Arnold, V. ``{\sl Mathematical Methods of Classical Mechanics }'' Springer. \vskip1em} \def\EKW{[EKW]} \def\BBEKWBB{ \item{\EKW} Eckmann, J. P., Koch, H. and Wittwer, P. ``{\sl A computer Assisted Proof of Universality in Area Preserving Maps}'' Memoirs, A.M.S., Vol 289 (1984). \vskip1em} \def\EckmannWittwer{[EW]} \def\BBEckmannWittwerBB{ \item{\EckmannWittwer} Eckmann, J. P. and Wittwer, P. ``{\sl Computer Methods and Borel Summability Applied to Feigenbaum's equation}'' Lecture Notes in Mathematics {\bf 227}, Springer Verlag (1985). \vskip1em} \def\FeffermanLlave{[FL]} \def\BBFeffermanLlaveBB{ \item{\FeffermanLlave} Fefferman, C. and Llave, R., ``{\sl Relativistic Stability of Matter, I\/}'', {Revista Mate{\-}m\'atica Ibero{\-}ameri\-cana} Vol {2} no.1\&2, pp. 119-213 (1986) \vskip1em} \def\KaucherMiranker{[KM]} \def\BBKaucherMirankerBB{ \item{\KaucherMiranker} Kaucher, E. W. and Miranker, W. L., ``{\sl Self-validating Numerics for Function Space Problems\/}'', { Academic Press, New York (1984)}. \vskip1em} \def\LanfordLlave{[LL]} \def\BBLanfordLlaveBB{ \item{\LanfordLlave}Lanford, O. and Llave, R. ``{\sl Solution of the Functional Equation for Critical Circle Mappings with Golden Rotation Number}'' {\it In preparation}. \vskip1em} \def\Llave{[Ll]} \def\BBLlaveBB{ \item{\Llave} Llave, R. ``{\sl Computer Assisted Bounds in Stability of Matter}'' Computer Aided Proofs in Analysis, IMA Series in Math. and Appl. Vol {\bf 28}. Cincinnati (1989). Springer. \vskip1em} \def\Moore{[Mo]} \def\BBMooreBB{ \item{\Moore} Moore, R. E., ``{\sl Methods and Applications of Interval Analysis\/}'', \break S.I.A.M., Philadelphia (1979). \vskip1em} \def\Rana{[Ra]} \def\BBRanaBB{ \item{\Rana} Rana , D., ``{\sl Proof of Accurate Upper and Lower Bounds for Stability Domains in Denominator Problems\/}'', Thesis, Princeton University (1987) \vskip1em} \def\Seco{[Se2]} \def\BBSecoBB{ \item{\Seco} Seco, L., ``{\sl Computer Assisted Lower Bounds for Atomic Energies\/}'' Computer Aided Proofs in Analysis, IMA Series in Math. and Appl. Vol {\bf 28}, 241---251. Cincinnati (1989). Springer. \vskip1em} \def\Lohner{[Lo]} \def\BBLohnerBB{ \item{\Lohner} Lohner, R., ``{\sl Einschlie\ss ung der L\"osung gew\"ohnlicher Anfangs-- und Rand\-wert\-auf\-gaben und Anwendungen \/}'' Dissertation, Universit\"at Karlsruhe (TH), 1988. \vskip1em} \def\BBIvriiBB{\refno{\Ivrii} Ivrii, V. ``{\sl Weyl's Asymptotics for the Laplace--Beltrami Operator in Riemann Polyhedra}'' Dokl. A. N. SSSR {\bf 38}, 35 -- 38. \vskip1em} \def\BBSigalnBB{ \refno{\Sigaln .} Sigal, I. M. (1982) ``{\sl Geometric Methods in the Quantum Many--Body Problem. Nonexistence of Very Negative Ions \/}'' Comm. Math. Phys. 85, 309--324. \vskip1em} \def\BBRuskaiBB{ \refno{\Ruskai .} Ruskai, M.B. (1982) ``{\sl Absence of Discrete Spectrum in Highly Negative Ions\/}'' {\bf I \& II} Comm. Math. Phys. 82, 457---469 and 85, 325---327. \vskip1em} \def\BBSolovejhfBB{ \refno{\Solovejhf .} Solovej, J.P. (1991) ``{\sl Proof of the Ionization Conjecture in a Reduced Hartree--Fock Model\/}'' Inv. Math. 104, 291--311. \vskip1em} \def\HKSW{[HKSW]} \def\BBHKSWBB{ \item{\HKSW} Helffer, B., Knauf, A., Siedentop, H., Weikard, R. ``{\sl On the Absence of a First--Order Correction for the Number of Bound States of a Schr\" odinger Operator with Coulomb Singularity \/}'' To appear in ``{\it Comm. P.D.E.}'' \vskip1em} \def\secder{1} \def\fa{2} \def\tf{3} \def\sdzero{4} \def\sdtero{4} \def\inv{5} \def\sdone{4} \def\sdlema{\secder.2} \def\sdcorola{\secder.3} \def\sdtha{6.1} \def\sda{(\secder.3)} \def\sdb{(\secder.5)} \def\sdc{(\secder.2)} \def\sdd{(\secder.4)} \def\sde{(\secder.6)} \def\sdf{(\secder.7)} \def\sdg{(\secder.8)} \def\faa{(\fa.1)} \def\fab{(\fa.2)} \def\fac{(\fa.3)} \def\faaa{\fa.2} \def\faala{\fa.3} \def\falema{\fa.1} \def\tfeq{(\tf.3)} \def\tfa{(\tf.6)} \def\tfb{(\tf.7)} \def\tfc{(\tf.8)} \def\tfd{(\tf.4)} \def\tfe{(\tf.5)} \def\tff{(\tf.2)} \def\tfeqy{(\secder.1)} \def\tfeqzero{(\tf.10)} \def\tfg{(\tf.12)} \def\tfh{(\tf.13)} \def\tfj{(\tf.9)} \def\tfala{\tf.1} \def\tfalb{\tf.2} \def\tfald{\tf.5} \def\tfalc{\tf.3} \def\tfale{\tf.6} \def\tfalf{\tf.7} \def\tfalg{\tf.8} \def\tfalh{\tf.10} \def\tfalj{\tf.11} \def\tfalgi{\tf.12} \def\tfalk{\tf.13} \def\tfall{\tf.19} \def\tfalm{\tf.16} \def\tflema{\tf.14} \def\tflemb{\tf.21} \def\tflemxa{\tf.4} \def\tflemxb{\tf.9} \def\tflemxc{\tf.17} \def\tfdefw{(\tf.1)} \def\tfaldd{\tf.15} \def\tfxa{(\tf.11)} \def\tfxb{(\tf.15)} \def\tfeqinf{(\tf.14)} \def\sdonea{(\sdone.2)} \def\sdxxa{(\sdone.1a)} \def\sdxxb{(\sdone.1b)} \def\sdxxc{(\sdone.1c)} \def\invlema{\inv.1} \def\invlemb{\inv.2} \def\invalga{\inv.3} \def\inva{(\inv.1)} \def\invca{(\inv.2a)} \def\invcb{(\inv.2b)} \def\invcc{(\inv.2c)} \def\invcd{(\inv.2d)} \def\invce{(\inv.2e)} \def\invxa{(\inv.5)} \def\invxb{(\inv.4)} \def\invxc{(\inv.3)} \def\sdrt{\sdzero.6} \def\sdeqa{(\sdzero.16)} \def\sdeqb{(\sdzero.17)} \def\sdzdefba{(\sdzero.13a)} \def\sdzdefb{(\sdzero.13b)} \def\sdza{(\sdzero.15)} \def\sdzb{(\sdzero.19)} \def\sdzc{(\sdzero.18)} \def\sdzup{(\sdzero.14a)} \def\sdzupp{(\sdzero.14b)} \def\sdzfa{(\sdzero.21a)} \def\sdzfb{(\sdzero.21b)} \def\sdzg{(\sdzero.20)} \def\sdzh{(\sdzero.23)} \def\sdzalb{\sdzero.5} \def\sdzlema{\sdzero.4} \def\sdzconda{(\sdzero.22)} \def\sdtrt{\sdtero.2} \def\sdtdefb{(\sdtero.3)} \def\sdta{(\sdtero.5)} \def\sdtb{(\sdtero.10)} \def\sdtc{(\sdtero.9)} \def\sdtalb{\sdtero.3} \def\sdtalc{\sdtero.1} \def\sdteqa{(\sdtero.7)} \def\sdteqb{(\sdtero.8)} \def\sdtup{(\sdtero.4a)} \def\sdtupp{(\sdtero.4b)} \def\sdtconda{(\sdtero.11c)} \def\sdtxxa{(\sdtero.12)} \feffsectwo{Aperiodicity of the Hamiltonian Flow}{in the Thomas--Fermi Potential} \hfill \vtop{\hsize 200pt \obeylines \medtype \it \hfill``...que para sacar una verdad en limpio \hfill menester son muchas pruebas y repruebas.'' \vskip10pt \hfill\sl ``Don Quijote de la Mancha'', \bf M. de Cervantes.} \vskip3em In \FeffSeca\ we announced a precise asymptotic formula for the ground--state energy of a non--relativistic atom. The purpose of this paper is to establish an elementary inequality that plays a crucial role in our proof of that formula. The inequality concerns the Thomas--Fermi potential $V_{TF}(r)= -y(a r)/r$, %$a=(3\pi/2)^{\ffra 2,3}$ $a>0$, where $y(r)$ is defined as the solution of $$\left.\eqalign{y''(x)&=x^{-\frac 1,2}y^{\frac 3,2}(x)\cr y(0)&=1\cr y(\infty)&=0\cr}\right\}\eqno\tfeqy$$ (Without loss of generality, in what follows we will take $a=1$.) Define $$F(\Omega)=F_y(\Omega) =\int\bra{{y(x)\over x}-{\Omega^2\over x^2}}_+^{\frac 1,2}\,dx\qquad \Omega\in \ooint 0,{\Omega_c}$$ where $$\Omega_c^2=\sup_{r>0}u(r)=u(r_c)\qquad u(r)=ry(r)$$ The subscript for $F$ will be used whenever we want to emphasize the dependence of $F$ on $y$. \vskip1em Then, $F(\Omega)$ depends smoothly on $\Omega$ (\SiedentopWeikardLB), and our main result here is as follows: \theorem{1.1} $$F''(\Omega)\le c<0\qquad \hbox{for all\ } \Omega\in \ooint 0,{\Omega_c}\eqno\sdc$$ \endt This is a quantitative form of the non--periodicity of almost all zero--energy orbits for the Hamiltonian $$H=|\xi|^2+V_{TF}(|x|)$$ on $$\reals 6=\{(x,\xi)\mid x\in \reals 3\quad \xi\in \reals 3\}$$ In fact, an easy computation shows that a zero--energy orbit with angular momentum $\Omega$ is periodic if and only if the derivative $F'(\Omega)$ is a rational multiple of $\pi$ (see \Arnold.) Hence, Theorem~1.1 shows that closed zero--energy orbits arise for only countably many $\Omega$. \vskip1em Theorem~1.1 will be used in our later papers (\FeffSece\ and \FeffSecf) to control the density and eigenvalue sum arising from the three dimensional Schr\"odinger operator $$H_Z=-\lapl+Z^{\frac 4,3}V_{TF}\Bracket{Z^{\frac 1,3}|x|}$$ for large $Z$. Aperiodicity of zero--energy Hamiltonian paths is well-known to play a crucial role in the study of eigenvalues and eigenfunctions. In our setting, Theorem~1.1 enters because our formulas for the eigenvalue sum and density involve expressions of the form $$S=\sum_{1\le l0$, $y(0)=1$ and $\lim_{x\to\infty}y(x)=0$. \item{b.} There exists a point $r_c$ s.t. $u(x)0 $ for $0\le x\le r_c$, and $u'(x)<0 $ for $r_c\le x$. Also, $u''(r_c)<0$. } \vskip1em We will denote the two solutions of $u(r)=\Omega^2$ by $r_1(\Omega)< r_2(\Omega)$. We start by giving convenient formulas for the derivatives of $F$. We point out that similar formulas were given in \SiedentopWeikardLB. One of the reasons we need formulas of the kind stated below is to obtain expressions such as \sdf\ and \sdg\ below. Also, we will see that in the case of an analytic $y$, not only is $F$ analytic on $\ooint 0,{\Omega_c}$, but it admits an analytic extension beyond $\Omega_c$. However, 0 will be in general an essential singularity. \lem{\sdlema} Let $y$ be as above. The following formulas hold: $$\eqalignno{ F(\Omega)&=\int\bra{u(x)-\Omega^2}_+^{\frac 1,2}\,{dx\over x}\cr F'(\Omega)&=-\Omega\int\bra{u(x)-\Omega^2}_+^{-\frac 1,2}\,{dx\over x}\cr F''(\Omega)&= -\lim_{\delta\to 0} \bra{\int_ {r_1(\Omega)+\delta} ^{r_2(\Omega)-\delta} \bra{u(x)-\Omega^2}^{-\frac 3,2}y(x)\,dx +c(\Omega)\delta^{-\frac 1,2}}\cr}$$ where $c(\Omega)$ is uniquely specified by requiring the finiteness of the limit. Moreover, if $b$ is any number less than $r_2(\Omega)$, then $${d^2\over d\Omega^2} \int_{r_1(\Omega)}^b\bra{u(x)-\Omega^2}_+^{\frac 1,2}\,{dx\over x} $$ equals $$ -\lim_{\delta\to 0} \bra{\int_ {r_1(\Omega)+\delta} ^{b} \bra{u(x)-\Omega^2}^{-\frac 3,2}y(x)\,dx +c_1(\Omega)\delta^{-\frac 1,2}}$$ again, for a constant $c_1$ that makes the limit finite. The corresponding symmetric case also holds. \proof The first two formulas are trivial. For the third, let $$H(\delta,\Omega)=\Omega\int_{r_1(\Omega)+\delta} ^{r_2(\Omega)-\delta}\bra{u(r)-\Omega^2}^{-\frac 1,2}\>{dr\over r}$$ Note that the formula for $F''$ amounts to showing that $${d\over d\Omega}\lim_{\delta\to 0}H(\delta,\Omega)=\lim_{\delta\to 0} {d\over d\Omega}H(\delta,\Omega)\eqno\sda$$ Indeed, the left hand side equals $-F''$, whereas the right hand side equals $$\displaylines{ \quad\lim_{\delta\to 0}\Biggl\{ \Omega^2\int_{r_1(\Omega)+\delta} ^{r_2(\Omega)-\delta}\bra{u(r)-\Omega^2}^{-\frac 3,2}\>{dr\over r} +\int_{r_1(\Omega)+\delta} ^{r_2(\Omega)-\delta}\bra{u(r)-\Omega^2}^{-\frac 1,2}\>{dr\over r} \hfill\cr \hfill +\Omega\Bracket{{\bra{u\bra{r_2-\delta}-u(r_2)}^{-\frac 1,2}\over r_2-\delta}\>r'_2(\Omega) -{\bra{u\bra{r_1+\delta}-u(r_1)}^{-\frac 1,2}\over r_1+\delta}\>r_1'(\Omega)}\Biggr\}\quad\cr \qquad\qquad=\lim_{\delta\to 0}\Biggl\{ \int_{r_1(\Omega)+\delta} ^{r_2(\Omega)-\delta}\bra{u(r)-\Omega^2}^{-\frac 3,2}u(r)\>{dr\over r} \hfill \cr\hfill -\Omega\sum_{i=1,2}{|u'(r_i)|^{-\frac 1,2} |r'_i(\Omega)|\delta^{-\frac 1,2}(1+\bigo\delta)\over r_i(\Omega)}\Biggr\}\qquad\cr}$$ which agrees with the formula asserted for $-F''$, provided that this previous expression for $c(\Omega)$ $$ c(\Omega)=-\Omega\sum_{i=1,2}{|u'(r_i)|^{-\frac 1,2} |r'_i(\Omega)|\over r_i(\Omega)}$$ actually makes the limit above finite. \vskip1em Therefore, the lemma will follow if we show that both $H(\delta,\Omega)$ and ${\partial\over \partial\Omega}H(\delta, \Omega)$ converge uniformly on compact subsets of $\ooint 0,{\Omega_c}$ to $C^1$ functions. This will imply, first, that we can interchange limits in \sda, and, second, that the expression for $c(\Omega)$ above is the right one. \vskip1em In order to see this, consider the change of variables given by $$t(r)=\cases{\bra{\Omega_c^2-u(r)}^{\frac 1,2}&if $r\ge r_c$\cr -\bra{\Omega_c^2-u(r)}^{\frac 1,2}&if $r\le r_c$\cr}\eqno\sdd$$ Note that $t$ is smooth and strictly increasing in the range $\ooint 0,\infty$. We can therefore consider its inverse, $r(t)$, and use it to rewrite $$H(\delta,\Omega)=\Omega\int_{t_1(\delta,\Omega)}^{t_2(\delta,\Omega)}\bra{D^2 -t^2}^{-\frac 1,2}w(t) \,dt$$ where $$t_1=t(r_1+\delta)\qquad t_2=t(r_2-\delta) \qquad D^2=\Omega_c^2-\Omega^2\qquad w(t)={r'(t) \over r(t)}$$ Note that $w$ is smooth on $\ooint 0,{\Omega_c}$, and that $$t_1=-D\bra{1+\tau_1(\delta)} \qquad t_2=D\bra{1+\tau_2\bra{\delta}} \qquad c\delta\le|\tau_i|\le C\delta\quad{\rm\ for\ }i=1,2 \eqno\sdb$$ uniformly on compact subsets of $\ooint 0,{\Omega_c}$, which implies that $$H(\delta,\Omega)=\Omega\int_{D^{-1}t_1}^{D^{-1}t_2}\bra{1-t^2}^{-\frac 1,2} w(tD) \,dt$$ converges uniformly to the $C^1$ function $$H(0,\Omega)=\Omega\int_{-1}^{1}\bra{1-t^2}^{-\frac 1,2} w(tD) \,dt=-F'(\Omega).\eqno\sde$$ As for ${d\over d\Omega}H(\delta,\Omega)$, $${d\over d\Omega}H(\delta,\Omega)=\int_{D^{-1}t_1}^{D^{-1}t_2} (1-t^2)^{-\frac 1,2} {\partial\over \partial\Omega}\bbracket{\Omega w(tD)} \,dt+\Omega \sum_{i=1,2}G_i(\delta,\Omega)$$ with $$ G_i(\delta,\Omega)=\pm\bra{1-D^{-2}t_i^2}^{-\frac 1,2}w(t_i){\partial \over \partial\Omega}\bra{D^{-1}t_i}$$ The first term above converges with $\delta$ to the smooth function $$ \int_{-1}^{1} (1-t^2)^{-\frac 1,2} {\partial\over \partial\Omega}\bbracket{\Omega w(tD)} \,dt$$ uniformly on compact subsets of $\ooint 0,{\Omega_c}$. Thus, the lemma will follow if we prove that $G_i$ goes to zero with $\delta$ uniformly in $\Omega$. By \sdb, this will in turn follow if we prove that $${\partial \over \partial\Omega}\bra{D^{-1}t_i}=\bigo\delta$$ By \sdb\ again, it is enough to prove that $${\partial \over \partial\Omega}\bra{D^{-1}t_i}^2=\bigo\delta$$ But, for $i=1$, $$\eqalignno{ {\partial \over \partial\Omega}\bra{D^{-1}t_1}^2&= {\partial \over \partial\Omega}\bra{{u(r_1+\delta)-\Omega_c^2\over u(r_1)-\Omega_c^2}}\cr &= {\bra{u(r_1)-\Omega_c^2}u'(r_1+\delta)r_1'(\Omega)- \bra{u(r_1+\delta)-\Omega_c^2}u'(r_1)r_1'(\Omega) \over\bra{ u(r_1)-\Omega_c^2}^2}\cr &= {r_1'(\Omega)\over\bra{ u(r_1)-\Omega_c^2}^2}\cdot\cr &\qquad\cdot \Bracket{\bra{u(r_1)u'(r_1+\delta)- u(r_1+\delta)u'(r_1)}-\Omega_c^2 \bra{u'(r_1+\delta)- u'(r_1)}}\cr}$$ The first factor above is trivial. The other is clearly bounded by $C\delta$, and, doing the same for $i=2$, the lemma follows. \vskip1em The last remark in the statement of the lemma follows in exactly the same way, with the only modification that one of the $G_i$ is in fact constant in $\delta$, which of course does not affect the uniform approach to a $C^1$ function. \endpf A closer look at \sde\ yields the following remark: \corol{\sdcorola} Define $w(t)$ as in the proof of the previous lemma. Then $$ -F''(\Omega) = \int_{-1}^{1} (1-t^2)^{-\frac 1,2} {\partial\over \partial\Omega}\bbracket{\Omega w(tD)} \,dt$$ In particular, if $y\in C^{k}\ooint 0,\infty$, then $F_y\in C^{k-1}\ooint 0,{\Omega_c}$, $k\ge 2$. Also, if $y$ is analytic $F(\Omega)$ admits an analytic extension to a complex neighborhood of $\ocint 0,{\Omega_c}$. \proof If $y\in C^k$, the same is true for $u$. Therefore, $t\in C^{k-1}$, thus $r\in C^{k-1}\ooint -\Omega_c,{\Omega_c}$ and $r(t)\ne 0$, which implies $w\in C^{k-2}\ooint -\Omega_c,{\Omega_c}$, and, by (1.6), $F'\in C^{k-2}$. \vskip1em In the case of an analytic $y$, since $w$ is analytic in some neighborhood around 0, it admits a convergent power series expansion $$w(t)=\sum_{n=0}^\infty w_nt^n\qquad t0$$ exists, is finite and not zero. This shows, in particular, that $F$ has an essential singularity at $0$ and that $F$ is not a linear function. \vskip2em The proof of \sdc\ will now go as follows: We make an initial division of $\ooint 0,{\Omega_c}$ into two intervals $\ooint 0,{\bar\Omega}$ and $\ccint \bar\Omega, {\Omega_c}$, that we will refer to as Zone I and Zone II, respectively. In Zone I, we will use the formula in Lemma \sdlema\ to prove \sdc\ uniformly on very little subintervals of $\ooint 0,{\bar\Omega}$. We will deal with this in Section~\sdone. Then, formula \sdg\ will allow us to show \sdc\ uniformly on Zone II, as explained in Section \inv. \vskip1em Our proof will rely on a very precise knowledge of the solution to the Thomas--Fermi equation. For this, we will use computer assisted techniques. The next section deals with a description of how the computer will be used to yield theorems. \title{Acknowledgments} We wish to express our deepest gratitude to R. de la Llave: in addition to stimulating conversations, he taught us everything we know about computer--assisted proofs, gave us useful advice concerning the presentation of the paper, and went through the excruciating pain of checking our computer programs. We are also grateful to D. Rana for providing us with his interval arithmetic package. Finally, we thank the Department of Mathematics of the University of Texas at Austin for their help with the electronic distribution of the computer programs. \Title{\fa. Computer--Assisted Analysis} \vskip1em Let ${\cal R}$ be the set of ``representable numbers'' in a computer, that is those numbers that the computer can represent exactly. Depending on the specific machine, they are usually real numbers with some finite binary expansion. \vskip1em It is well known that computers can only perform arithmetic in an approximate way: the addition ---for example--- of two representable numbers is another representable number that will probably be close to the true sum, but is not exactly the true sum. \vskip1em The idea to perform rigorous arithmetic is to instruct the computer on how to produce upper and lower bounds to the true results of arithmetic operations between representable numbers; in other words, we work with intervals with endpoints in ${\cal R}$, and we implement arithmetic operations on intervals in such a way that given two intervals, the computer will produce a third that is guaranteed to contain the result of all arithmetic operations between points in the initial intervals. This is usually called ``interval arithmetic''. We denote the set of all these intervals by ${\cal I}$. Also, given a real function $f(x)$, we denote $$f(I)=\{f(x)\mid x\in I\}\qquad I\in{\cal I}$$ Binary functions of intervals are defined accordingly. In particular, a statement like $I_1>I_2$ means that $x>y$ for all pairs $(x,y)$, $x\in I_1$, $y\in I_2$. Also, given $I=\ccint a,b$ and $\epsilon\ge 0$, we introduce the shorthand notation $I\pm \epsilon$ to denote an interval containing $\ccint a-\epsilon,{b+\epsilon}$. We also point out, although it really is redundant, that in what follows, finite decimal expressions for numbers represent the rational numbers with exactly those decimal expansions. \vskip1em The next step is to perform a similar kind of arithmetic, but where objects are functions in some Banach space, not numbers. A convenient Banach space to use in this theory is the space of piecewise analytic functions, with a lower bound on the size of the domains of analyticity. \vskip1em Occasionally, it will be convenient to switch to genuine real variable theory, for which we will do our work on $C^0\ccint -1,1$. The reason for this is that inversion of functions in $\reals 1$ is a little easier than the complex counterpart, mainly because the domain of definition problem is trivial in the real case. We remark though, that the use of $C^0$ is not essential, and the same analysis could be carried over to $H^1$ with a little more work. \vskip2em More precisely, consider the Banach Algebras $$H^1=\left\{f(z)\>\mid\> f(z)=\sum_{n=0}^\infty a_n\,z^n,\quad \sum_{n=0}^\infty |a_n|<\infty\right\}$$ and $$C^0=\left\{f(x)\>\mid\> f{\rm\ is\ continuous\ on\ }\ccint -1,1\right\}$$ with norms $$\lnorm{f}_1=\sum_{n=0}^\infty \abs{a_n}\qquad \lnorm f_\infty=\sup|f(x)|$$ respectively. $H^1$ is a subspace of the set of analytic functions in the unit disk. Then, our substitute for intervals are sets ${\cal U}^1(\vsz I,N;C_h,C_g;k)$ of the form $$ \biggl\{f(z)=\sum_{n=0}^\infty a_n\,z^n+z^kg(z)\>\biggl| \> a_n\in I_n,\quad 0\le n\le N, \sum_{n=N+1}^\infty \abs{a_n}\le C_h \>,\>\lnorm g_1\le C_g\biggr\}\eqno\faa$$ where $C_h$ and $C_g$ are positive real numbers and $I_n$ are intervals in the real line. The parameter $k$ will generally be problem--dependent and fixed. For the computer implementation, $C_h$ and $C_g$ will run over the set of computer--representable numbers, and the intervals will be those with representable endpoints. We refer to $C_h$ and $C_g$ as high and general order error terms respectively, for obvious reasons. If intervals have nonempty interior and $C_h>0$, or if $k=0$ and $C_g>0$, then these sets are in fact a neighborhood basis for the topology induced by $\lnorm{\ }_1$. For this reason, we will refer to these ${\cal U}$ as ``neighborhoods'', even if in general they will not be. We will refer to them as neighborhoods of {\bf type} $k$ whenever we want to emphasize the integer $k$ in definition \faa. If $C_g=0$, we refer to them as type $\infty$. In general, ${\cal U}(k)$ means that ${\cal U}$ is a neighborhood of type $k$. Also, we will refer to them as being of {\bf order} $N$ to indicate that they consist of $N+1$ intervals. In our implementation, $N$ will not be fixed, but chosen adaptatively during the execution of the programs. \vskip1em The reason why this is a convenient space to work in is because elementary operations, such as addition, product, integration, differentiation (composed with a slightly contracting dilation), evaluation at a point and integration of initial value problems in ordinary differential equations can be conveniently bounded by elementary formulas in terms of this set of neighborhoods. By trivial scaling, we will be able to do analysis on $$H^1(|z-z_0|\le r)= \left\{f(z)\>\mid\> f(z)=\sum_{n=0}^\infty a_n\,\bra{z-z_0\over r}^n,\quad \sum_{n=0}^\infty |a_n|<\infty\right\}$$ a subspace of the set of analytic functions on the disk of center $z_0$ and radius $r$. \vskip2em As for $C^0$, we will use sets (that we will also refer to as ``neighborhoods'') of the type: $$\displaylines{\quad {\cal U}^0(\vsz I,N;C_h,C_g;k;S) = \biggl\{f(z)=\sum_{n=0}^N a_n\,z^n + z^{N+1}h(z) + z^kg(z) \>\biggl| \> \hfill\cr \hfill a_n\in I_n,\quad 0\le n\le N, \qquad\sup_{z\in S}\abs{h(z)}\le C_h\>,\> \sup_{z\in S} \abs{g(z)}\le C_g \biggr\}\qquad\qquad\llap\fab}$$ where $S$ is a subset of $\ccint -1,1$, and $h$ and $g$ are continuous functions on $S$. We will use the superscript 0 or 1 whenever we want to emphasize in which topology we are taking these ``neighborhoods''. Note the natural inclusion $${\cal U}^1(\vsz I,N;C_h,C_g;k)\subset {\cal U}^0(\vsz I,N;C_h,C_g;k;S)$$ for any $S\subset\ccint -1,1$. These sets of neighborhoods ${\cal U}^0(k)$ will not allow us to perform as many operations as their smaller brothers the ${\cal U}^1(k)$, but we can still add, multiply, raise to fractional powers and integrate (among others) in terms of them; furthermore, the formulas for these neighborhood operations are exactly the same as those for the ${\cal U}^1(k)$. \vskip1em We illustrate this neighborhood analysis describing how we can raise neighborhoods to real powers. At this point, we make the following remark concerning our use and description of algorithms: Algorithms describe a procedure that, if successful, will allow us to construct (usually upper and lower bounds for) certain numbers. When we describe these algorithms, we will state under which conditions they {\it fail}; a failure means that the procedure is stopped, an error reported, and no theorem proved. Obviously, if during the description of an algorithm, we use another algorithm, a failure in the execution of the latter implies also a failure of the former algorithm. \lem{\falema} Let $0 , \> 1+|\alpha|{\lnorm y\over 1-\lnorm{y}}} & $-1\le\alpha\le 2$\cr {\vbox{\vskip1em}} \cr \displaystyle (1-\lnorm y)^{-|\alpha|} & otherwise\cr} \cr \vbox{\vskip2em} \cr C_{\faaa}(\alpha,r) &\eqbydef \cases{ \displaystyle \min\bra{ |\alpha|+r{|\alpha|\,|\alpha-1|\over (1-r)^2}\> , \> |\alpha|(1-r)^{-|\alpha-1|}} & $-1\le\alpha\le 2$\cr \vbox{\vskip1em} \cr \displaystyle |\alpha|(1-r)^{-|\alpha-1|} & otherwise\cr} \cr }$$ \proof First, if $-1\le\alpha\le 2$ and $\norm {y_1}$, $\norm {y_2}\le r$, $$(1+y_1)^\alpha-(1+y_2)^\alpha=\sum_{n=1}^\infty{\alpha\choose n} \bra{y_1^n-y_2^n}$$ Now, $$ y_1^n-y_2^n = (y_1-y_2)\sum_{k=0}^{n-1}y_1^ky_2^{n-1-k}$$ and $$\lnorm{\sum_{k=0}^{n-1}y_1^ky_2^{n-1-k}}\le nr^{n-1}$$ Since $2\ge\alpha\ge -1$, $\abs{\alpha\choose n}$ is a decreasing sequence in $n$, for $n\ge 1$. Therefore, $$\eqalignno{ {\lnorm{T(y_1)-T(y_2)}\over\lnorm{y_1-y_2}}&\le \sum_{n=1}^\infty\abs{\alpha\choose n}nr^{n-1}\cr &\le |\alpha|+ r{|\alpha|\,|\alpha-1|\over (1-r)^2}\cr}$$ \vskip1em On the other hand, for $\alpha$ in the same range, $$\eqalignno{\lnorm{T(y)}&\le\sum_{n=0}^\infty \abs{{\alpha\choose n}} \lnorm y^n\cr &\le 1+\abs{\alpha}{\lnorm y\over 1-\lnorm y} \cr}$$ \vskip1em Now, for general $\alpha$, and $\norm {y_1}$, $\norm{y_2}\le r$, note that $$\lnorm{e^y}\le e^{\lnorm{y}}$$ and $$\lnorm{\log (1+y)}\le -\log (1-\lnorm{y})$$ Therefore, $$\lnorm{T^\alpha(y)}=\lnorm{e^{\alpha \log(1+y)}} \le (1-\lnorm{y})^{-|\alpha|}$$ and $$\eqalignno{ \lnorm{T^\alpha(y_1)-T^\alpha(y_2)}&= \abs{\alpha}\lnorm{(y_1-y_2)\int_0^1T^{\alpha-1}\bracket{ ty_1+(1-t)y_2}\,dt}\cr &\le \abs{\alpha}\lnorm{y_1-y_2} (1-r)^{-|\alpha-1|} &\mathendpf \cr}$$ \algorithm{\faala} Given a neighborhood ${\cal U}(\vsz I,N;C_h,C_g;k)$ satisfying {\itemize \item{1.} $I_0>0$. \item{2.} $|I_0|> \sum_{n>0}|I_n|+C_h+C_g$. \item{3.} if $\alpha>2$, then $2N>\alpha-1$. } \vskip1em we construct another, $\tilde{\cal U}(k)$, such that, if $f\in{\cal U}$ then $f^\alpha\in\tilde{\cal U}(k)$. The algorithm is independent of $k$, and of whether the neighborhoods are in $H^1$ or $C^0$. If $C_g=0$ for ${\cal U}$, then the same is true for $\tilde {\cal U}$. \description Assume first that $\alpha \ge -1$. Let $f\in{\cal U}$. Put $\tilde f=\bracket{f(0)}^{-1}\cdot f$, so $\tilde f=1+y(z)+z^kg(z)$, where $y(z)=z\tilde y(z)$, $$1+y(z)\in{\cal U}(\vsz {I'},N;C'_h,0;0)\eqno\fac$$ for $I'_i=f(0)^{-1}\cdot I_i$, $C'_h=f(0)^{-1}\cdot C_h$, and $\lnorm g\le C_g/f(0)$. Bounds for all this can be computed easily since we know that $f(0)\in I_0$. Now, $$\bracket{1+y(z)}^\alpha=\sum_{n=0}^N{\alpha\choose n}y(z)^n+h(z)$$ where $h(z)=z^{N+1}\tilde h(z)$. In the $H^1$ topology, $\norm h_1 =\norm {\tilde h}_1$, and $$\eqalignno{ \norm{\tilde h}_1&\le\sum_{n>N}\abs{{\alpha \choose n}}\lnorm{y}_1^n\cr & \le\abs{{\alpha\choose N+1}}{r^{N+1}\over 1-r}\cr}$$ for $$r=\sum_{i=1}^N\abs{I'_i}+C'_h$$ where we have used condition 3. in the statement of the algorithm. In the $C^0$ topology, $$\eqalignno{ \abs{\tilde h(z)}&\le\sum_{n>N}\abs{{\alpha \choose n}}\abs{{y(z)\over z}}^n\cr &\le \abs{{\alpha\choose N+1}}{r^{N+1}\over 1-r}\cr}$$ for $$r=\sum_{i=1}^N\abs{I'_i}+C'_h.$$ As a result of this, the computation of $\norm{\tilde h}$ is done in exactly the same way whether we are in the $C^0$ or $H^1$ topologies. Concerning the computation of the factor ${1\over 1-r}$, it is done as follows: we first check that $r\in\ooint 0,1$ the check for $r>0$ being unnecessary, harmless but convenient; then, we compute an upper bound for ${1\over 1-r}$ with our interval arithmetic package, knowing that an overflow will be reported and the program terminated if we cannot find such upper bound with machine--numbers. Also, $$(1+y(z))^\alpha-\bra{\tilde f(z)}^\alpha=\bigo{ z^k}$$ which implies that general errors are of type $k$. In the case that we are in $H^1$, since multiplication by $z$ is an isomorphism, by Lemma~\faaa, we see that general errors are bounded by $$\lnorm{(1+y(z))^\alpha-{\tilde f(z)}^{\alpha}}\le \lnorm{g} C_{\faaa}(\alpha,\lnorm{y}+\lnorm{g})$$ If, however, we are in $C^0$, apply Lemma 2.2 to $(\reals 1$,$+$,$\cdot)$, to get $$\eqalignno{ \abs{\bra{1+y(z)}^\alpha-{\tilde f(z)}^\alpha} &\le \abs{1+y(z)-\tilde f(z)}\cdot C_{2.2}\bra{\alpha, \lnorm g_\infty+\lnorm y_\infty} \cr &\le |z^k|\cdot\lnorm{g}_\infty \cdot C_{2.2}\bra{\alpha,\lnorm g_\infty+\lnorm y_\infty} \cr}$$ since $\abs{y(z)}$, $|\tilde f(z)-1|\le \lnorm{g}_\infty+\lnorm y_\infty$ and $C_{2.2}(\alpha,t)$ is increasing in $t$. \vskip1em Therefore, say that $$\sum_{n=0}^N{\alpha \choose n}y(z)^n\in{\cal U}_1(\vsz \tilde I,N; {\tilde C_h},0;\infty)$$ by \fac. Then, $$\bra{\tilde f}^{\alpha}\in{\cal U}(\vsz \tilde I,N; \tilde{\tilde C_h},\tilde{\tilde C_g};k)$$ with $$\tilde{\tilde C_h}=\tilde C_h+ \abs{{\alpha\choose N+1}}{r^{N+1}\over 1-r}$$ and $$\tilde{\tilde C_g}= f(0)^{-1}C_g\cdot C_{\faaa}\bra{\alpha,\lnorm{y}+C_gf(0)^{-1}}$$ and $$f^\alpha\in f(0)^\alpha\cdot {\cal U}(\vsz \tilde I,N; \tilde{\tilde C_h} ,\tilde{\tilde C_g}; k)$$ \vskip1em In the case $\alpha< -1$, we can find an integer $k$ such that $2^{-k}\alpha\ge -1$. Then, we can find a neighborhood containing $f^{2^{-k}\alpha}$. By ordinary multiplication we can thus construct a neighborhood containing $f^{\alpha}=\bra{f^{2^{-k}\alpha}}^{2^{k}}$. \endpf Although computer--assisted analysis has become fairly standard, we refer the reader to \Moore\ and \KaucherMiranker \ for a description of the basic ideas. The technique for solving ODE's is adapted from \Seco\ and \thesis, and is tailored to handle our particular ODE. See \Lohner\ for a thorough discussion on ODE solving techniques, with very good general algorithms. Also, we refer the reader to \EKW, \EckmannWittwer, \FeffermanLlave, \LanfordLlave, \Llave\ and \Rana\ for a sample of computer--assisted proofs of a wide variety of problems. Main ideas in our approach go back to those proofs. \vskip1em Our interval arithmetic package is an adaptation of the one used in \thesis\ and \Seco, which in turn is an adaptation of the one developed by D. Rana. See \Rana\ and \thesis\ for details on the software. \Title{\tf. The Thomas--Fermi Equation} In this section we will be concerned with the problem of getting good bounds for the solution of the Thomas--Fermi equation \tfeqy. It is well known (\Hille) that $$-w_0=\lim_{r\to 0}y'(r)<0\eqno\tfdefw$$ exists, and that $y$ admits a power series expansion $$y(r)=144 r^{-3}\Bracket{\sum_{n=0}^\infty b_nr^{-n\alpha}}\eqno\tff$$ convergent for $r$ large enough, with $b_0=1$, $b_1<0$ and $\alpha=\oh(\sqrt{73}-7)$. \vskip1em Also, $y$ is always positive, decreasing, and it is the only such solution of the ODE satisfying \tfdefw\ and \tff. \title{The Initial Value Problem away from the Singularities.} In this section we will be concerned with the solution to the Initial Value Problem $$\left.\eqalign{u''(x)&=x^{-\frac 1,2}u^{\frac 3,2}(x)\cr u(x_0)&=u_0\cr u'(x_0)&=u_1\cr}\right\}\eqno\tfeq$$ for $x_0, u_0>0$ The solution to this problem will be in terms of a function $f\in H^1$ satisfying $$u(x)=u_0+u_1\cdot r\cdot z+z^2f(z)$$ where $z=(x-x_0)/r$ and $r$ is a small positive representable number (in particular, $rds}\>dt$$ and that $T$ induces in a trivial way an operator $\tilde T$ of which $f$ is its fixed point. \vskip1em Throughout this section, we will do our work on $H^1$, and $\lnorm\,$ will always denote $\lnorm\,_1$. \algorithm{\tfala} We deduce conditions on $u_0$, $u_1$, $x_0$, $r$ and $\alpha$ under which $\tilde T$ is a well--defined contraction in $B(0,\alpha)\subset H^1$, and we compute an upper bound for $\lipnorm {\tilde T}$. \description Let $g=\sum a_nz^n$. Consider the operators $$\eqalign{ T_1(f)&=r\,u_1\,z+z^2f(z)\cr T_2(g)&=\bra{u_0+g(z)}^{\frac 3,2}\cr T_3(g)&=(rz+x_0)^{-\frac 1,2}\cdot g\cr T_4(g)&=r^2\sum_{n\ge 0}{a_nz^{n}\over (n+1)(n+2)}\cr}\eqno \tfd$$ It is clear that $$T(u)=u_0+ru_1z+z^2\,(T_4\circ T_3\circ T_2\circ T_1)(f)\eqno\tfe$$ and thus, $\tilde T= T_4\circ T_3\circ T_2\circ T_1$. Now, $T_1$ is affine with an isometry as the linear part, and $$\lipnorm{T_4}\le \oh r^2\eqno\tfa$$ Using Lemma \faaa, and putting $\beta=r/x_0$, we can see that $$\eqalignno{ \lipnorm{T_3}&\le \lnorm{(rz+x_0)^{-\frac 1,2}}_1\cr &\le x_0^{-\frac 1,2}K_{\faaa}(-\oh,\beta) &\tfb\cr}$$ Here we assume $\beta<1$, otherwise we say the algorithm fails. For $T_2$, we have $$\lipnorm{T_2}\le u_0^{\frac 1,2} C_{\faaa}(\fra 3,2,\gamma)\eqno\tfc$$ whenever $$\gamma\ge u_0^{-1}\sup_{\lnorm f\le \alpha}\lnorm{T_1(f)} $$ Since $$ u_0^{-1}\sup_{\lnorm f\le \alpha}\lnorm{T_1(f)} \le{r|u_1|+\alpha\over u_0} \eqbydef \gamma_0$$ we have $$\lipnorm{\tilde T}\le \oh r^2 x_0^{-\frac 1,2} u_0^{\frac 1,2} K_{\faaa}(-\oh,\beta) C_{\faaa}(\fra 3,2,\gamma_0)\eqno\tfj$$ Also, here we assume $\gamma_0<1$, or else the algorithm fails. \vskip1em Next, we need to show that $\tilde T$ maps $B(0,\alpha)$ into itself. In order to do this, note that $\lnorm{T_4(g)}\le\oh r^2\lnorm{g}$, which implies $$\eqalignno{ \lnorm{\tilde T(0)}&\le \oh r^2\lnorm{(rz+x_0)^{-\frac 1,2}}\cdot\lnorm{(u_0+ru_1z)^{\frac 3,2}} \cr &\le \oh r^2 x_0^{-\frac 1,2} u_0^{\frac 3,2} K_{\faaa}(-\oh,\beta) K_{\faaa}\bra{\fra 3,2,{r|u_1|\over u_0}}\cr}$$ Note that our assumption on $\gamma_0$ guarantees that the last term above is well--defined. Then, since $$\lnorm{\tilde T(f)}\le \lnorm{\tilde T(0)}+\alpha\lipnorm{\tilde T}$$ we see that $\tilde T$ maps $B(0,\alpha)$ into itself provided $$\oh r^2 x_0^{-\frac 1,2} u_0^{\frac 3,2} K_{\faaa}(-\oh,\beta) K_{\faaa}\bra{\fra 3,2,{r|u_1|\over u_0}} \le \alpha\bra{1-L}$$ whenever $L$ is an upper bound for $\lipnorm{\tilde T}$. The algorithm also reports a failure if the upper bound $L$ obtained using (3.9) is not strictly less than 1. \endpf Note that if the previous conditions are satisfied, we also know that the solution $u$ is strictly positive on $\ccint x_0-r,{x_0+r}$. Also, we know that it is defined as an analytic function on $|z-x_0|N}\abs{a_n}\le \lnorm{f-p}\le C \eqno\mathendpf$$ \vskip1em \title{Remark:} Note that the previous algorithm enables us to construct a neighborhood of type 2 that contains $u$ as a function of $z$. \algorithm\tfalc Given disjoint intervals $x_0^\ast$ and $x_1^\ast$, and representable $u_0$ and $u_1$, we construct intervals $y_0^\ast$ and $y_1^\ast$ such that the solutions $u$ to \tfeq\ with initial values $u_0$ and $u_1$ for $x\in x_0^\ast$ satisfy $$u(x')\in y^\ast_0\qquad u'(x')\in y^\ast_1$$ for any $x'\in x_1^\ast$. \description Choose a representable $r$ such that $r\ge |x_0^\ast-x_1^\ast|$, (if we can't, we report a failure) and run the previous algorithm for this $r$. Then, $y_0^\ast$ can be readily obtained by simply evaluating the neighborhood ${\cal U}$ produced by the algorithm at the interval $x_1^\ast$. In order to obtain $y_1^\ast$, we note that $$u'(x')=u_1+\int_x^{x'}u^{\frac 3,2}(s)s^{-\frac 1,2}\,ds$$ and this can be also easily computed. For a sharp bound, note that by the previous remark, we have $u(s) [\hbox{as a function of\ }{z={s-x\over r}}] \in{\cal U}(\vsz I,N;0,C;2)$, and thus, we also have $$u(s)^{\frac 3,2}s^{-\frac 1,2} [\hbox{as a function of\ }z]\in{\cal U}(\vsz I,N;C_h,C_g;2)$$ After integration, this reduces general error terms by a factor 3 compared to the ones that would follow from the weaker statement $$u(s)^{\frac 3,2}s^{-\frac 1,2} [\hbox{as a function of\ }z] \in{\cal U}(\vsz I,N;C_h,C_g;0) \eqno\mathendpf $$ The following lemma has a trivial proof. \lem{\tflemxa} Say $y_1$ and $y_2$ are positive solutions of $y''=x^{-\frac 1,2}y^{\frac 3,2}$ on the interval $\ccint x_1,{x_2}$, with $x_1>0$. {\itemize \item{1.} If $y_1(x_1)\ge y_2(x_1)$ and $y_1'(x_1)\ge y_2'(x_1)$ for all $x\in\ccint x_1,{x_2}$, then we have that $y_1(x)\ge y_2(x)$ and $y_1'(x)\ge y_2'(x)$ for all $x\in\ccint x_1,{x_2}$. \item{2.} If $y_1(x_2)\ge y_2(x_2)$ and $y_1'(x_2)\le y_2'(x_2)$ for all $x\in\ccint x_1,{x_2}$, then we have that $y_1(x)\ge y_2(x)$ and $y_1'(x)\le y_2'(x)$ for all $x\in\ccint x_1,{x_2}$. } \endt \definition Let $x_i^\ast=\ccint x_i^{\rm dn},{x_i^{\rm up}}$ for $i=1,2$ be two intervals. Then, we define $$x_1^\ast\cup_I x_2^\ast=\ccint \min_{i=1,2} x_i^{\rm dn},{\max_{i=1,2}x_i^{\rm up}}$$ \algorithm\tfald Given disjoint intervals $x_0^\ast$ and $x_1^\ast$, and intervals $u_0^\ast$ and $u_1^\ast$, we construct intervals $y_0^\ast$ and $y_1^\ast$ such that all solutions $u$ to \tfeq\ with initial values equal to any $u_0\in u_0^\ast$ and any $u_1\in u_1^\ast$, for any $x\in x_0^\ast$ are guaranteed to exist as positive solutions on $\ccint x,{x'}$, and furthermore satisfy $$u(x')\in y^\ast_0\qquad u'(x')\in y^\ast_1$$ for all $x'\in x_1^\ast$. \description Assume first that $x_0^\astx_1^\ast$, then we run the previous algorithm, first, for $u_0=u_0^{\rm dn}$ and $u_1=u_1^{\rm up}$, to obtain intervals $w^\ast_0$ and $w_1^\ast$, and, second, for $u_0=u_0^{\rm up}$ and $u_1=u_1^{\rm dn}$, two obtain intervals $z^\ast_0$ and $z_1^\ast$. It is then clear as before, that we can put $$ y^\ast_0 = w^\ast_0\cup_I z^\ast_0 \qquad y^\ast_1 = w^\ast_1\cup_I z^\ast_1 \eqno\mathendpf $$ \title{The Initial Value Problem at 0.} Here we will be concerned with the solution to the Initial Value Problem $$\left.\eqalign{u''(x)&=x^{-\frac 1,2}u^{\frac 3,2}(x)\cr u(0)&=1\cr u'(0)&=-w\cr}\right\}\eqno\tfeqzero$$ for $w>0$. In this case, the solution to this problem will be in terms of a function $f\in H^1$ satisfying $$u(x)=1-w\cdot r\cdot z^2+z^3f(z)\eqno\tfxa$$ where $z=(x/r)^{\frac 1,2}$ and $r$ is a small positive representable number. The solution of (3.10) can be viewed as the fixed point of $$T(u)=1+\int_{0}^x\bra{-w+\int_{0}^t{u^{\frac 3,2}(s) \over s^{\frac 1,2}}\>ds}\>dt$$ and again $T$ induces in a trivial way an operator $\tilde T$ of which $f$ is its fixed point. \algorithm{\tfale} We deduce conditions on $w$, $r$ and $\alpha$ under which $\tilde T$ is a contraction in $B(0,\alpha)$, and we compute an upper bound for $\lipnorm {\tilde T}$. \description Let $g=\sum a_nz^n$. Consider the operators $$\eqalign{ T_1(f)&=-r\,w\,z^2+z^3f(z)\cr T_2(g)&=\bra{1+g(z)}^{\frac 3,2}\cr T_3(g)&=4r^{\frac 3,2}\sum_{n\ge 0}{a_nz^{n}\over (n+1)(n+3)}\cr}\eqno \tfg$$ It is clear that $$T(u)=1-wx+z^3(T_3\circ T_2\circ T_1)(f)\eqno\tfh$$ and thus, $\tilde T= T_3\circ T_2\circ T_1$. Just as in Algorithm \tfala, $T_1$ is affine with an isometryt as the linear part, $\lipnorm{T_3}\le \fra 4,3 r^{\frac 3,2}$ and, for $T_2$, we have $$\lipnorm{T_2}\le C_{\faaa}(\fra 3,2,\gamma_0) $$ where, in this case $$\sup_{\lnorm f\le \alpha}\lnorm{T_1(f)} \le{rw+\alpha} \eqbydef \gamma_0$$ We check that $\gamma_0<1$; otherwise, the algorithm fails. Therefore, $$\lipnorm{\tilde T}\le \fra 4,3 r^{\frac 3,2} C_{\faaa}(\fra 3,2,\gamma_0) $$ Then we check that the upper bound for $\norm{\tilde T}_{\rm Lip}$ thus obtained is strictly less than 1; otherwise, the algorithm fails. \vskip1em Next, note that $\lnorm{T_3(g)}\le\fra 4,3 r^{\frac 3,2}\lnorm{g}$, which implies $$\eqalignno{ \lnorm{\tilde T(0)}&\le \fra 4,3 r^{\frac 3,2} \lnorm{(1-wrz^2)^{\frac 3,2}}\cr &\le \fra 4,3 r^{\frac 3,2}K_{\faaa}(\fra 3,2,wr)\cr }$$ Then we see that $\tilde T$ maps $B(0,\alpha)$ into itself provided $$\fra 4,3 r^{\frac 3,2} K_{\faaa}(\fra 3,2,wr)\le \alpha\bra{1-\lipnorm{\tilde T}} \eqno\mathendpf$$ \algorithm\tfalf Given representable $w$ and $r$ we construct a neighborhood $${\cal U}(\vsz I,N;0,C_g,0)$$ such that the solution of \tfeqzero\ is well--defined on $\ccint 0,r$ and satisfies $$u(x)=1-wx+z^3f(z)\qquad z=\bra{{x\over r}}^{\frac 1,2}$$ for $f\in{\cal U}$. \description Similar to Algorithm \tfalb. \algorithm\tfalg Given representable $w$ and $r$, we construct intervals $y_0^\ast$ and $y_1^\ast$ such that the solution $u$ of \tfeqzero\ satisfies $$u(r)\in y_0^\ast \qquad u'(r)\in y_1^\ast$$ \description $y_0^\ast$ can be obtained with a trivial variant of Algorithm~\tfalc, via Algorithm~3.7. For $y^\ast_1$, note that, if we put $$u^{\frac 3,2}(x)=(T_2\circ T_1)f(z)=\sum_{n\ge 0}a_nz^n$$ then $$\eqalignno{ u'(r)&=-w+\int_0^r{u^{\frac 3,2}(x)\over x^{\frac 1,2}}\>dx\cr &=-w+r^{\frac 1,2}\sum_{n=0}^\infty{2a_n\over n+1}\cr}$$ Note now that in our representation $z=(x/r)^{\frac 1,2}$, we have a neighborhood of type 3 containing $u(x)$ as a function of $z$. We can thus construct another neighborhood of type 3 such that $$\sum_{n=0}^\infty a_nz^n\in{\cal U}(\vsz I,N;C_h, C_g; 3)$$ Thus, $$u'(r)\in -w+r^{\frac 1,2}\bra{\sum_{n=0}^N {2I_n\over n+1}\pm\epsilon}$$ whenever $$\abs{\epsilon}\ge {2C_h\over N+2}+\oh C_g \eqno\mathendpf $$ \lem{\tflemxb} Let $u_1$ and $u_2$ be the solutions of \tfeqzero, with values $w_1$ and $w_2$, $w_1u_2(x)$ and $u_1'(x)>u'_2(x)$ for $x\in\ccint 0,R$. \proof Let $f_1$ and $f_2$ be associated with $u_1$ and $u_2$ as in \tfxa. Since $$u_1(x)\ge 1-w_1x-z^3\lnorm{f_1}$$ and $$u_2(x)\le 1-w_2x+z^3\lnorm{f_2}$$ for all $x$ small enough, we have that $u_1(x)>u_2(x)$ and thus $u_1''(x)>u_2''(x)$. Since the $u''_i$ are integrable at the origin, we conclude that $$ u_1'(x)=\int_0^xu''_1(t)\,dt-w_1 > \int_0^xu''_2(t)\,dt-w_2 = u_2'(x)$$ for all $x$ small enough. The lemma now follows from Lemma \tflemxa. \endpf \algorithm\tfalh Given representable $r$ and $t$, and an interval $ w^\ast$, we construct intervals $y_0^\ast$ and $y_1^\ast$ such that any solution $u$ of \tfeqzero\ for any $w\in w^\ast$ can be continued to $\ccint 0,t$ and satisfies $$u(t)\in y_0^\ast \qquad u'(t)\in y_1^\ast.$$ \description Run Algorithm 3.8 twice, once for each endpoint of $w^\ast$, to obtain two pairs of intervals $w_0^\ast$, $w_1^\ast$ and $z_0^\ast$, $z_1^\ast$. Lemma 3.9 then shows that all solutions of (3.10) with $w\in w^\ast$ are bounded above and away from 0, and can thus be extended as well--defined positive functions over $\ccint 0,t$. Then, Lemma~3.9 again allows us to put $$y_0^\ast=w_0^\ast\cup_I z_0^\ast \qquad y_1^\ast=w_1^\ast\cup_I z_1^\ast \eqno\mathendpf $$ \vskip2em \title{The Initial Value Problem at Infinity.} Here we will be concerned with the solution to the Initial Value Problem $$\left.\eqalign{u''(x)&=x^{-\frac 1,2}u^{\frac 3,2}(x)\cr u(\infty)&=0\cr b_1&=b\cr}\right\}\eqno\tfeqinf$$ where the last condition is interpreted in the sense of \tff. The solution to this problem in this case will be expressed as $$u(x) = {144\over x^3}\bra{1+bx^{-\alpha}+z^2f(z)}\eqno\tfxb$$ where $f\in H^1$, $z=R^\alpha x^{-\alpha}$, for some $R$ large. In this case, the operators involved are not so obvious. Define $$\eqalignno{ T_1(f)&= bR^{-\alpha}z+z^2f(z)\cr T_2(g) &= (1+g)^{\frac 3,2}\cr T_3(g) &= 12\sum_{n=2}^\infty{a_n z^{n-2}\over (n\alpha+3)(n\alpha+4)} \cr}$$ where, in the last formula, $g(z)=\sum_{n\ge 0}a_nz^n$. Then, put $$\tilde T=T_3\circ T_2\circ T_1$$ We now check that if $f$ is a fixed point of $\tilde T$ in $H^1$, then $u$ defined as in (3.15) solves (3.14). Note first that $$\eqalignno{ {u^{\frac 3,2}(x)\over x^{\frac 1,2}}&= {12\cdot 144\over x^5}\bra{T_2\circ T_1}(f)\cr}$$ where $$\bra{T_2\circ T_1}(f) = \sum_{n=0}^\infty {a_n}z^n\qquad a_0=1\quad a_1= \fra 3,2bR^{-\alpha}$$ Therefore, since $u$ and its derivatives vanish at $\infty$, $$\eqalignno{ u(x)&=\int_x^\infty\int_r^\infty {u^{\frac 3,2}(t)\over t^{\frac 1,2}}\,dt\, dr \cr &={144\over x^3}\bra{1+{12\,a_1\over (3+\alpha)(4+\alpha)}z+z^2{\tilde T}(f)} \cr} $$ Since $\alpha=\oh(\sqrt{73}-7)$ satisfies the equation $(\alpha+3)(\alpha+4)=18$, $u$ satisfies (3.14). \vskip1em The problem here is considerably more subtle than in the previous cases, due to the fact that $T_3$ does not scale with $R$. As a consequence, contraction properties of $\tilde T$ either hold or don't, and taking large $R$ won't help much. We are lucky, however, that the norm of $T_2$ is essentially $\fra 3,2$, and that the norm of $T_3$ is essentially $${12\over (2\alpha+3)(2\alpha+4)}<\oh$$ which says that the Lipschitz norm of $\tilde T$ will approximately be $\fra 3,4$. We make this precise now. \lem{\tfalj} Put $\beta=0.3$. Assume that $\abs{\bar b }= R^{-\alpha}|b|\le 0.23$. Then $\tilde T$ is a contraction in $B(0,\beta)$, and $\lipnorm {\tilde T}\le 0.8652$. \endt \title{Proof: (Calculator--Assisted)} Let $f_1$, $f_2\in B(0,\beta)$, and put $\bar f=f_1-f_2$. $$(T_2\circ T_1)(f)=1+\fra 3,2\bra{\bar bz+z^2f}+\fra 3,8 \bra{\bar b^2 z^2+2\bar bz^3f+z^4f^2}+\sum_{n\ge 3}{\fra 3,2\choose n} \bra{\bar b z+z^2f}^n$$ So, $$\displaylines{ \quad(T_2\circ T_1)(f_2)-(T_2\circ T_1)(f_2)=\fra 3,2z^2 \bar f+\fra 3,4\bar bz^3\bar f+\fra 3,8 \bra{z^4(f_1^2-f_2^2)}\hfill\cr \cr\hfill+\sum_{n\ge 3}{\fra 3,2\choose n} \bra{\bra{\bar b z+z^2f_1}^n -\bra{\bar b z+z^2f_2}^n}\cr}$$ Now, since $T_3$ is linear, bounded, and the sum converges absolutely, we have $$\displaylines{ \quad\tilde T(f_1)-\tilde T(f_2)= \fra 3,2 T_3(z^2\bar f)+\fra 3,4\bar bT_3(z^3\bar f)+\fra 3,8 T_3\bra{z^4(f_1^2-f_2^2)}\hfill\cr \hfill+\sum_{n\ge 3}{\fra 3,2\choose n} T_3\bra{\bra{\bar b z+z^2f_1}^n -\bra{\bar b z+z^2f_2}^n}\cr}$$ Note now that, for any $f\in H^1$, we have $$\lnorm{T_3(z^kf)}\le {12\over (k\alpha+3)(k\alpha+4)}\lnorm{z^kf}$$ and that $$\eqalignno{ \bra{\bar b z+z^2f_1}^n -\bra{\bar b z+z^2f_2}^n&=z^{n+1}h(z)\cr \lnorm{\bra{\bar b z+z^2f_1}^n -\bra{\bar b z+z^2f_2}^n}&\le n\lnorm{f_1-f_2}(|\bar b|+\beta)^{n-1}\cr}$$ Thus, $$\eqalignno{ \lipnorm{\tilde T}&\le {3\over 2}{12\over (2\alpha+3)(2\alpha+4)}+ {3|\bar b|\over 4}{12\over (3\alpha+3)(3\alpha+4)}+ {3\over 8}{12\cdot 2\cdot\beta\over (4\alpha+3)(4\alpha+4)}\cr &\qquad+ \sum_{n\ge 3}\abs{\fra 3,2\choose n} {12n \over \bracket{(n+1)\alpha+3}\bracket{(n+1)\alpha+4}} (|\bar b| +\beta)^{(n-1)}\cr &\le 0.72+.27|\bar b|+0.21\beta+ X+ Y{(|\bar b|+\beta)^{20}\over 1-|\bar b|-\beta} \cr&\le 0.8652\cr }$$ where we have set $$X=\sum_{n=3}^{20} \abs{\fra 3,2\choose n} {12n \over \bracket{(n+1)\alpha+3}\bracket{(n+1)\alpha+4}} (|\bar b| +\beta)^{(n-1)}$$ and $$ Y\eqbydef \abs{\fra 3,2\choose 21} {12\cdot 21 \over \bracket{22\alpha+3}\bracket{22\alpha+4}} \ge\abs{\fra 3,2\choose n} {12n \over \bracket{(n+1)\alpha+3}\bracket{(n+1)\alpha+4}} $$ for $n\ge 21$, and we have used $$X\le 0.019\qquad Y{(|\bar b|+\beta)^{20}\over 1-|\bar b|-\beta} \le 9\cdot 10^{-10}$$ \vskip1em On the other hand, $${(T_2\circ T_1)(0)}=\sum_{n\ge 0}{\fra 3,2 \choose n}{|\bar b|}^nz^n$$ Thus, $$\eqalignno{ \lnorm{\tilde T(0)}&\le\sum_{n\ge 2}\abs{\fra 3,2 \choose n} {12{|\bar b|}^n\over (n\alpha+3)(n\alpha+4)}\cr &\le \fra 3,{16}{|\bar b|}^2+ 0.0225{|\bar b|}^3 +0.0066{{|\bar b|}^4\over 1-{|\bar b|}}\cr &\le 0.01022\cr}$$ Therefore, $$\lnorm{\tilde T(f)}\le 0.01022+0.8652\beta \le \beta$$ and $\tilde T$ maps $B(0,\beta)$ into itself. \endpf \algorithm{\tfalgi}Given $b^\ast$ (interval) and $R$ (representable), we produce ${\cal U}_1$ such that, for any $b\in b^\ast$, the solution $u$ of \tfh\ is given by $$y(x)={144\over x^3}\bra{1+bx^{-\alpha}+z^2f(z)}\qquad z=R^\alpha x^{-\alpha}$$ with $f\in{\cal U}_1$. Here, ${\cal U}_1$ depends only on $b^\ast$, i.e., it is independent of which particular $b$ in $b^\ast$ we are considering. \description We first check that we are in the hypothesis of Lemma \tfalj. In this case, $\tilde T$ has a fixed point $f$, and, as we saw before, $y$ defined as above satisfies the ODE. \vskip1em In order to obtain bounds for $f$, we first look for a heuristic guess $p$: for example, we iterate $\tilde T$ (and truncate) a few times, starting with the function 0. Then, since computing rigorously $\tilde Tp$ for all $b\in b^\ast$ poses no difficulty in view of Algorithm~\faala, we conclude that $$\lnorm{f-p}\le {\lnorm{\tilde Tp-p}\over 1-0.8652} \le 7.5\lnorm{\tilde Tp-p}\qquad{\rm all\ }b\in b^\ast $$ Note that $p$ is the same for all $b\in b^\ast$, but $\tilde Tp$ still depends on $b$. However, the computation of $$\sup_{b\in b^\ast}\lnorm{\tilde Tp-p}$$ poses no problem, since it is less than or equal to $\lnorm{\tilde Tp-p}$ in the interval arithmetic sense. The algorithm fails if the hypothesis of Lemma~3.11 are not met, or if $\lnorm p > 0.3$. \endpf \algorithm{\tfalk}Given $b$ and $R$, we produce two intervals $u_0^\ast$ and $u_1^\ast$ such that, if $u$ is the solution to $\tfh$, we have $$u(R)\in u_0^\ast \qquad u'(R)\in u_1^\ast$$ \description First, run Algorithm~3.12 for these values of $b$ and $R$. Again, it is easy to obtain $u_0^\ast$. Let $f$ be related to $u$ as in \tfxb. Then, say $$\bra{1+bx^{-\alpha}+z^2f(z)}^{\frac 3,2} =\sum_{n\ge 0} a_nz^n \in {\cal U}(\vsz I,N;C_h,C_g;2)$$ Then, $$\eqalignno{ u'(R)&=-\int_{R}^\infty{144\cdot 12\over x^5}\sum a_nz^n\>dx\cr &= {-144\cdot 12\over R^4}\sum {a_n\over n\alpha+4}\cr &\in {-144\cdot 12\over R^4}\bra{\sum_{n=0}^N {I_n\over 4+n\alpha} \pm\epsilon}\cr}$$ with $$\abs{\epsilon}\le {C_h\over 4+(N+1)\alpha}+{C_g\over 4+2\alpha} \eqno\mathendpf $$ \vskip2em \title{Remark:} Note that it is enough to run this algorithm for representable values of $b$, due to the monotonicity of the T--F equation (Lemma 3.4). We omit the trivial details, which are similar to those in Algorithm~3.5 \vskip1em \title{The Boundary Value Problem} Next we discuss how to solve the Boundary Value Problem $$\left.\eqalign{u''(x)&=x^{-\frac 1,2}u^{\frac 3,2}(x)\cr u(0)&=1\cr u(\infty)&=0\cr}\right\}$$ \vskip1em We first describe how to obtain bounds for $w_0$. \lem{\tflema} Let $u$ be the solution of \tfeq, with $u_1<0$. If $${2u_0^{\frac 5,2}\over x_0^{\frac 1,2}}\le u_1^2$$ then, there exists a point $t> x_0$ such that $u$ can be extended as a well--defined positive solution of the ODE to $\coint x_0,t$ and, furthermore, $\inf_{x\in\ooint x_0,t} u(x)=0$. \proof Assume the lemma is false. It follows from general ODE considerations that, either $u$ can be extended as a positive well--defined solution of the ODE, or else there exists a $T$ such that $\sup_{x\in\ooint x_0,T} u(x)=\infty$. Let $$d={|u_1|x_0^{\frac 1,2}\over u_0^{\frac 3,2}}$$ and note that in both of the two cases above $u$ extends to a well--defined positive solution of the ODE to $\ooint x_0,{x_0+d}$ and furthermore, $u\le u_0$ on $\ccint x_0,{x_0+d}$. Indeed, consider two cases: {\itemize \item{a.}$u$ can be extended as a positive solution of the ODE all the way up to $\infty$. Then, if $u'(x)<0$ it is trivial. Otherwise, let $x_1>x_0$ be the first (and only) zero of $u'$; this means in particular that $u\le u_0$ on $\ccint x_0,{x_1}$. Then, our claim follows by noting that $$ |u_1|\le \sup_{\ooint x_0,{x_1}}u''\cdot|x_0-x_1|\le {u_0^{\frac 3,2} x_0^{-\frac 1,2}}\abs{x_0-x_1}$$ which implies $\ccint x_0,{x_0+d}\subset\ccint x_0,{x_1}$. \item{b.}$u$ can be extended as a positive solution of the ODE all the way up to $T$, where it blows up. Since $u'(x_0)<0$, there exists $x_1$, such that $x_0w_0$. \description By repeated applications of the previous algorithms, we can determine points $x_i$ and intervals $I_i$, $I'_i$, for $i=0,\ldots,n$, for $n$ large, such that the solution to the TF equation with initial values $u(0)=1$, $u'(0)=-w$ satisfies $u(x_i)\in I_i$ and $u'(x_i)\in I'_i$. These algorithms also guarantee us that $u$ does not vanish on $\ccint 0,{x_n}$. \vskip1em If, for some $i$, we have $I_i0$, this implies that for some $r_0w$. \vskip1em On the other hand, we know that if $u$ becomes arbitrary small on $\ooint 0,t$ for some $t$, then $w_00$ on $\coint M,\infty$, then we have that $u_1(x)\le u_2(x)$ and $u_1'(x)\ge u_2'(x)$ for all $x\in\coint M,\infty$. \proof Obviously it is enough to assume $a_1R$ and thus $u''_1(x)R$. Now, note that $$u_i'(x)=-\int_x^\infty u''_i(t)\,dt\qquad{\rm for\ }i=1,2$$ which implies that, not only do we have $0R$, but also $0>u'_1(x)>u'_2(x)$ for all $x>R$. Finally, if $R$ is larger than $M$, we apply Lemma~{\tflemxa} to guarantee that $0b_1$. Also, assuming bounds for $b_1$, and given $w$, we indicate whether $ww_0$. \description Let $y$ be the Thomas--Fermi function, and $u$ be the solution of \tfh. Assuming bounds for $w_0$, Algorithm~3.16 allows us to produce representable $x_i$ and intervals $I_i$ and $I'_i$, such that $y(x_i)\in I_i$ and $y'(x_i)\in I'_i$. For these $x_i$, using Algorithm~3.13 and repeated applications of Algorithm~3.5 (going backwards), we can produce intervals $J_i$ and $J'_i$ such that $u(x_i)\in J_i$ and $u'(x_i)\in J'_i$. In this situation we can again guarantee that $u>0$. Then, if for some $i$ $$ I_i>J_i \qquad\hbox{or}\qquad I'_iJ_i'$$ then we have $b>b_1$. We report a failure if $$I_i\cap J_i\ne \emptyset \qquad I_i'\cap J_i'\ne \emptyset $$ for all $i$, in which case no relation is claimed between $b$ and $b_1$. \vskip1em The rest of the algorithm follows along the same lines. \endpf Note that the last part of the previous algorithm constitutes a refinement of Algorithm~3.16, but it requires bounds for $b_1$. Also, Algorithm~3.15 allows us to obtain an initial, probably wasteful, bound for $w_0$. This initial bound allows us to obtain a bound for $b_1$, which in turn will allow us to improve our initial bound for $w_0$. Iterating this last algorithm in this way allows us to obtain improved bounds for both $w_0$ and $b_1$. The intersection of the bounds produced by Algorithms~3.16 and 3.18 are improved bounds for the Thomas--Fermi function and its derivative at points $x_i$. These translate immediately to better bounds for the solution of the Thomas--Fermi equation, and related constants. \algorithm{3.20} We can produce $x_i$, $r_i\in {\cal R}$, and $${\cal U}_i(\vsz {I^i},N;C_{h,i},C_{g,i},2)\qquad i=1,\ldots,m$$ such that $$y(x_i+z\cdot r_i)\in {\cal U}_i(\vsz {I^i},N;C_{h,i},C_{g,i},2) \qquad i=1,\ldots,m$$ and $$\cup_{i=1}^m \ooint {x_i-r_i},{x_i+r_i}= \ooint {x_1-r_1},{x_m+r_m}\subset\ooint 0,\infty$$ \description Our previous remark gives us the $x_i$, $r_i$, $I^i_0$ and $I^i_1$. The rest follows by applying Algorithm~3.2 for every $i$. \lem\tflemb The following inequalities hold: $$\eqalignno{ 1.588071022611278 &\le w_0\le 1.588071022611471 \cr -13.270973847925352 &\ge b_1\ge -13.270973848125353 \cr 0.486348538043594 &\le\Omega_c^2\le 0.486348538046869 \cr 2.104025280219502 &\le r_c\le 2.104025280273837 \cr}$$ Needless to say, the decimal numbers quoted above stand for the exact rational numbers they represent. \endt \cproof The inequalities for $w_0$ and $b_1$ follow by carrying out previous algorithms. The inequality for $r_c$ follows by checking that $$u'( 2.104025280219502 )\ge 0\ge u'( 2.104025280273837 )$$ The bounds for $\Omega_c$ are then trivial. \endpf \Title{\sdone. Zone I.} The purpose of this section is to prove (1.2) for all $\Omega$ in Zone I, as defined at the end of Section \secder. We will do this as follows: \vskip1em First, we partition Zone I into ``fat'' intervals $\{W_i\}_{i=1}^n$. Note that the first such interval will have the form $\ocint 0,{\Omega_{\epsilon}}$, for an $\Omega_{\epsilon}$ to be picked (much) later in our proof. In fact, the role of the $W_i$ will change as they approach zero: the larger ones (most of them, by the way) will receive identical treatment. Then, there will be a family of them, rather close to zero, which will receive a sort of special treatment, and then the single $W_1=\ocint 0,{\Omega_{\epsilon}}$ which will be on its own. Second, each fat interval $W$ is divided into a finite partition of (lots of) suitably small subintervals $\Omega^\ast$ (except $W_1$ which will be both a ``fat'' and ``thin'' interval at the same time.) Our aim is to produce uniform bounds for $-F''(\Omega)$ for all $\Omega\in\Omega^\ast$: for $W_1$ we will be able to produce only lower bounds, since $-F''$ is unbounded there; for the others, we will be able to produce both upper and lower bounds. \vskip1em Say $\Omega^\ast=\ccint z_1,{z_2}$ is contained in the fat interval $W=\ccint w_1, {w_2}$. We construct two functions $a(\Omega^\ast)$ and $b(\Omega^\ast)$, constant on each subinterval $\Omega^\ast$, such that $$r_1(\Omega)< a< bi_1$ meaning that the computation of the $J_i$ is always done directly, without using the faster method below. Then, we take $t_i$ for $i=i_0,\ldots,i_1$ to be the same for all $\Omega^\ast=\ccint z_1,{z_2}\subset W=\ccint w_1,{w_2}$, and we compute once and for all the following numbers: $$ a_{k,i}=\int_{t_i}^{t_{i+1}} \bra{u(r)-w_k^2}^{-\frac 3,2}y(r)\,dr \qquad b_{k,i}=3w_k\int_{t_i}^{t_{i+1}} \bra{u(r)-w_k^2}^{-\frac 5,2}y(r)\,dr $$ for $k=1,2$, and $i=i_0,\ldots,i_1$. Next, note that the functions $$f_i(w)=\int_{t_i}^{t_{i+1}} \bra{u(r)-w^2}^{-\frac 3,2}y(r)\,dr$$ are increasing and convex on $W$. Therefore, if $w\in\ccint w_1,{w_2}$, $$\max_{k=1,2}\bra{f_i'(w_k)\bra{w-w_k}+f_i(w_k)} \le f_i(w)\le {f_i(w_1)-f_i(w_2)\over w_1-w_2}\bra{w-w_1}+f_i(w_1)$$ Thus, $$\max_{k=1,2}\bra{b_{k,i}\bra{\Omega-w_k}+a_{k,i}} \le J_i(\Omega)\le {a_1-a_2\over w_1-w_2}\bra{\Omega-w_1}+a_1\qquad \Omega\in W$$ This gives us intervals $\tilde J_i(\Omega^\ast)$, such that $$J_i(\Omega)\subset \tilde J_i(\Omega^\ast) \qquad i_0\le i\le i_1\qquad \Omega\in\Omega^\ast\eqno\sdonea$$ \vskip1em In practice, $\tilde a$ and $\tilde b$ will be far from $r_1$ and $r_2$. They will enclose a region which is safely away from the singularities of the integrand in our formula for $F''$, for which we can expect \sdonea\ to be sharp. \vskip1em For $i$ outside of the range $\ccint i_0,{i_1}$, we compute $f_i(z_1)$ and $f_i(z_2)$ directly, and, by our previous remark, $$J_i(\Omega)\in \tilde J_i(\Omega^\ast)\eqbydef \ccint {f_i(z_1)},{f_i(z_2)}$$ Thus, we have defined $\tilde J(\Omega^\ast)$ for all $i=1,\ldots,n$, and we conclude that $$I_1(\Omega)\in \sum_{i=1}^n\tilde J_i(\Omega^\ast)\qquad \Omega\in\Omega^\ast $$ \vskip1em \vskip1em \title{Computation of $\bf I_2$.} Consider a small number $\bar \Omega_2\muchsmaller\Omega_c$, that we can make coincide with one of the endpoints of the fat intervals $W_i$. We distinguish two cases: $\Omega>\bar\Omega_2$ and $\Omega\le\bar\Omega_2$. \vskip1em If $\Omega>\bar\Omega_2$, we use Algorithm \tfalb\ to compute ${\cal U}_1$ such that $$u(x)=\Omega^2+zf(z)\qquad z={x-r_1(\Omega)\over r}\qquad f\in{\cal U}_1$$ where $r\ge\abs{a-r_1(\Omega)}$ and ${\cal U}_1$ is uniform for all $\Omega\in\Omega^\ast$. Note that $f(0)>0$. Also, in order to apply Algorithm~3.2, we need to obtain bounds for $r_1(\Omega)$ and for $u'(r_1)$; the first can be done by obtaining heuristic bounds $r_{\rm dn}$ and $r_{\rm up}$, and checking that $u(r_{\rm dn})\le\Omega^2$ and $u(r_{\rm up})\ge\Omega^2$, which can be easily checked using the information given by Algorithm~3.20. Bounds for $u'(r_1)$ can be obtained using the bounds for $r_1$ and the information on $y_{\rm TF}$ (hence on $u$) given by Algorithm~3.20. See the section on implementation for more details. Therefore, $$\eqalignno{ \int_{r_1(\Omega)+\delta}^a \bra{u(x)-\Omega^2}^{-\frac 3,2}y(x)\,dx &= \int_{r_1(\Omega)+\delta}^a z^{-\frac 3,2}f^{-\frac 3,2}(z)y(x)\,dx\cr &= \int_{r_1(\Omega)+\delta}^a z^{-\frac 3,2}\tilde f(z)\,dx\cr}$$ for a new function $\tilde f(z)=y(x)f^{-\frac 3,2}(z)$, that can also be enclosed in a computable ${\cal U}_2$. Note that $\tilde f(0)>0$ also. Thus, if $$\tilde f(z)=\sum_{n\ge 0}a_nz^n\in{\cal U}(\vsz J,N;C_h,C_g;1)$$ we see that $$\eqalignno{ \int_{r_1(\Omega)+\delta}^a \bra{u(x)-\Omega^2}^{-\frac 3,2}y(x)\,dx &= \int_{r_1(\Omega)+\delta}^a \sum_{n\ge 0}a_nz^{n-\ffra 3,2}\,dx\cr &= r \left.\sum_{n\ge 0}{a_n\over n-\fra 1,2}z^{n-\ffra 1,2}\right| _{z={\delta\over r}}^{z={a-r_1(\Omega)\over r}}\cr}$$ This implies that $$\eqalignno{I_2&= r \sum_{n\ge 0}{a_n\over n-\fra 1,2}\bra{{a-r_1(\Omega)\over r}}^{n-\ffra 1,2} \cr &\in r \sum_{n= 0}^N{J_n\over n-\fra 1,2}\bra{{a-r_1(\Omega)\over r}}^{n-\ffra 1,2} \pm\epsilon\cr}$$ with $$\abs{\epsilon}\le r\bra{{C_h\over N+\oh}+2C_g}$$ \vskip1em \vskip1em \vskip1em When $\Omega\le\bar\Omega_2$, we proceed as follows: \vskip1em Consider the change of variables given by $r(t)$, the inverse of $u$. Then, by the last remark in Lemma \sdlema, $$I_2={d\over d\Omega}\bra{\Omega\int_{r_1(\Omega)}^a \bra{u(r)-\Omega^2}^{-\frac 1,2}\>{dr\over r}} ={d\over d\Omega}\bra{\Omega\int_{\Omega^2}^{u(a)} \bra{t-\Omega^2}^{-\frac 1,2}w(t)\,dt}$$ for $$ w(t)={r'(t)\over r(t)}$$ \vskip1em In order to compute $w$, we consider the following: \vskip1em Let $x_0$ and $\epsilon'$ be small numbers satisfying {\itemize \item{a.} $u'(x)>1-\epsilon'$ for $x\in\ccint 0,{x_0}$. \item{b.} For a sequence $\{\bar b_n\}\in l^1$, we have $$u(x)=x\bra{1+\sum_{n=2}^\infty\bar b_n\bar x^{\frac n,2}} \quad x\le x_0$$ Furthermore, we know that $$ 1+\sum_{n=2}^\infty\bar b_n z^{n}\in{\cal U}(\vsz I,m;0,C_g;2) \eqno\sdtdefb$$ with $ I_0=\ccint 1,1$ and $ I_1=\ccint 0,0$. Here, $\bar x$ denotes $x/x_0$, and $\bar b_n=b_n\cdot x_0^{\frac n,2}$. } \vskip1em Define $$\bar t = \bra{{t\over x_0}}^{\frac 1,2}$$ We also consider a small number $\eta\le u(x_0)$. It will be chosen so that \sdtb\ below holds. We start by obtaining expressions for $u'(r)$ and $u''(r)$ similar to the one for $u$ in \sdtdefb. Note first that \sdtdefb\ is equivalent to an expression for $y(x)$. Then, by the Thomas--Fermi equation, and by integration, we have $$\eqalignno{ y''(x)&=x^{-\frac 1,2}\bra{\sum_{n=0}^\infty \bar b_n\bar x\>^{\frac n,2}} ^{\frac 3,2} =x^{-\frac 1,2}\sum_{n=0}^\infty y''_n\bar x\>^{\frac n,2} =x^{-\frac 1,2}y_{pp}(\bar x^{\frac 1,2}) \qquad y''_0=1\cr y'(x)&=\sum_{n=0}^\infty y'_n\bar x\>^{\frac n,2} =y_{p}(\bar x^{\frac 1,2}) \qquad y'_n=\cases{ -w_0 & $n=0$\cr \vbox{\vskip1em} \fra 2,nx_0^{\frac 1,2}y''_{n-1} &$n>0$\cr}\cr}$$ from which we obtain $$\eqalignno{ u'(x)&=xy'(x)+y(x) = \sum_{n=0}^\infty u'_n\bar x\>^{\frac n,2} =u_{p}(\bar x^{\frac 1,2}) &\sdtup\cr &\qquad u'_n=\cases{ 1 & $n=0$\cr 0 & $n=1$\cr \bar b_n+y'_{n-2}\cdot x_0 &$n\ge 2$\cr}\cr u''(x)&= xy''(x)+2y'(x) =u_{pp}(\bar x^{\frac 1,2}) = \sum_{n=0}^\infty u''_n\bar x\>^{\frac n,2} &\sdtupp\cr &\qquad u''_n=\cases{ -2w_0&$n=0$\cr 2y_n'+x_0^{\frac 1,2}y''_{n-1}&$n>0$\cr} \cr}$$ Note that since we know a neighborhood of type 2 that contain $y$, we can enclose $y_{pp}$ and $u_{p}$ also in neighborhoods of type 2, and $y_{p}$ and $u_{pp}$ in neighborhoods of type 3. \vskip1em We start our analysis understanding $r(t)$. First, a technical algorithm. \algorithm{\sdtalc} Given a function $r(t)=t\cdot R(\bar t)$ with $$R(z)\in{\cal U}^0(\vsz {a^\ast},N; C_N,0;\infty;S)\qquad a^\ast_0=\ccint 1,1$$ valid for $\bar t\in S\subset\ccint 0,1$, and given a neighborhood ${\cal U}^1$ in $H^1$, we can compute another neighborhood ${\cal U}^0_2$, also of type $\infty$ in $C^0$, also valid on $\bar t\in S$, such that if $$f(t) = \sum_{n=0}^\infty \bar c_n \bar t\>^{n}\in {\cal U}^1(\vsz I,N;C_h,C_g,m)$$ then $G(\bar t)\in {\cal U}^0_2$ for $$G(\bar t)= f\bracket{r(t)}=\sum_{n=0}^\infty \bar c_n \bra{{r(t)\over x_0}}^{\frac n,2}$$ We assume that $0^n+\bar t^{N+1}h(\bar t)}^\gamma = \sum_{n=0}^{n_0} a_{n,\gamma} \bar t\>^n+\bar t\>^{n_0+1} h(\bar t; \gamma,n_0+1) $$ with $a_{n,\gamma}\in a^\ast_{n,\gamma}$, the $a^\ast_{n,\gamma}$ easily determined intervals, $$\lnorm{h(\bar t;\gamma,n_0)}_{C^0}\le \epsilon_{\gamma,n_0}.$$ We can see, on one hand, that $$\eqalignno{ \sum_{n=0}^N \bar c_n \bra{{r(t)\over x_0}}^{\frac n,2} &= \sum_{n=0}^N \bar c_n\bar t\>^n \bbracket{\sum_{k=0}^{N-n} a_{k,\frac n,2}\bar t\>^k+\bar t\>^{N+1-n} h(\bar t; \fra k,2, N-n)} \cr &=\sum_{n=0}^Nd_n\bar t\>^n+\bar t\>^{N+1}h(t) \cr}$$ with $$\eqalignno{ \abs{h(t)}&\le \sum_{n=0}^N\abs{\bar c_n}\epsilon_{\ffra n,2,N-n+1} \cr &\le \sum_{n=0}^N\abs{I_n}\epsilon_{\ffra n,2,N-n+1} +C_g\max_{n=m,\ldots,N}\epsilon_{\ffra n,2,N-n+1}\cr}$$ and $$\eqalignno{ d_n&=\sum_{i+j=n}\bar c_i\cdot a_{j,\ffra i,2}\cr &\in\sum_{i+j=n}I_i\cdot a^\ast_{j,\ffra i,2}\pm\epsilon_n\eqbydef d_n^\ast \cr}$$ where $$\epsilon_n\le\cases{C_g\sup_{m\le i\le n}\abs{a_{n-i, \ffra i,2}} &if $n\ge m$\cr \vbox{\vskip1em}\cr \hfil 0 \hfil & otherwise\cr}$$ On the other hand, since $$\eqalignno{ \sum_{n=N+1}^\infty \abs{\bar c_n} \bra{{r(t)\over x_0}}^{\frac n,2} &\le \bra{C_g+C_h}\bra{r(t)\over x_0}^{\fra (N+1),2}\cr &\le \bra{C_g+C_h}\bar t\>^{N+1}\bra{1+\sum_{n=1}^N\abs{a_n}+C_N}^{ \fra (N+1),2} \cr}$$ we conclude that $$f(r(t))=G(\bar t)\in {\cal U}^0(\vsz {d^\ast},N;\tilde C_h,0;\infty)$$ with $$\displaylines{ \quad\tilde C_h=\bra{C_g+C_h} \bra{1+\sum_{n=1}^N\abs{a_n}+C_N}^{\fra (N+1),2}+ \sum_{n=0}^N\abs{I_n}\epsilon_{\frac n,2,N-n+1} \hfill\cr \hfill +C_g\max_{n=m,\ldots,N}\epsilon_{\frac n,2,N-n+1} \qquad\qquad\llap{\mathendpf}\cr}$$ \algorithm{\sdtrt} Given $N\ge 0$, we produce intervals $a_2^\ast,\ldots,a_N^\ast$, and a constant $C_N$, such that $$\abs{r(t)-t\Bracket{1+\sum_{n=2}^Na_n\bar t\>^{n} }}\le C_N\,t\cdot\bar t\>^{ N+1}$$ for constants $a_i\in a^\ast_i$, $i=2,\ldots,N$, and for $t\le \eta$. \description First, we will construct an inductive procedure to define numbers $a_n$ such that $$u\bra{r_N(t)}=t\bra{1+\bigo{\bar t\>^{N+1}}}\qquad t\to 0\eqno\sdta$$ where $$r_N(t)=t\bra{1+\sum_{n=2}^Na_n\bar t\>^n}$$ By induction. For $N=1$, let $r_0(t) = t$. Note that $r(t)\ge r_0(t)$, that $r_0(t)\le x_0$ for $\bar t\le 1$, and that, if $t\le \eta\le u(x_0)$ then $\bar t\le 1$. Therefore, we have $$u(r_0(t)) = t \bra{1+\sum_{n\ge 2} b_n t^{\frac n,2}}$$ Thus, $$\abs{r_0(t)-r(t)}\le t\>{\abs{\sum_{n\ge 2}b_n t^{\frac n,2}} \over \inf_{r\in \ccint r_0(t),{r(t)}}|u'(r)|}\eqno(4.6)$$ Since, for $t$ small enough, $r(t)\le x_0$, the denominator is bounded below by $(1-\epsilon')$, and we conclude $$r_0(t)-r(t) = \bigo{t^2}$$ \vskip1em For general $N$, we set $$r_N(t)=t\bra{1+\sum_{n=2}^Na_n\bar t\>^n}$$ where $a_2,\ldots,a_{N-1}$ satisfy the induction hypothesis. Note that we have $${\sum_{n=1}^\infty b_n r_N(t)^{\frac n,2}- \sum_{n=1}^\infty b_n r_{N-1}(t)^{\frac n,2}}= \bigo{\bar t\>^{N+1}}\eqno\sdteqa$$ Thus, if for any real number $\gamma$ we put $$\bra{1+\sum_{n=2}^{N-1} a_n \bar t\>^n}^\gamma = \sum_{n=0}^{N} a_{n,\gamma}\bar t\>^{n}+ \bigo{\bar t\>^{N+1}} \eqno\sdteqb$$ we see that $$\eqalignno{ u(r_{N-1}(t)) &= t\bra{1+\sum_{n=2}^{N-1} a_{n}\bar t\>^n} \cr &\qquad\cdot\bra{\sum_{n=0}^N \bar b_n \bar t\>^n\Bracket{\sum _{i=0}^{N}a_{i,\frac n,2}\bar t\>^{i} +\bigo{\bar t\>^{N+1}}}+\bigo{\bar t\>^{N+1}}}\cr &= t\bra{1+\sum_{n=2}^{N-1} a_{n}\bar t\>^n} \cdot\bra{\sum_{n=0}^N c_n\bar t\>^{n}+ \bigo{\bar t\>^{N+1}}}&\sdtc \cr}$$ where $$c_k = \sum_{n=0}^k\bar b_n\cdot a_{k-n,\frac n,2}\qquad c_0=1.$$ By the induction hypothesis, \sdtc\ is equal to $t(1+\bigo{\bar t\>^N})$. Thus, using \sdteqa, we can see that $$\eqalignno{ u(r_{N}(t)) &= t\bra{1+\sum_{n=2}^{N-1} a_{n}\bar t\>^n + a_N\bar t\>^N} \cdot\bra{\sum_{n=0}^N c_n\bar t\>^{n}+ \bigo{\bar t\>^{N+1}}}\cr}$$ where the $c_n$ here are the same as those in (4.9). Therefore, by putting $$a_N=-\sum_{k=1}^{N-2}a_{N-k}c_k-c_N$$ we get rid of all $\bar t\>^N$ terms, thus obtaining \sdta. So far, we have proved the existence of numbers $a_n$ such that \sdta\ is satisfied. This procedure also gives us an algorithm to compute the $a_n^\ast$. Indeed, bounds $a^\ast_{n,\gamma}$ for the $a_{n,\gamma}$ can be computed explicitly, since by the induction hypothesis we already know $a^\ast_i$, for $i=1,\ldots,N-1$. As for the $c_k$, recalling \sdtdefb, $$ c_k=\sum_{n=0}^k \bar b_n\cdot a_{k-n,\frac n,2} \in \sum_{n=0}^k I_n\cdot a^\ast_{k-n,\frac n,2}\pm\epsilon_k$$ with $$|\epsilon_k |\le\cases{ \sup_{k\ge n\ge 2}\abs{a_{k-n,\frac n,2}}\cdot C_g & if $k\ge 2$\cr 0 & if $k\le 1$\cr}$$ In particular, it follows immediately that $a_2=-\bar b_2=x_0\cdot w_0\in -I_2$. \vskip2em To obtain a good value for the constant $C_N$, we proceed as follows: First, check that $$\eta\cdot \bra{1+\sum_{n=2}^N\abs{a_n}}\le x_0\eqno(4.9a)$$ (See also \sdtb\ below.) This allows us to invoke Algorithm~\sdtalc, with $S=\ccint 0,{\sqrt{{ \eta/x_0}}}$, to obtain $$y(r_N(t))=f(\bar t)\in {\cal U}^0_1(\cdots; \cdots ;\infty)$$ and thus we can write $$u\bra{r_N(t)}=r_N(t)\cdot f(\bar t)\equiv t\cdot g(\bar t)$$ with $g$ belonging to ${\cal U}^0(\vsz I,N; \tilde{\tilde C}_h,0;\infty;t\le\eta)$, the product neighborhood of $${\cal U}_2(\vsz a^\ast,N; 0,0;\infty;t\le\eta)$$ and ${\cal U}^0_1$. Now, since \sdta\ implies that $g(\bar t)=1+\bigo{\bar t\>^{N+1}}$, we can take $I_0=\ccint 1,1$ and $I_i=\ccint 0,0$, for $i=1,\ldots,N$. Note that this is relied crucially on the fact that ${\cal U}^0$ is of type $\infty$; in fact, since $$g(\bar t)\in {\cal U}^0(\vsz I,N; \tilde{\tilde C}_h,0;\infty;t\le\eta)$$ we can find constants $p_i\in I_i$ such that $$\abs{g(\bar t)-\sum_{k=0}^Np_i\bar t\>^k}\le \tilde{\tilde C_h}\bar t\>^{N+1}$$ On the other hand, since $g(\bar t)=1+\bigo{\bar t\>^{N+1}}$, we must have $p_0=1$ and $p_i=0$ for $i=1,\ldots,N$, thus $$\abs{g(\bar t)-1}\le \tilde{\tilde C_h}\bar t\>^{N+1}.$$ If Algorithm~4.1 had produced a neighborhood of any other type, it would have been harder to conclude this without changing $\tilde{\tilde C_h}$. More precisely, if we have $\phi(t)$ defined on $t\in\ccint -1,1$ such that $$\abs{\phi(t)-\sum_{n=0}^Na_nt^n}\le C t^k \quad{\rm and}\quad \phi(t)-\sum_{n=0}^Na_nt^n=\bigo{t^{N+1}}$$ for $t\in\ccint -1,1$, we cannot conclude $$\abs{\phi(t)-\sum_{n=0}^Na_nt^n}\le C t^{N+1}$$ with the same constant $C$ unless $k> N$. Counterexamples with $k\le N$ are readily available (simply take $N=k=0$, $\phi(t)=t^3-t$, $a_0=0$; then, $|\phi|\le 2^{-\frac 1,2}$ but $\phi(t)$ is not bounded by $2^{-\frac 1,2}|t|$.) \vskip1em So, we have that $$\abs{u\bracket{r_N(t)}-t}\le \tilde{\tilde C}_h\cdot t\cdot\bar t\>^{N+1}\qquad t\le\eta$$ Next, note that $$\abs{r_N(t)-r(t)}\le{\abs{u(r_N(t))-t}\over \inf_{0\le r\le \max( r(t),r_N(t))}\abs{u'(r)}}$$ Then, the fact that $\eta\le u(x_0)$ and (4.9a) imply that $$\abs{u'(r)}\ge (1-\epsilon')\qquad 0\le r\le \max\bra{r(t),r_N(t)} \quad t\le\eta $$ from which the lemma follows by taking $$C_N={\tilde{\tilde C_h}\over (1-\epsilon')} \eqno\mathendpf $$ \algorithm{\sdtalb} We produce a neighborhood ${\cal U}^0(\vsz I,N; C_h,0;\infty)$ such that $$h(t) \eqbydef tw'(t)+w(t) = f(\bar t)\qquad f\in{\cal U}$$ for $t\le \eta$. \description We note that $$r'(t)={1\over u'\bracket{r(t)}}\qquad r''(t)=-\bra{r'(t)}^3u''\bracket{r(t)}$$ Therefore, we check that $$\eta\cdot\bra{1+\sum_{n=1}^N\abs{a_n}+C_N}\le x_0\eqno\sdtb$$ and as a result of this, we can apply Algorithm \sdtalc\ and obtain neighborhoods of type $\infty$ in $C^0$ containing functions $f$, $f_p$ and $f_{pp}$ s.t. $$r(t)=tf(\bar t) \qquad r'(t)=f_p(\bar t)\qquad r''(t)=f_{pp}(\bar t)$$ which are valid for $t\le \eta$. These functions are obtained by putting $$f_p(\bar t)={1\over u_p\bra{\bracket{r( t)/x_0}^{\frac 1,2}}} \qquad f_{pp}(\bar t)=-f_p^3\cdot u_{pp}\bra{\bracket{r(t)/x_0}^{\frac 1,2}}$$ Note that with this definition, $f$ and $f_p$ are normalized to be 1 at 0, and $f_{pp}(0)=2w_0$. Thus, $$ \eqalignno{ w(t)&={r'(t)\over r(t)}={1\over t}\bra{{f_p(\bar t)\over f(\bar t)}} \cr w'(t)&= {r''(t)\over r(t)} -\bra{r'(t)\over r(t)}^2 = {1\over t}\bra{{f_{pp}(\bar t)\over f(\bar t)}} -{1\over t^2}\bra{{f_p(\bar t)\over f(\bar t)}}^2 \cr}$$ and $$\eqalignno{h(t) = tw'(t)+w(t)&=t^{-1}\bra{ {f_p(\bar t)\over f(\bar t)} -\bra{{f_p(\bar t)\over f(\bar t)}}^2} +\bra{f_{pp}(\bar t)\over f(\bar t)}\cr &= f_h(\bar t)\cr}$$ for a function $f_h$ belonging to an easily computable neighborhood of type $\infty$ in $C^0$. Note that by our normalization, the $t^{-1}$ terms drop out. Furthermore, there are no $t^{-1}\bar t$ terms since neither $f$ nor $f_p$ have $\bar t$ terms, and in fact $f_h(0)=w_0$, which we can easily see as follows: first, $f(\bar t)=1+w_0t+\bigo{\bar t\>^3}$, $f_p(\bar t)=1+2w_0t^2+\bigo{\bar t\>^3}$ and $f_{pp}(\bar t)=2w_0+\bigo{\bar t}$, therefore $$t^{-1}\bra{ {f_p(\bar t)\over f(\bar t)} -\bra{{f_p(\bar t)\over f(\bar t)}}^2}=-w_0+\bigo{t^{-1}\bar t\>^3}=- w_0+\bigo{\bar t}$$ and $${f_{pp}(\bar t)\over f(\bar t)}=2w_0+\bigo{\bar t}$$ thus $f_h(0)=w_0$. Moreover, if we set $${f_p(\bar t)\over f(\bar t)} -\bra{{f_p(\bar t)\over f(\bar t)}}^2 +t\cdot\bra{f_{pp}(\bar t)\over f(\bar t)} \in {\cal U}^0(\vsz I,{N+2};\epsilon_h,0;\infty)$$ then $$f_h\in {\cal U}^0(x_0^{-1}\cdot I_2,\ldots,x_0^{-1}I_{N+2}; x_0^{-1}\epsilon_h,0;\infty;t\le\eta)$$ valid for $t\le\eta$. \vskip1em Now, let $$f_h(\bar t) = \sum_{n=0}^Na_n\bar t\>^n+H(\bar t)$$ with $$\abs{H(\bar t)}\le \epsilon_h\abs{\bar t}^{N+1} \qquad t\le \eta$$ \vskip2em Finally, then, let $\delta$ be a small number such that $u(\delta)\le\eta$, set $\bar\Omega_2\le \sqrt{u(\delta)}$, and consider $\Omega\le\bar\Omega_2$ for which we set $a(\Omega)\equiv \delta$: $$\displaylines{ {d\over d\Omega}\bra{ \Omega\int^{\delta}_{r_1(\Omega)}\bra{u(r)-\Omega^2}^ {-\frac 1,2}{dr\over r}}= {d\over d\Omega}\bra{\Omega^2\int_{1}^{\Omega^{-2}u(\delta)}\bra{t-1}^ {-\frac 1,2}w(t\Omega^2)\,dt}\cr \hfill\eqalign{&= 2\Omega\int_{1}^{\Omega^{-2}u(\delta)}\bra{t-1}^ {-\frac 1,2}h(t\Omega^2)\,dt\cr &\qquad -2\bra{u(\delta)-\Omega^2}^{-\frac 1,2}w(u(\delta))u(\delta)\cr &= 2\Omega\sum_{n=0}^N{a_n x_0^{-\frac n,2}}\Omega^{n} \int_{1}^{\Omega^{-2}u(\delta)}\bra{t-1}^{-\frac 1,2}t^{\ffra n,2}\,dt \cr &\qquad+\tilde h(\Omega) -\bra{u(\delta)-\Omega^2}^{-\frac 1,2}{2u(\delta)\over \delta u'(\delta)}\cr }\qquad\qquad\llap{(4.11a)}\cr}$$ with $$\abs{\tilde h(\Omega)}\le 2\Omega^{N+2}\epsilon_h {x_0}^{-\fra (N+1),2} \int_{1}^{\Omega^{-2}u(\delta)} \bra{t-1}^{-\frac 1,2}t^{\fra (N+1),2}\,dt \eqno{(4.11b)} $$ At this point, we introduce another small number, $\Omega_\epsilon$, on which we impose, first, the condition $$u(\delta)\ge 2\Omega_\epsilon^2\eqno\sdtconda$$ Expression (4.11a) above can be computed easily for all $\Omega\ge \Omega_\epsilon$. The evaluation of integrals of the type $\int (t-1)^{-\frac 1,2} t^\gamma\,dt$ can be done by enclosing the integrand locally in neighborhoods in $H^1$. We omit the trivial details. \vskip1em When $\Omega\le\Omega_\epsilon$, consider first the following trivial Lemma. \lem{\sdzlema} If $R\ge 2$, then {\itemize \item{1.} $\displaystyle\int_1^R(t-1)^{-\ffra 1,2}t^\gamma\,dt\> \le 2\,R^{\gamma+\ffra 1,2}$ when $\gamma\ge 0$. \item{2.} $\displaystyle\int_1^R(t-1)^{-\frac 1,2}t^\gamma\,dt\> \ge 1 $ when $-1\le \gamma$. \item{3.} $\displaystyle\int_1^R(t-1)^{-\ffra 1,2}t^\gamma\,dt\> =\bigo{ R^{\gamma+\ffra 1,2}}$ when $\gamma> -\fra 1,2$. } \endt Then, by \sdtconda, a., b., c. and Lemma~4.4, $$ {d\over d\Omega}\bra{\Omega\int^{\delta}_{r_1(\Omega)}\bra{u(r)-\Omega^2}^ {-\frac 1,2}{dr\over r}}$$ is bounded below by $$T_1(\Omega)\eqbydef 4\sqrt{u(\delta)}\sum_{a_n<0}a_n\bra{u(\delta)\over x_0}^{\ffra n,2} -\bra{u(\delta)-\Omega^2}^{-\frac 1,2}{2u(\delta)\over \delta u'(\delta)} \eqno\sdtxxa $$ where we have set $a_{N+1}=-\epsilon_h$. \title{Computation of $I_3$} Here, we also consider two cases: $\Omega\ge{\bar \Omega_3}$ and $\Omega<{\bar \Omega_3}$. The first case is dealt with in a similar manner to $I_2$. We omit the trivial modifications. The second case is also treated in much the same way, with a few differences coming mainly from the different powers in the asymptotic expansion of $u$ at 0 and at $\infty$. We include the details, although many of the differences are basically typographical considerations, because conclusions are somewhat different. In particular, as will be noted below, $I_3$ is mainly responsible for the singularity of $F''$ at $\Omega=0$. \vskip1em Let $M$ be a large number satisfying {\itemize \item{a.} $u'(M)<0$ and $u''(x)\ge 0$ on $\coint M,\infty$. \item{b.} For a sequence $\{\bar b_n\}\in l^1$, we have $$u(x)= {144\over x^2}u_0(x)= {144\over x^2}\bra{1+\sum_{n=1}^\infty\bar b_n\bar x^{-n\alpha}} \quad x\ge M\eqno\sdzdefba$$ Furthermore, we know that $$ 1+\sum_{n=1}^\infty\bar b_n z^{n}\in{\cal U}(\vsz I,m;0,C_g;2) \qquad I_0=\ccint 1,1\eqno\sdzdefb$$ Here, $\bar x$ denotes $x/M$, and, as a rule, we set $\bar b_n=b_n/M^{n\alpha}$. \item{c.} $\abs{u'(x)}\ge 2\cdot 144\,x^{-3}(1-\epsilon')$ for $x\ge M $. } \vskip1em Define $$\bar t = \bra{{M\,t^{\frac 1,2}\over 12}}^\alpha$$ We also consider a small number $\eta\le u(M)$. It will be chosen so that \sdzb\ below holds. We start by obtaining expressions for $u'(r)$ and $u''(r)$ similar to the one for $u$ in \sdzdefb. Note first that \sdzdefb\ is equivalent to an expression for $y(x)$. Then, by the Thomas--Fermi equation, and by integration, we have $$\eqalignno{ y''(x)&={12\cdot 144\over x^{5}}\bra{\sum_{n=0}^\infty \bar b_n\bar x\>^{ -n\alpha}}^{\frac 3,2}= {12\cdot 144\over x^{5}}\sum_{n=0}^\infty y''_n\bar x\>^{ -n\alpha} \qquad y''_0=1\cr y'(x)&={-3\cdot 144\over x^4}\sum_{n=0}^\infty \bra{{4\over 4+n\alpha}} y''_n\bar x\>^ {-n\alpha} ={-3\cdot 144\over x^4}\sum_{n=0}^\infty y'_n\bar x\>^ {-n\alpha} \qquad y'_0=1\cr}$$ from which we obtain $$\eqalignno{ u'(x)&=xy'(x)+y(x) = -{2\cdot 144\over x^3}\sum_{n=0}^\infty\bra{{3y'_n-\bar b_n\over 2}} \bar x\>^{-n\alpha}\cr &= -{2\cdot 144\over x^3}u_{p}(\bar x\>^{-\alpha}) &\sdzup\cr u''(x)&= xy''(x)+2y'(x) ={6\cdot 144\over x^4} \sum_{n=0}^\infty \bra{2y''_n-y'_n}\bar x\>^{-n\alpha} \cr&= {6\cdot 144\over x^4} u_{pp}(\bar x\>^{-\alpha}) &\sdzupp\cr}$$ for $u_p$ and $u_{pp}$ in $H^1$, normalized so $u_p(0)=u_{pp}(0)=1$. The strategy will be, as in the case for $I_2$, to change variables to the inverse function of $u$, $r(t)$. \vskip1em \algorithm{\sdzalb} Given a function $$r(t)=12 t^{-\frac 1,2}\cdot R(\bar t) $$ with $$R(z)\in{\cal U}^0(\vsz {a^\ast},N;C_N,0;\infty;S)\qquad a^\ast_0=\ccint 1,1$$ satisfying the hypothesis $$r(t)\ge M\qquad {\rm whenever\ }\bar t\in S\subset\ccint -1,1$$ and given $$ g(x) = \sum_{n=0}^\infty \bar c_n z^{n} \in {\cal U}^1(\vsz I,N;C_h,C_g,m)\qquad z = (x/M)^{-\alpha}$$ we compute another neighborhood of type $\infty$ in $C^0$, such that, if we set $$G(\bar t) = g\bra{r(t)}= \sum_{n=0}^\infty \bar c_n \bra{{r(t)\over M}}^{-n\alpha} $$ then $$G(\bar t) \in {\cal U}^0(\vsz d^\ast,N; C,0;\infty;S)$$ valid for $\bar t\in S$. \description Consider $a_j\in a^\ast_j$, for $j=0,\ldots,N$. Put $$\bra{\sum_{n=0}^N a_n \bar t\>^n+\bar t^{N+1}h(\bar t)}^\gamma = \sum_{n=0}^{n_0} a_{n,\gamma} \bar t\>^n+\bar t\>^{n_0+1} h(\bar t; \gamma,n_0+1) $$ with $a_{n,\gamma}\in a^\ast_{n,\gamma}$, and $$\lnorm{h(\bar t;\gamma,n_0)}_0\le \epsilon_{\gamma,n_0}$$ We can see, on one hand, that $$\eqalignno{ \sum_{n=0}^N \bar c_n \bra{{r(t)\over M}}^{-n\alpha} &= \sum_{n=0}^N \bar c_n\bar t\>^n \bbracket{\sum_{k=0}^{N-n} a_{k,-n\alpha}\bar t\>^k+\bar t\>^{N+1-n} h(\bar t; -n\alpha, N-n+1)} \cr &=\sum_{n=0}^Nd_n\bar t\>^n+\bar t\>^{N+1}h(t) \cr}$$ with $$\eqalignno{ \abs{h(t)}&\le \sum_{n=0}^N\abs{\bar c_n}\epsilon_{-n\alpha,N-n+1} \cr &\le \sum_{n=0}^N\abs{I_n}\epsilon_{-n\alpha,N-n+1} +C_g\max_{n=m,\ldots,N}\epsilon_{-n\alpha,N-n+1}\cr}$$ and $$\eqalignno{ d_n&=\sum_{i+j=n}\bar c_i\cdot a_{j,-i\alpha}\cr &\in\sum_{i+j=n}I_i\cdot a^\ast_{j,-i\alpha}\pm\epsilon_n\eqbydef d_n^\ast \cr}$$ where $$\epsilon_n\le\cases{C_g\sup_{i=m,\ldots,n}\abs{a_{n-i,-i\alpha}} &if $n\ge m$\cr \vbox{\vskip1em}\cr \hfil 0 \hfil & otherwise\cr}$$ On the other hand, since $$\eqalignno{ \abs{\sum_{n=N+1}^\infty \bar c_n \bra{{r(t)\over M}}^{-n\alpha}} &\le \bra{C_g+C_h}\bra{r(t)\over M}^{-(N+1)\alpha}\cr &\le \bra{C_g+C_h}\bar t\>^{N+1}\bra{1-\sum_{n=1}^N\abs{a_n}-C_N}^{-(N+1)\alpha} \cr}$$ we conclude that $$g(r(t))=G(\bar t)\in {\cal U}^0(\vsz {d^\ast},N;\tilde C_h,0;\infty)$$ with $$\displaylines{ \quad\tilde C_h=\bra{C_g+C_h} \bra{1-\sum_{n=1}^N\abs{a_n}-C_N}^{-(N+1)\alpha}+ \sum_{n=0}^N\abs{I_n}\epsilon_{-n\alpha,N-n+1} \hfill\cr \hfill +C_g\max_{n=m,\ldots,N}\epsilon_{-n\alpha,N-n+1} \quad\cr}$$ If we cannot check that $$\bra{1-\sum_{n=1}^N\abs{a_n}-C_N}> 0$$ the algorithm fails. \endpf Now, we analyze $r(t)$. \algorithm{\sdrt} Given $N\ge 0$, we produce intervals $a_1^\ast,\ldots,a_N^\ast$, and a constant $C_N$, such that $$\abs{r(t)-12\,t^{-\frac 1,2}\Bracket{1+\sum_{n=1}^Na_n\bar t\>^{n} }}\le C_N\,t^{-\frac 1,2}\bar t\>^{ N+1}$$ for constants $a_i\in a^\ast_i$, $i=1,\ldots,N$, and for $t\le \eta$. \description First, we will construct an inductive procedure to define numbers $a_n$ such that $$r_N(t)=12t^{-\frac 1,2}\bra{1+\sum_{n=1}^Na_n\bar t\>^n}$$ satisfies $$u\bra{r_N(t)}=t\bra{1+\bigo{\bar t\>^{N+1}}}\qquad t\to 0\eqno\sdza$$ By induction. For $N=0$, let $r_0(t) = 12 t^{-\frac 1,2}$. Note that, since $u(x)<144 x^{-2}$, then $r(t){\abs{\sum_{n\ge 1}b_n 12^{-n\alpha}t^{n\ffra \alpha,2} } \over \inf_{r\in \ccint r(t),{r_0(t)}}|u'(r)|}$$ Note that hypotheses a. and c. imply $$\abs{u'(r)}\ge \abs{u'(r_0(t))}\ge {t^{\ffra 3,2}\over 6}(1-\epsilon') \qquad{\rm provided\ } r\in \ccint r(t),{r_0(t)}\cap\coint M,\infty$$ Our previous remarks, and our assumption on $\eta $ then imply $$\abs{r_0(t)-r(t)}\le C\,t^{\ffra \alpha-1,2}\qquad t \le \eta.$$ Here we have used the fact that $r(t)\ge M$ for $t<\eta$. \vskip1em For general $N$, we set $$r_N(t)=12\,t^{-\frac 1,2}\bra{\sum_{n=0}^Na_n\bar t\>^n}\qquad a_0=1$$ where $\vs a,{N-1}$ satisfy the induction hypothesis. Note that we have $${\sum_{n=1}^\infty b_n r_N(t)^{-n\alpha}- \sum_{n=1}^\infty b_n r_{N-1}(t)^{-n\alpha}}= \bigo{\bar t\>^{N+1}}\eqno\sdeqa$$ Thus, if for any real number $\gamma$ we put $$\bra{\sum_{n=0}^{N-1} a_n \bar t\>^n}^\gamma = \sum_{n=0}^{N} a_{n,\gamma}\bar t\>^{n}+ \bigo{\bar t\>^{N+1}} \eqno\sdeqb$$ we see that $$\eqalignno{ u(r_{N-1}(t)) &= t\bra{\sum_{n=0}^N a_{n,-2}\bar t\>^n+\bigo{\bar t\>^{N+1}}} \cr &\qquad\cdot\bra{\sum_{n=0}^N \bar b_n \bar t\>^n\Bracket{\sum _{i=0}^{N}a_{i,-n\alpha}\bar t\>^{i} +\bigo{\bar t\>^{N+1}}}+\bigo{\bar t\>^{N+1}}}\cr &= t\bra{\sum_{n=0}^N a_{n,-2}\bar t\>^n+\bigo{\bar t\>^{N+1}}} \cdot\bra{\sum_{n=0}^N c_n\bar t\>^{n}+ \bigo{\bar t\>^{N+1}}}\qquad\qquad &\sdzc \cr}$$ where $$c_k = \sum_{n=0}^k\bar b_n\cdot a_{k-n,-n\alpha}\qquad c_0=1$$ By the induction hypothesis, \sdzc\ is equal to $t(1+\bigo{\bar t\>^N})$. Thus, using \sdeqa, we can see that $$\eqalignno{ u(r_{N}(t)) &= t\bra{\sum_{n=0}^N a_{n,-2}\bar t\>^n - 2a_N\bar t\>^N+\bigo{\bar t\>^{N+1}}} \cdot\bra{\sum_{n=0}^N c_n\bar t\>^{n}+ \bigo{\bar t\>^{N+1}}}\cr}$$ for exactly the same $c_n$ as in \sdzc. Therefore, by putting $$a_N=\oh\sum_{m+n=N}a_{n,-2}c_m$$ we get rid of all $\bar t\>^N$ terms, thus obtaining \sdza. So far, we have proved the existence of numbers $a_n$ such that \sdza\ is satisfied. This procedure also gives us an algorithm to compute the $a_n^\ast$. Indeed, bounds $a^\ast_{n,\gamma}$ for the $a_{n,\gamma}$ can be computed explicitly, since by the induction hypothesis we already know $a^\ast_i$, for $i=1,\ldots,N-1$. As for the $c_k$, recalling \sdzdefb, $$ c_k=\sum_{n=0}^k \bar b_n\cdot a_{k-n,-n\alpha} \in \sum_{n=0}^k I_n\cdot a^\ast_{k-n,-n\alpha}\pm\epsilon_k$$ with $$|\epsilon_k |\le\cases{ \sup_{2\le n\le k}\abs{a_{k-n,-n\alpha}}\cdot C_g & if $k\ge 2$\cr 0 & if $k\le 1$\cr}$$ In particular, it is easy to see that $a_1=\bar b_1/2\in \oh I_1$ (recall (4.13)). \vskip2em To obtain a good value for the constant $C_N$, we proceed as follows: By Algorithm \sdzalb, we can construct a neighborhood ${\cal U}_1^0$ such that $$f(\bar t)=u_0\bra{r_N(t)}\in {\cal U}^0_1(\vsz I,N; \tilde{\tilde C}_h,0;\infty;t\le\eta) $$ (see (4.13a)) provided $r_N(t)\ge M$ for $t\le \eta$. At this point then we check that $$12\eta^{-\fra 1,2}\bra{1-\sum_{n=1}^N\abs{a_n}}\ge M$$ See also \sdzb\ below. If we put $$ g(\bar t)=\bra{\sum_{n=0}^N a_n \bar t\>^n}^{-2}\in{\cal U}_2(\infty)$$ then, $$u\bra{r_N(t)}=t\cdot g(\bar t)\cdot f(\bar t)$$ with $$F(\bar t)=g(\bar t)\cdot f(\bar t)\in {\cal U}^0(\vsz \hat I,N;\hat C_h,0;\infty)$$ and ${\cal U}^0 $ is the product neighborhood of ${\cal U}^0_1$ and ${\cal U}_2$. Note that \sdza\ implies that $F(\bar t)=1+\bigo{\bar t\>^{N+1}}$, therefore, we can take $\hat I_1=\ccint 1,1$ and $\hat I_i=\ccint 0,0$, for $i=1,\ldots,N$. This implies that $$\abs{u\bracket{r_N(t)}-t}\le \hat C_h\cdot t\cdot\bar t\>^{N+1}\qquad t\le\eta$$ Now, note that $$\abs{r_N(t)-r(t)}\le{\abs{u(r_N(t))-t}\over \inf_{M\le r\le \max( r(t),r_N(t))}\abs{u'(r)}}$$ At this point, we check that $a_i<0$ for all $i=1,\ldots,N$, which implies that $r_N(t)\le r_0(t)$. We then conclude, by hypothesis c., that $$\abs{u'(r)}\ge \abs{u'\bracket{r_0(t)}}\ge \fra 1,6 t^{\frac 3,2}(1-\epsilon')\qquad M\le r\le \max\bra{r(t),r_N(t)} $$ from which the algorithm follows by taking $$C_N={6\hat C_h\over (1-\epsilon')} \eqno\mathendpf $$ Now we compute bounds for the derivatives of $r'(t)$, also in the $C^0$ topology; this will allow us to compute bounds for $w(t)=-{r'(t)\over r(t)}$. We check first that $$12\eta^{-\frac 1,2}\bra{1-\sum_{n=1}^N\abs{a_n}-\tilde C_N}\ge M \qquad \tilde C_N=\fra 1,{12}C_N \eqno\sdzb$$ \algorithm{4.7} We produce a neighborhood ${\cal U}^0(\vsz I,N; C_h,0;\infty;t\le\eta)$ such that $$h(t) \eqbydef tw'(t)+w(t) = t^{-1}\cdot f(\bar t)\qquad f\in{\cal U}$$ for $t\le \eta$, where we define $w(t)=-r'(t)/r(t)$. \description We note that $$r'(t)={1\over u'\bracket{r(t)}}\qquad r''(t)=-\bra{r'(t)}^3u''\bracket{r(t)}$$ As a result of this, in view of \sdzup\ and \sdzupp\ and Algorithm \sdzalb, we obtain neighborhoods containing functions $f$, $f_p$ and $f_{pp}$ s.t. $$r(t)=12t^{-\frac 1,2}f(\bar t) \qquad r'(t)=-6t^{-\ffra 3,2}f_p(\bar t)\qquad r''(t)=9t^{-\ffra 5,2}f_{pp}(\bar t)$$ which are valid for $t\le \eta$, and where $$f_p(\bar t)=f^3(\bar t)\cdot\bra{u_p\Bracket{\bracket{{r(t)\over M}} ^{-\alpha}}}^{-1} \qquad f_{pp}(\bar t)=f_p^3(\bar t)\cdot f^{-4}(\bar t) \cdot u_{pp}\bra{\bracket{{r(t)\over M}}^{-\alpha}}$$ Thus, $$ \eqalignno{ w(t)&={-r'(t)\over r(t)}={1\over 2t}\bra{{f_p(\bar t)\over f(\bar t)}} \cr w'(t)&=\bra{r'(t)\over r(t)}^2 -{r''(t)\over r(t)}={1\over 4t^2}\bra{{f_p(\bar t)\over f(\bar t)}}^2 -{3\over 4t^2}\bra{{f_{pp}(\bar t)\over f(\bar t)}} \cr}$$ and $$\eqalignno{h(t) = tw'(t)+w(t)&=t^{-1}\bra{ \oh{f_p(\bar t)\over f(\bar t)}+\fra 1,4\bra{f_p(\bar t)\over f(\bar t)}^2 -\fra 3,4{f_{pp}(\bar t)\over f(\bar t)}}\cr &= {1\over 4t}f_h(\bar t)\cr}$$ for a function $f_h$ belonging to an easily computable neighborhood in $C^0$. \endpf \vskip1em Note now that our choice of functions was normalized so that $f(0)=f_p(0)=f_{pp}(0)=1$, which implies that $f_h(0)=0$. This is important: it says that if the Thomas--Fermi potential were {\it equal} to $cx^{-3}$, then $F'(\Omega)$ (which still makes sense) would be constant (recall Section~1), and would make Theorem~1.1 completely wrong. But it is not, and one can easily see that $$f_h(\bar t) = -{\alpha^2b_1\over 2\cdot 12^\alpha}t^ {\frac\alpha,2} +\bigo {t^ {\alpha} }\eqno\sdzg$$ as follows: recall that $$r(t)=12t^{-\frac 1,2}\bra{1+\fra \bar b_1,2\bar t+ \bigo{\bar t\>^{2}}}$$ Note that $$\eqalignno{r'(t)&=-6t^{-\frac 3,2}\bra{1-\fra (\alpha-1)b_1,{2\cdot 12^\alpha}\>t^{\frac\alpha,2}+\bigo{t^{\alpha}}}\cr r''(t)&=9t^{-\frac 5,2}\bra{1+\fra (\alpha-1)(\alpha-3)b_1,{6\cdot 12^{\alpha}}t^{\frac\alpha,2}+\bigo{t^{\alpha}}}\cr}$$ which are easily guessed by termwise differentiation of the expression for $r(t)$, and ---not so easily--- checked using the formulas for $r'(t)$ and $r''(t)$ above. This yields $$\displaylines{\quad h(t)={1\over 4t}\biggl(2\bracket{1-\fra (\alpha-1)b_1,{2\cdot 12^{\alpha}}t^{\frac\alpha,2}}\bracket{1-\fra b_1,{2\cdot 12^\alpha}t^{\frac\alpha,2}}+ \bracket{1-\fra(\alpha-1)b_1,{12^{\alpha}}t^{\frac\alpha,2}} \bracket{1-\fra b_1,{12^\alpha}t^{\frac\alpha,2}} \hfill\cr\hfill - 3\bracket{1+\fra(\alpha-1)(\alpha-3)b_1,{6\cdot 12^{\alpha}}t^{\frac\alpha,2}} \bracket{1-\fra b_1,{2\cdot 12^\alpha}t^{\frac\alpha,2}} +\bigo{t^{\alpha}}\biggr)\quad\cr}$$ which immediately implies (4.20). \vskip1em Now, let $$f_h(\bar t) = \sum_{n=1}^Na_n\bar t\>^n+H(\bar t)$$ with $$\abs{H(\bar t)}\le \epsilon_h\abs{\bar t}^{N+1} \qquad t\le \eta$$ Finally, then, let $L$ be a large number such that $u(L)\le\eta$: we set $\bar\Omega_3\le \sqrt{u(L)}$, $b(\Omega)=L$ for all $\Omega\le\bar\Omega_3$, and, arguing as before, we have $$\eqalignno{ I_3&= {d\over d\Omega}\bra{\Omega\int_{L}^{r_2(\Omega)}\bra{u(r)-\Omega^2}^ {-\frac 1,2}{dr\over r}}\cr &= {d\over d\Omega}\bra{\Omega\int_{\Omega^2}^{u(L)}\bra{t-\Omega^2}^ {-\frac 1,2}w(t)\,dt}\cr &= {d\over d\Omega}\bra{\Omega^2\int_{1}^{\Omega^{-2}u(L)}\bra{t-1}^ {-\frac 1,2}w(t\Omega^2)\,dt}\cr &= 2\Omega\int_{1}^{\Omega^{-2}u(L)}\bra{t-1}^ {-\frac 1,2}h(t\Omega^2)\,dt\cr &\qquad -2\bra{u(L)-\Omega^2}^{-\frac 1,2}w(u(L))u(L)\cr &= \oh\Omega^{-1}\sum_{n=1}^N{a_nM^{n\alpha}\over 12^{n\alpha}}\Omega^{n\alpha} \int_{1}^{\Omega^{-2}u(L)}\bra{t-1}^{-\frac 1,2}t^{\ffra \alpha,2n-1}\,dt \cr &\qquad+\tilde h(\Omega) +\bra{u(L)-\Omega^2}^{-\frac 1,2}{2u(L)\over Lu'(L)} &\sdzfa\cr}$$ with $$\abs{\tilde h(\Omega)}\le \oh\Omega^{-1+\alpha(N+1)}\epsilon_h \bra{M\over 12}^{(N+1)\alpha} \int_{1}^{\Omega^{-2}u(L)}\bra{t-1}^{-\frac 1,2}t^{\ffra \alpha,2(N+1)-1}\,dt \eqno\sdzfb $$ Now, we recall $\Omega_\epsilon$, on which we impose now the extra condition $$u(L)\ge 2\Omega_\epsilon^2\eqno\sdzconda$$ Both \sdzfa\ and \sdzfb\ can be computed easily for all $\Omega\ge \Omega_\epsilon$. The evaluation of integrals of the type $\int (t-1)^{-\frac 1,2} t^\gamma\,dt$ can be done by the same method as in the previous section. \vskip1em When $\Omega\le\Omega_\epsilon$, note that the first term in (4.21a) goes to infinity as $\Omega\to 0$, while all the others remain bounded. We use this to obtain a uniform lower bound for the absolute value of this derivative. By \sdzg\ we know that $a_1>0$; thus, we have $$\displaylines{ {d\over d\Omega}\bra{\Omega \int_{L}^{r_2(\Omega)}\bra{u(r)-\Omega^2}^ {-\frac 1,2}{dr\over r}}\ge \hfill\cr \hfill\eqalign{ &\ge \oh\Omega^{-1+\alpha}a_1\bra{M\over 12}^\alpha +{u(L)}^{-\frac 1,2} \sum_{{a_n<0}\atop{n\ge 3}}{a_n\bra{u(L)M^2\over 144}^{n\fra \alpha,2}} \cr &\qquad +\bra{u(L)-\Omega^2}^{-\frac 1,2}{2u(L)\over Lu'(L)}\cr }\qquad\llap\sdzh\cr}$$ where we have put $a_{N+1}=-\epsilon_h$. Note that, since the exponent $\gamma=\alpha-1$ does not fall under the cases considered in Lemma~4.4, \sdzh\ is only correct provided $a_2>0$. Of course, one can try to modify Lemma~4.4 to include the case $n=2$, but since it so happens that $a_2>0$ there is no need. By this we mean that we check $a_2>0$: if the check fails, our proof of Theorem~1.1 fails, and we claim no theorem. \vskip1em Putting together now \sdxxa---\sdxxc, \sdtxxa, \sdtconda, \sdzconda\ and \sdzh, we conclude that, if $$\Omega^2\le \oh\min\bracket{u(L),u(\delta)}$$ then $$-F''(\Omega)\ge \oh a_1\bra{M\over 12}^\alpha\Omega^{-1+\alpha} +T_1(\Omega)+T_2(\Omega)+T_3\eqno{(4.24)}$$ for $$\eqalignno{ T_2(\Omega) &= {u(L)}^{-\frac 1,2} \sum_{a_n<0}{a_n\bra{u(L)M^2\over 144}^{n\fra \alpha,2}} +\bra{u(L)-\Omega^2}^{-\frac 1,2}{2u(L)\over Lu'(L)} <0 \cr T_3&=\int_\delta^L\bracket{y(r)}^{-\frac 1,2}\>{dr\over r^{\frac 3,2}}>0 \cr}$$ \vskip1em The following is a consequence of formulas (4.21a,b), (4.11a,b) and Lemma~4.4. \propo{4.8} We have $$F''(\Omega)=-c_0\Omega^{-1+\alpha}+\bigo{1}\qquad c_0>0$$ as $\Omega\to 0$. \endt \vskip1em We now organize the main results in this section in the following algorithm. \algorithm{4.9}Given representable $\delta$ (small) and $L$ (large), and given neighborhoods in $C^0$ containing the functions $h(t)$ in Algorithm 4.3 and 4.7, valid for $00$ be given. Condition 3 implies that $B(w_0,T-\epsilon)$ does not intersect $\Gamma_s$ for $R-\epsilon'< s0$ and $u_2R^2>h$. \item{2.}For a constant $M$ we have $$\abs{{d^4\over dx^4}u(x)}\le M\qquad |z|\le 1$$ } \vskip1em Then, $t(x)$ as in \sdd\ can be extended analytically to $B(r_c,R)$, and there is an inverse $r(w)$ of $t(x)$, analytic in $\abs{w}< T$ where $$T\le \sqrt{u_2R^2-h}$$ and $$\eqalignno{ \sup_{|w|\le T}\abs{r'(w)}&\le {2\sqrt{u_2+hR^{-2}}\over 2u_2-3\abs{u_3}R -\fra 1,6MR^2 } \cr \abs{{d^{n+1}r\over dw^{n+1}}(0)}&\le n!\>T^{-n}\> {2\sqrt{u_2+hR^{-2}}\over 2u_2-3\abs{u_3}R -\fra 1,6MR^2} \qquad n\ge 0\cr }$$ \proof First, note that 1. implies that $$t(x)=z\sqrt{u_2R^2-zf(z)}$$ exists as an analytic function in $x\in B(r_c,R)$ (i.e., $|z|\le 1$), since the radicand never vanishes and a ball is simply connected, and note also that this definition agrees with \sdd\ if $x$ is real. Also, note that $t(x)$ satisfies the hypothesis of the previous lemma in the circle $x\in B(r_c,R)$. Indeed, $t(x)\ne 0$ unless $x=r_c$ (by 1), and if $|x-r_c|=R$, then $|z|=1$ and $$|t(x)|\ge \sqrt{u_2R^2-|f(z)|}\ge T$$ Therefore, by the previous lemma, $r(w)$ exists for all $|w|h$ and $\beta < 1$. Otherwise, the algorithm fails. By the previous lemma, $r'(t)$ can be written as $r'(t)=\sum_{n\ge 0}r'_n(t/T)^n$, where $r'_n$ can be computed by power--matching, for $n=1,\ldots,{2N+1}$, because we have $C_g=0$, and $$\sum_{n>{2N+1}}\abs{r'_n}\le {2R^{-1}\sqrt{I_0+h}\over R^{-2}\bra{2\abs{I_0}-3\abs{I_1}}-\fra 1,6MR^2} \cdot {\beta^{2N+2}\over 1-\beta}\eqno\invxb$$ A neighborhood of type $\infty$ and order $2N+2$ containing $r(t)$ can be obtained by integration. This allows us to construct a neighborhood ${\cal U}_1$ of type $\infty$ and order $2N+1$ containing the function $g$ in the statement of the algorithm, by simply dividing the neighborhood for $r'$ by the neighborhood for $r$. \endpf \algorithm{5.4} We compute a bound for $F''$ in Zone II. \description Note that $$\eqalignno{ \alpha_n &= {1\over \pi}\int_{-1}^1(1-t^2)^{-\frac 1,2}t^{2n}\,dt\cr &={1\over 2^{2n+1}i^{2n}\pi}\int_0^{2\pi}\bra{ e^{i\theta}-e^{-i\theta}}^{2n}\,d\theta\cr &={2n \choose n}2^{-2n} &\invxa \cr}$$ so their computation poses no difficulty. Note also that $\alpha_n>\alpha_{n+1}$ for all $n\ge 0$. Therefore, by (1.8), if we set $$ \bar w_n=T^n\cdot w_n$$ we can see that $$\eqalignno{ -\fra 1,\pi F'(\Omega)&=\Omega\sum_{n=0}^\infty \bar w_{2n}\bra{D\over T}^{2n}\alpha_n\cr &=\Omega\sum_{n=0}^{N} \bar w_{2n}\bra{D\over T}^{2n}\alpha_n +\Omega\sum_{n>{N}} \bar w_{2n}\bra{D\over T}^{2n}\alpha_n \cr}$$ The first term is a polynomial in $\Omega$, so we can easily compute its derivative anywhere. In fact, its derivative equals $$\sum_{n=0}^{N}\bar w_{2n}\alpha_n\gamma^n+ \Omega\sum_{n=1}^{N}\bar w_{2n}n\alpha_n\gamma^{n-1}\bra{{ -2\Omega\over T^2}}$$ with $$\gamma=\bra{D\over T}^2$$ Here we check that Zone II is included in the set of $\Omega$ that make $\gamma<1$. Otherwise, we report a failure and we quit the proof. As for the other term, using Lemma~\falema, and taking $$C_h\ge \sum_{n>N}\abs{\bar w_{2n}}$$ we have $$\displaylines{ \quad\abs{{d\over d\Omega}\bra{\Omega\sum_{n>{N}}{ \bar w_{2n}} \bra{D\over T}^{2n}\alpha_n}}\hfill\cr \hfill\eqalign{ &\le C_h\alpha_{N+1}\gamma^{N+1} +2\Omega_c^2\sum_{n> {N}}{\alpha_n \abs{\bar w_{2n}}n\gamma^{n-1}\over T^2}\cr &\le C_h\alpha_{N+1}\bra{\gamma^{N+1} +{2\Omega_c^2\over T^2}\sup_{n> N}{n\gamma^{n-1}}}\cr &\le \cases{ {\displaystyle C_h\alpha_{N+1}\gamma^{N}\bra{\gamma +{2(N+1)\Omega_c^2\over T^2}}}& if $N+1\ge {1\over |\log \gamma|}$\cr \noalign{\vskip1em}\cr {\displaystyle C_h\alpha_{N+1}\bra{\gamma^{N+1}+ {2\Omega_c^2\over eT^2\gamma|\log\gamma|}}}& in any case\cr}\cr} \cr}$$ All previous expressions can be easily computed using \invxa. Also, note the slight improvement in the result as a consequence of taking the neighborhoods in the previous algorithm to be of odd order. \endpf \vskip2em This concludes the description of all algorithms needed for the proof of Theorem~1.1. We summarize its computer--assisted proof in the following algorithm. \algorithm{5.5\quad(Proof of Theorem 1.1)} We produce a constant $c$ such that Theorem~1.1 holds. \description Run Algorithm~3.20 (and algorithms thereof) to obtain all necessary knowledge of the Thomas--Fermi function. \vskip1em Take $\bar\Omega$ as explained at the end of Section~1. to define Zone I and Zone II. As stated earlier in this section, we check that $\gamma(\bar\Omega)<1$. Then, we compute an upper bound for $F''$ in Zone II, and check that it is strictly negative. \vskip1em Choose $\Omega_\epsilon$, $\bar \Omega_2$ and $\bar \Omega_3$ as in Section~4, and a partition consisting of fat subintervals of $\ccint {\Omega_\epsilon},{\bar\Omega}$ whose endpoints contain both $\bar\Omega_2$ and $\bar \Omega_3$ and a subpartition of thin intervals $\Omega_i^\ast$. We compute the numbers $a_{k,i}$ and $b_{k,i}$. Next, we check that $F''$ is bounded above by a strictly negative number on the interval $\ocint 0,{\Omega_\epsilon}$ and on $\ccint \Omega_\epsilon,{\bar\Omega}$ as described in Algorithm 4.9. \vskip1em Theorem~1.1 then follows by taking the maximum of all these ---finitely many--- strictly negative constants. \endpf \Title{6. Some Extensions.} The purpose of this Section is to extend Theorem~1.1 to a neighborhood of the Thomas--Fermi potential, in an appropriate topology. As pointed out earlier, the fact that Theorem~1.1 holds is a rather delicate one. The following theorem shows this precisely. \theorem{\sdtha} Given any two large numbers $N$ and $R$, and given $\epsilon$ small, there exists a smooth function $f(x)$ such that { \itemize \item{a.} $f(x)=y(x)$ for $0\le x\le R$. \item{b.} For all $x\ge R$, and all $n\ge 0$, we have that $$\abs{{d^n\over dx^n}f(x)- {d^n\over dx^n}y(x)}\le\epsilon C_n x^{-3-n}$$ } \vskip1em \vskip1em and, however, we also have that $F_f(\Omega)$ vanishes at least $N$ times in $\ooint 0,{\Omega_c}$. (Note that, if $R$ is large enough, $\Omega_c$ is independent of $f$.) The $C_n$ are universal constants. In particular, they are independent of $\epsilon$, $R$ and $f$. \proof Note that we can assume $R$ to be as large as we need. \vskip1em By Corollary~1.3, $F''_{f}(\Omega)$ is bounded in the range $\Omega_\epsilon\le\Omega<\Omega_c$. Also, $F_f''(\Omega)=F_y''(\Omega)<0$ in the same range. >From a trivial adaptation of Section~\sdone, it follows that if $$f(x)={144\over x^3}\bra{1+bx^{-\alpha}}\qquad x\ge R$$ then $$F_f''(\Omega)= c_1b\Omega^{-1+\alpha}+\bigo{1}\qquad \alpha=\oh(\sqrt{73}-7) <1$$ for $c_1>0$, uniformly in $\Omega$. Therefore, taking $b=\epsilon$, there is $\Omega_1\muchsmaller\Omega_\epsilon$ such that $F_f''(\Omega_1)>0$. This gives a function $f$ such that $F_f''$ has at least one zero. To get more zeros, take $R_1$ large depending on $\Omega_1$ so that $F_f(\Omega)$, $\Omega\in\coint\Omega_1,{\Omega_c}$, is independent of $f(x)$ outside of $x\in \ooint 0,{R_1}$. Then, define $$f(x)={144\over x^3}\bra{1-\epsilon x^{-\alpha}}\qquad x\ge 2R_1$$ and smooth. Then, for $\Omega_2$ small enough $F_f''(\Omega_2)<0$. This gives us two zeros for $F''_f$. And so on. \endpf >From this theorem it is then clear that if we want Theorem~1.1 to hold, we need a stronger grip of the behavior of the function $f$ at infinity. \vskip1em The following theorem is just a consequence of the rest of this article. Its proof is computer assisted. \theorem{6.2} There exist $N$ large integer, $C$ and $x_1$ large constants, and $\epsilon>0$ and $x_0>0$ small, such that if $f(x)$ satisfies {\itemize \item{1.} $\lnorm{f-y}_{C^N\ccint x_0,{x_1}}\le\epsilon$. \item{2.} $f(x)=1-w\cdot x+x^{\frac 3,2}g\bra{(x/x_0)^{\frac 1,2}}$ with $\abs{w-w_0}\le\epsilon$, $g$ analytic and $\lnorm{g-g_{0}}_1\le\epsilon$, where $y_{TF}(x)=1-w_0\cdot x+x^{\frac 3,2}g_{0}\bra{x(x/x_0){\frac 1,2}}$. \item{3.} Recall formula {\rm \sdzdefba}. Then, $$f(x)={144\over x^3}\bra{1+\sum_{n=1}^\infty \bar a_n \bar x\>^{-n\alpha}}\qquad x\ge R$$ with $$\sum_{n=1}^\infty\abs{\bar b_n-\bar a_n}\le\epsilon$$ } \vskip2em Here, $\epsilon$ is assumed to be small enough so that our assumptions on $f$ stated at the beginning of Section~\secder\ are satisfied. \vskip1em Then, $$F_f(\Omega)\le c< 0\qquad \Omega\in\ooint 0,{\max\abs{rf(r)}}$$ \cproof Take $\eta$ any small number. If $\epsilon$ is small enough, hypothesis 2. and 3. imply that formulas (4.11ab) and (4.21ab) remain valid for $f(x)$ by perturbing the $a_n$ and $\epsilon_h$ by at most $\eta$ percent. Also, for $\epsilon$ small enough, hypothesis 1. implies that the value of integral $I_1$ in (4.1a) remains valid for $f$ also with an error at most $\eta$ percent. As a consequence, $T_1$, $T_2$ and $T_3$ will change by at most $C\eta$ percent. Therefore, $$-F''_f(\Omega)\ge \oh \tilde a_1\bra{\tilde M\over 12}^\alpha\Omega^{-1+\alpha} -C$$ where $\tilde a_1$ and $\tilde M$ differ from the ones in (4.20) by at most $\eta$ percent. In fact, we only need $\tilde a_1>\oh a_1>0$. Therefore, taking $\Omega_0\muchsmaller\Omega_\epsilon$, we see that $F''(\Omega)0$$ (This is the only point where we use a computer--assisted result.) >From formula (1.6), it follows that hypothesis 1., for $\epsilon$ small enough, implies that $$\abs{F_f''(\Omega)-F_y''(\Omega)}\le \fra 1,{10}\delta \qquad\Omega\in\coint \Omega_0,{\Omega_c}$$ which concludes the proof of the theorem. \endpf \Title{7. The Implementation} The aim of this section is to provide details about the way algorithms were implemented. The section will be organized as follows: {\smallitemize \item{1.} General remarks; in particular, the choice of several heuristic parameters is of special importance for a successful run of the computer proof: we list the approximate values. \item{2.} The second deals with the computer programs, which can divided into two groups. \itemitem{a.} One is a general package that performs general arithmetic and functional operations on certain general objects. This basic interval arithmetic package is a variation on the one used in \thesis\ and \Seco, which in turn is an adaptation of the one developed by D. Rana in \Rana. It is too long to present here, but we will give enough information about it so that a similar package can be built with little thought. In particular, we will list all function names with a very brief description of each. \itemitem{} Such packages are quite common already, and probably they will soon be standard. \itemitem{b.} The other is a package which takes care of the specific functions needed to prove our theorem. It follows very closely the algorithmic presentation in the present paper. We will list all these programs, preceded by a short explanation for each function, which will relate each of them to the corresponding algorithm in the text above. } \vskip1em \title{7.1 General Remarks.} According to the general package, (see below) we can store functions locally using the neighborhoods in function space ${\cal U}(\vsz I,N; C_h,C_g;k)$ introduced in Section~2 as follows: say $f(t)=\tilde f(\fra t-x,r)$, where $\tilde f\in{\cal U}$. Then, our knowledge of $f$ can be stored as a structure variable, consisting of { \itemize \item{1.} A pointer to an array of intervals: it is used to store the $I_j$. \item{2.} Two integers: one has value $N$, the order of the Taylor approximation. The other has value $k$, the type of the neighborhood. \item{3.} Two doubles, to store $C_h$ and $C_g$. \item{4.} Two doubles, to store $x$ and $r$. \medskip} By considering arrays of the structures above, we can store our {\it global} knowledge of functions as a single structure variable, consisting of a pointer to an array of the structure variables above. As a consequence, objects like the Thomas--Fermi function $y(x)$, are represented as a single variable. This gives a special computational meaning to Algorithm~3.20, the main result of Section~3. We divide our remarks according to the section they are related to. \vskip1em \title{Section 3.} In Algorithm~3.20, note that the choice of the $x_i$ and $r_i$ is in principle arbitrary, but in practice, it is very important that they are chosen carefully. Main points to take into account are: {\itemize\item{1.} All runs of parent algorithms should be successful \item{2.} Error bounds $C_{h,i}$ and $C_{g,i}$ obtained when we run the algorithm are sensitive to our choice of $x_i$ and $r_i$. The proof Theorem~1.1 is in turn sensitive to these error bounds. In principle, the smaller the $r_i$ the better. It is important that these error bounds are small enough so that we can prove our theorem. \item{3.} The number $m$ is important also: a large $m$ is a consequence of small $r_i$ which will give small error terms for the $C_{{h\atop g},i}$, but will make the computation of $I_1$ in Section~4 very slow, maybe too slow to prove our theorem in a finite time. On the other hand, a small $m$ will speed up the computation of $I_1$, but will yield bad bounds for the $C_{{h\atop g}},i$. Similar considerations hold for the choice of $N$. \medskip} The choice of $N$ is fixed on a trial and error basis. $N\sim 10$ works. About the $x_i$ and $r_i$ note that their choice was made in Algorithms~3.16 and 3.18. They were picked adaptatively inside the program, in the sense that if during the execution of Algorithm~3.2 (a parent algorithm), one of the error bounds grows outside a pre--specified range, then we make the next $r_i$ a little smaller. And viceversa, if that error goes below a certain range, then we make the next $r_i$ a little bigger. The error we look at in deciding this is $\norm{p-\tilde Tp}/\norm p$ in Algorithm~3.2, and we wanted it to be within the bounds $\ccint {\sim 10^{-17}},{\sim 10^{-15}}$. We chose $x_0\sim 0.008$ and $r_0\sim 0.0008$. The radii grow as we leave the origin. In carrying out this procedure we made $x_{i+1}\sim x_i+r_i$. This gave enough overlap between intervals to capture the behavior of our functions all over $\ooint x_0-r_0,{x_m+r_m}$. As a result of this method, we obtained $m\sim 800$ and $x_m\sim 300$. The following remark is of mild interest: when choosing the $x_i$, we instructed the computer to include the point $\sim$2.10 with the idea in mind that $r_c$ is close to this number. Since the information given by Algorithm~3.20 is the only one we would like to use when computing $F''$, the neighborhoods for $y_{\rm TF}(x)$ around $r_c$ which would have to be computed in Section~5 turn out a little better. \vskip1em \title{Section~4.} The actual value for $\bar\Omega$ is about $0.6956$, only $\sim 10^{-3}$ from $\Omega_c$. Thus, Zone II will turn out to be very small. This is unavoidable using our complex--variable methods for Zone II, since with radii larger than that, we cannot exclude the existence of other zeros of $u(z)-\Omega_c^2$ in the vicinity of $r_c$ in the complex plane. \vskip1em The first heuristic choice we have to make is the numbers $a$ and $b$ as a function of $\Omega$. We describe the choice of $a$. The choice of $b$ is similar. In the notation of Algorithm~3.20, choose the $x_i$ closest to $r_1$ such that $$\sum_{k=1}^N\abs{I^i_k}\le \delta\cdot\abs{I^i_0-\Omega^2}\qquad\delta<1$$ This is a trivial prerequisite if we want to understand $\bra{u(r)-\Omega^2}^{-\frac 3,2}$ in $H^1$. The choice of $\delta$ is delicate, though: if it is very close to 1, then the fractional power operation will yield bad error bounds. If it is very small, we will be forced to take $x_i$ far away from $r_1$. This will hurt the error terms when we compute $I_2$, and it could even make the computation of $I_2$ not possible with our method. The problem is that we will be forced to solve O.D.E.'s at rather large distances (this is already dangerous), and, even worse, we will have to take fractional powers of Taylor expansions with large radii: this may be impossible if, for instance, the solution of the O.D.E. has zeros within our large radius in the complex plane. A value of about 0.3 for $\delta$ works most of the time, but it needs fixing for some values of $\Omega$. We refer the reader to functions {\tt alim()} and {\tt blim()} in the program listings for the specific choice of $\delta$ as a function of $\Omega$. \vskip1em We continue now with the computation of $I_1$, the most time--consuming procedure. The division into fat intervals is done with intervals of length $\sim 10^{-3}$. This is large enough so we can cover the all of Zone I = $\ccint 0,{\bar\Omega}$ with not so many of those fat intervals, around 1000 of them, and it is also small enough so that the approximation given by (4.2) in terms of the $a_{k,i}$ and $b_{k,i}$ is good enough. As we approach $\bar\Omega$, we made the length of these fat intervals smaller, about $10^{-4}$. Note that a reduction in the size of the $W_i$ results in having to compute the $a_i$ and $b_i$ more often, but if we are close to $\Omega_c$, the interval $\ccint r_1(\Omega),{r_2(\Omega)}$ is very small anyway which require few $i$, and thin $W_i$ won't hurt. Next, we have to choose $\tilde a$ and $\tilde b$, or, which is equivalent, we have to choose $i_0$ and $i_1$. We chose them to be $i_0=10$, $i_1=n-10$. This works. The choice of the $\Omega^\ast$ is the most delicate. What we did, is give the computer an initial interval of length $l$ and let it compute bounds for $I_1$ as well as $I_2$ and $I_3$: if these bounds are good enough so that we can show that $-F''>0$ on $\Omega^\ast$, we tell the computer happily to take another $\Omega^\ast$ inside $W_i$, and do the same, until all of $W_i$ is covered with tests. If for some interval we cannot produce the bound $-F''>0$, then we tell the computer to subdivide that interval into two halves, and try each half recursively until (hopefully) we finish. The process finished, so we conclude $-F''>0$ all over $W_i$. The length of the $\Omega^\ast$ that work is about $5\cdot 10^{-5}$, degenerating until about $10^{-7}$ near $\bar\Omega$: note that this will generate {\it a lot} of computations. In principle, we could have given the computer as a first try all of the interval $W_i$, even without hope, but let the computer figure out how much finer to go before getting the desired bounds. This is fine from the rigorous point of view, but we would be wasting a lot of precious time asking the computer to make checks that we are confident are going to fail. It is thus important to grind $W_i$ into finer intervals before feeding the computer with this recursive procedure. \vskip1em Concerning the computation of $I_2$ and $I_3$, they are analogous, and the only thing worth mentioning is our choice of the following heuristic parameters: $M\sim 291$, $L\sim 295$, $x_0\sim 0.012$ and $\delta\sim 0.0099$. The degrees of Taylor expansions we chose are 10 for $I_3$ and 20 for $I_2$. Also, $\Omega_\epsilon\sim 10^{-2}$, $\bar\Omega_2\sim 0{.}0932$ and $\bar\Omega_3\sim 0{.}03469$. See the implementation comments for functions {\tt secder0()} and {\tt secder1()} for more about the $\bar\Omega_{2,3}$. \vskip1em \title{Section 5.} We finally discuss the peculiarities of Zone II. Recall that the diameter of Zone II is about $10^{-3}$. Also, it follows from Section~5 that it is rather easy to obtain good bounds for the value for $-F''(\Omega_c)$. The analysis for Zone~II therefore looks unnecessarily complicated, since it would follow from the apparently easy, but in practice deep statement that $\abs{F''(x)}$ is bounded by about $1000$. Since it is numerically evident that $\abs{F''(x)}$ is bounded by a number much smaller than $1000$, maybe one can obtain a good bound for $F'''$ which would make the analysis of Zone II trivial. The only parameter of real importance is $R$. Too large $R$ are bad because, as mentioned before, it forces us to carry our fractional power analysis to large distances. Too small $R$ will force us to take small $\tilde T$ and therefore small $T$, and will result of restricting our knowledge of $w(t)=g(t/T)$ to a too small neighborhood of $\Omega_c$, thus being unable to cover all of Zone II. Our choice was $R\sim 0.462$. Once $R$ is chosen, we take $\tilde T$ as large as we can, still satisfying \invxc, and we are left with the choice of $T$ only. Of course, we would like to take $T$ as large as possible, but note that the closer we take $T$ to $\tilde T$, the closer $\beta$ will get to 1, which will give us a bad error estimate in \invxb. This negative effect can be neutralized by taking $N$, which until now was arbitrary, to be big, so that the power $\beta^{2N+2}$ makes the right hand side of \invxb\ very small. We took $T\sim 0.0605$ and $N\sim 26$, but larger $N$ will be even better. The only problem with large $N$ is that it will force us to invert a polynomial of large degree. Even with our sloppy implementation of the inversion procedure, errors and speed are of negligible importance. It follows from these remarks that the analysis of Section~5 will work on an interval around $\Omega_c$ whose length depends basically on how large we can take $R$. Without a more refined analysis, our choice of $R$ is imposed on us by the apparent complex solutions of $u(z)=\Omega_c^2$ around $r_c$, and thus is not subject to improvement. In other words, there is a good reason for taking $\bar\Omega\sim 0.6956$ and not smaller: Zone II is given to us by the problem, not by the computer's ability to compute fast or accurately. As a result, with a slower or less accurate computer, which would not be able to compute $-F''$ in Zone I all the way up to $\bar\Omega$, we wouldn't be able to prove our theorem in this way. One would need to perform a real variable analysis to a larger Zone II, in a similar way to the analysis of $I_2$ and $I_3$ for $\Omega < \Omega_2 , \Omega_3$. \vskip1em Finally, all programs are written in {\bf C}, and were run on several IBM RS600 simultaneously. As explained later, our problem can be naturally split into several independent processes, making it a very appropriate problem to run on different machines at the same time. Execution took about two days for the programs related to the Thomas--Fermi equation, and about 6 hours for the ones involving the actual computation of $F''$. Executable files averaged 4Mb each. \vskip1em \parskip=10truept \title{7.2 General Purpose Package.} The basic variable types in this package are the following: \vskip1em \halign{\tt # & \qquad\tt# &\qquad \tt #\cr typedef struct $\{$ &double dn;\cr &double up;$\}$ &INTERVL;\cr typedef struct $\{$&double b;$\}$ &BND;\cr typedef struct $\{$ &int deg;\cr &INTERVL *p; $\}$&POLY;\cr typedef struct $\{$&POLY p;\cr &BND center;\cr &BND r;\cr &BND g;\cr &int k;\cr &BND h;$\}$ &RSERIES;\cr typedef struct $\{$&int n;\cr &RSERIES *f;$\}$ &GRS;\cr union convert$\{$ &reps r;\cr &unsigned long int i[2];\cr &$\}$;\cr } \vskip1em {\medtype\parskip=0pt\obeylines\tt double mtwo = (double) -2; double mone = (double) -1; double zero = (double) 0; double half = (double) 0.5; double one = (double) 1; double two = (double) 2; double eight = (double) 8; BND bmone = $\{$(double) -1.$\}$; BND b0ero = $\{$(double) 0$\}$; BND bquarter = $\{$(double) .25$\}$; BND bhalf = $\{$(double) .5$\}$; BND bone = $\{$(double) 1$\}$; BND btwo = $\{$(double) 2$\}$; BND bthree = $\{$(double) 3$\}$; BND bfour = $\{$(double) 4$\}$; INTERVL imfour = $\{$(double) -4,(double) -4$\}$; INTERVL imthree = $\{$(double) -3,(double) -3$\}$; INTERVL imtwo = $\{$(double) -2,(double) -2$\}$; INTERVL imone = $\{$(double) -1,(double) -1$\}$; INTERVL izero = $\{$(double) 0,(double) 0$\}$; INTERVL ihalf = $\{$(double) 0.5,(double) 0.5$\}$; INTERVL imhalf = $\{$(double) -0.5,(double) -0.5$\}$; INTERVL ione = $\{$(double) 1,(double) 1$\}$; INTERVL itwo = $\{$(double) 2,(double) 2$\}$; INTERVL ithree = $\{$(double) 3,(double) 3$\}$; INTERVL ifour = $\{$(double) 4,(double) 4$\}$; INTERVL ifive = $\{$(double) 5,(double) 5$\}$; INTERVL isixteen = $\{$(double) 16,(double) 16$\}$; INTERVL imsix = $\{$(double) -6,(double) -6$\}$; INTERVL ieight = $\{$(double) 8,(double) 8$\}$; INTERVL ifortyeight = $\{$(double) 48,(double) 48$\}$; int n; double ln2; INTERVL iln2 = $\{$0.69314718055994484, 0.69314718055994584$\}$; } \vskip1em The following are the function descriptions. {\tt up(r)}. {Returns a representable strictly larger than {\tt r}. } {\tt dn(r)}. {Returns a representable strictly smaller than {\tt r}. } \vskip1em The functions to follow return variable of type {\tt BND}. Variables {\tt a} and {\tt b} are of type {\tt BND}, {\tt x} is of type {\tt INTERVL} and {\tt m} is of type {\tt int}. {\tt ucvtib(x)}. {Returns an upper bound for {\tt x}. } {\tt lcvtib(x)}. {Returns a lower bound for {\tt x}. } {\tt cvtdb(d)}. {Converts {\tt d} (a {\tt double}) into {\tt BND}. } {\tt cvtintb(m)}. {Converts {\tt m} into {\tt BND}. } {\tt uplusb(a,b)}. {Returns an upper bound for the sum of {\tt a} and {\tt b}. } {\tt lplusb(a,b)}. {Returns a lower bound for the sum of {\tt a} and {\tt b}. } {\tt neg(a)}. {Returns $-${\tt a}. } {\tt absb(a)}. {Returns $|{\tt a}|$. } {\tt minb(a,b)}. {Returns the minimum of {\tt a} and {\tt b}. } {\tt maxb(a,b)}. {Returns the maximum of {\tt a} and {\tt b}. } {\tt umultb(a,b)}. {Returns an upper bound for the product of {\tt a} and {\tt b}. } {\tt lmultb(a,b)}. {Returns a lower bound for the product of {\tt a} and {\tt b}. } {\tt uinvb(a)}. {Returns an upper bound for the inverse of {\tt a}. } {\tt linvb(a)}. {Returns a lower bound for the inverse of {\tt a}. } {\tt udivb(a,b)}. {Returns an upper bound for {\tt a}/{\tt b}. } {\tt ldivb(a,b)}. {Returns a lower bound for {\tt a}/{\tt b}. } {\tt usquareb(b)}. {Returns an upper bound for {\tt b}$^2$. } {\tt lsquareb(b)}. {Returns a lower bound for {\tt b}$^2$. } {\tt upowerb(b,m)}. {Returns an upper bound for {\tt b}$^{m}$. } {\tt lpowerb(b,m)}. {Returns a lower bound for {\tt b}$^{m}$. } The functions to follow return a variable of type {\tt int}. {\tt eqb(a,b)}. {Returns 1 if {\tt a}={\tt b}, 0 otherwise. } {\tt neqb(a,b)}. {Returns 0 if {\tt a}={\tt b}, 1 otherwise. } {\tt lsb(a,b)}. {Returns 1 if ${\tt a}<{\tt b}$, 0 otherwise. } {\tt lseqb(a,b)}. {Returns 1 if ${\tt a}\le{\tt b}$, 0 otherwise. } {\tt grtb(a,b)}. {Returns 1 if ${\tt a}>{\tt b}$, 0 otherwise. } {\tt grteqb(a,b)}. {Returns 1 if ${\tt a}\ge {\tt b}$, 0 otherwise. } \vskip1em The functions to follow return \ variable of type {\tt INTERVL}, unless stated otherwise. Variables {\tt x} and {\tt y} are of type {\tt INTERVL}, {\tt d} is {\tt double}, {\tt m}, {\tt i} and {\tt j} are of type {\tt int} and {\tt b} is of type {\tt BND}. {\tt cvtbi(b)}. Converts {\tt b} into {\tt INTERVL}. {\tt cvtdi(d)}. Converts {\tt d} into {\tt INTERVL}. {\tt cvtinti(m)}. Converts {\tt m} into {\tt INTERVL}. {\tt plus(x,y)}. Returns an interval containing the true set--theoretic sum of {\tt x} and {\tt y}. {\tt neg(x)}. Returns $-${\tt x}. {\tt iabs(x)}. Returns $|{\tt x}|$. {\tt uabs(x)}. Returns an upper bound to $|{\tt x}|$. The function returns a variable of type {\tt BND}. {\tt labsi(x)}. Returns a lower bound to $|{\tt x}|$. Returns a variable of type {\tt BND}. {\tt iequ(x,y)}. Returns 1 if the arguments are exactly the same, 0 otherwise. {\tt ienlarge(x,b)}. Returns an interval containing all points at distance at most {\tt b} from {\tt x}. {\tt mult(x,y)}. Returns an interval containing the true set--theoretic product of {\tt x} and {\tt y}. {\tt divi(x,y)}. Returns an interval containing the true set--theoretic division of {\tt x} by {\tt y}. If $0\in{\tt y}$, then we abort the program. {\tt inv(x)}. Returns an interval containing the true set--theoretic inverse of {\tt y}. If $0\in{\tt y}$, then we abort the program. {\tt square(x)}. Returns an interval containing the true set--theoretic square of {\tt x}. {\tt power(x,m)}. Returns an interval containing the true set--theoretic power {\tt x}$^m$. {\tt intersect(x,y)}. Returns ${\tt x}\cap{\tt y}$, which also belongs to ${\cal I}$. {\tt iunion(x,y)}. Returns ${\tt x}\cup_I{\tt y}\in{\cal I}$, the smallest interval containing the union of both arguments. {\tt ration(i,j)}. Returns an interval containing ${\tt i}/{\tt j}$. {\tt iexp(x)}. Returns an interval containing $e^x$. This can be easily constructed using the Taylor expansion for the exponential. {\tt ilog(x)}. Returns an interval containing $\log({\tt x})$. This can be easily constructed using the Taylor expansion for the exponential, in the case ${\tt x}\in\coint \oh,1$, and the general case follows trivially after we obtain upper and lower bounds for $\log 2$. These bounds can be obtained heuristically, and then checked using the function {\tt iexp()}. Alternatively, bounds for $\log 2$ are available in the literature, which are better than the ones we could check with {\tt iexp()}; we preferred our way since we simply don't know whether those bounds in the literature are rigorous. This is somewhat wasteful, since {\tt iexp()} is rather conservative (not much). But it did not affect our proof in any noticeable way. The functions to follow return variables of type {\tt POLY}, unless said otherwise. Arguments starting with {\tt p} are also of type {\tt POLY}, {\tt m} is of type {\tt int}, {\tt x}, {\tt y} and {\tt a} are of type {\tt INTERVL}, and variables starting with {\tt b} are {\tt BND}. {\tt make\_poly(m)}. Returns a {\tt POLY} of degree {\tt m} with zero coefficients. {\tt polycopy(p)}. Returns a {\tt POLY} identical to {\tt p}. {\tt coeff(p,m)}. Returns {\tt p.p[m]}, an {\tt INTERVL}. {\tt coeffmult(p1,p2,m)}. Returns the {\tt m}'th coefficient in the algebraic product (in the interval arithmetic sense) of {\tt p1 \rm and \tt p2}. {\tt polysca(p,a)}. Returns bounds for {\tt a}$\cdot${\tt p}. {\tt evalpoly(p,a)}. Returns an {\tt INTERVL} containing the algebraic evaluation of {\tt p} at {\tt a}. {\tt polynorm(p)}. Returns a {\tt BND}, which is an upper bound for the sum of the absolute value of the coefficients of {\tt p}. {\tt polyplus(p1,p2)}. Returns bounds for the algebraic sum of the arguments. {\tt polymult(p1,p2)}. Returns bounds for the algebraic product of the arguments. {\tt polyscale(p,a)}. Returns bounds for the polynomial in $x$ given by ${\tt p}({\tt a}\cdot x)$. {\tt polyder(p)}. Returns bounds for the algebraic derivative of {\tt p}. {\tt polyinteg(p)}. Returns bounds for the algebraic integral of {\tt p}. {\tt polycomp(p1,p2)}. Returns bounds for the algebraic composition {\tt p1}({\tt p2}). {\tt coeffcomp(p1,p2,m)}. Returns bounds for {\tt m}'th coefficient in the algebraic composition {\tt p1}({\tt p2}). {\tt polyinv(p)}. Returns bounds for the first {\tt p.deg}+1 Taylor coefficients of the functional inverse $p^{-1}$ such that {\tt p}$\circ (p^{-1})={\rm Id}$. The following functions return variables of type {\tt RSERIES}, with same radius, center, order and type as the arguments, unless stated otherwise. Variable names continue with the same type, except that those starting with {\tt s} and {\tt r} are now of type {\tt RSERIES}. {\tt rs(x,r,i)}. Returns a {\tt RSERIES}, with {\tt .center}={\tt x}, {\tt .r}={\tt r} and {\tt .p.deg} = {\tt i}. Polynomial coefficients are zero. {\tt rscopy(r)}. Returns a {\tt RSERIES} identical to {\tt r}. {\tt ichcoo(a,r)}. Returns bounds for $({\tt r.center}-{\tt a})/ {\tt r.r}$. {\tt geomrs(x,y,b1,m,b2)}. Returns a neighborhood for $1/({\tt x}t+ {\tt y})$ with center at {\tt b1}, radius {\tt b2} and degree {\tt m}. {\tt rstrunc(s,m)}. Returns a {\tt RSERIES} of degree {\tt m} which contains {\tt s}. It aborts if ${\tt m}>{\tt s.p.deg}$. {\tt rsplusc(r,a)}. Returns bounds for {\tt r}+{\tt a}. {\tt rsca(r,a)}. Returns bounds for {\tt a}$\cdot${\tt r}. {\tt rsplus(r,s)}. Returns bounds for the sum of the arguments. The order and type are the smaller of those of the arguments. It is assumed without check that the center of the arguments are identical. {\tt rsminus(r,s)}. Returns bounds for {\tt r}$-${\tt s}. The order and type are the smaller of those of the arguments. It is assumed without check that the center of the arguments are identical. {\tt rsmult(r,s)}. Returns bounds for the product of the arguments. The order and type are the smaller of those of the arguments. It is assumed without check that the center of the arguments are identical. {\tt rseval(r,a)}. Returns an {\tt INTERVL} with bounds for {\tt r}({\tt a}). {\tt rsinteg(r)}. Returns a neighborhood for the functions $\int_{{\rm r.center}}^x f(t)\,dt$ where $f\in{\tt r}$, and $|x-{\tt r.center}|<{\tt r.r}$. The polynomial order and type of the output is one more than those of the argument. {\tt rsdint(r,a,b)}. Returns a {\tt INTERVL} with bounds for $\int_b^a f(t)\,dt$ where $f\in{\tt r}$. {\tt rsoverx(s)}. Returns bounds for ${\tt s}(x)/(x-{\tt s.center})$. It is assumed here without check that {\tt s}({\tt s.center})=0 exactly and the type of {\tt s} is at least 1. {\tt rslog(x,y,b1,m,b2)}. Returns a neighborhood for $\log({\tt x}t+ {\tt y})$ with center at {\tt b1}, radius {\tt b2} and degree {\tt m}. {\tt ievders(s,x)}. Returns an {\tt INTERVL} containing bounds for the derivative of {\tt s} at {\tt x}. {\tt ievnders(s,x,m)}. Returns an {\tt INTERVL} containing bounds for the {\tt m}'th derivative of {\tt s} at {\tt x}. {\tt bl1nrs(s)}. Returns a {\tt BND} which contains an upper bound for $\lnorm s_1$. {\tt rstimesx(s)}. Returns bounds for the functions $x\cdot{\tt s}(x)$. {\tt frac22(x,b)}. Returns a {\tt BND} with an upper bound for $C_{2.2}(x,b)$, as in Section~2. {\tt frak22(x,b)}. Returns a {\tt BND} with an upper bound for $K_{2.2}(x,b)$. {\tt rsmatpower(s,*r)}. Returns the argument {\tt *r}, a pointer to an array of {\tt RSERIES}, containing bounds for all powers of {\tt s}, from 0 to {\tt s.p.deg}. {\tt rspower(r,x)}. Returns bounds for {\tt r}$^x$. {\tt polypower(p,x)}. Returns a {\tt POLY} containing bounds for the Taylor approximation of degree {\tt p.deg} of {\tt p}$^x$. The functions to follow perform operations on variables of type {\tt GRS}, represented by arguments starting with {\tt gs}. Functions {\obeylines\parskip=0pt\parindent=30pt {\tt grscopy(gs)} {\tt grsmult(gs1,gs2)} {\tt grseval(gs,x)} {\tt grsdereval(gs,x)} {\tt grstimesx(gs)} {\tt grspower(grs,x)} } perform the corresponding operations as their {\tt RSERIES} counterparts on each {\tt RSERIES} member of their structures. {\tt grsdint(gs,x,y)}. Returns bounds for the integral from {\tt x} to {\tt y} of all global functions in {\tt gs}. Note here that the role the of {\tt x} and {\tt y} is reversed with respect to {\tt rsdint()}. {\tt grs(n)}. This function simply returns a {\tt GRS} with {\tt .n} member equal to {\tt n}, and with space allocated for {\tt n}$+1$ variables of type {\tt RSERIES}. Note that the further allocation needed {\it in} the {\tt POLY} member of {\tt RSERIES} is not done here. This should be done using either {\tt rs()} or {\tt make\_poly()} above. {\tt grsintpt(gs,i)}. Returns a {\tt double}, a heuristic choice for the ``middle'' point between the centers of the {\tt i}'th and {\tt i}+1'th members of {\tt gs}. If we denote the centers by $x_1$ and $x_2$, and corresponding radii by $r_1$ and $r_2$, this function returns approximately the number $\displaystyle{r_1\cdot x_2+r_2\cdot x_1}\over \displaystyle {r_1+r_2}$. {\tt grsloc(gs,x)}. Another heuristic function. Returns an integer representing the member of the structure {\tt gs} which best captures the behavior of {\tt gs} near {\tt x}, i.e., the one that minimizes (in a heuristic way), the output of {\tt ichcoo(x,\dots)} above. The functions with names equal to the above followed by an {\tt f} perform the same operations, plus: they destroy the arguments containing pointers by freeing the memory they have allocated. In addition to these functions, we also have the following, which are of an entirely heuristic nature. They are designed to make the heuristic guesses of $p$ in Algorithms~3.2 and similar. They manipulate polynomials, this time defined simply as arrays of {\tt SIZE}$+1$ variables of type {\tt double} (we took {\tt SIZE}=50 in our programs); we will denote such variables here with names starting with {\tt po}. They also use the additional {\tt extern} variable {\tt DEGREE}, smaller than {\tt SIZE} at all times, intended to allow us to vary the effective degree of these polynomials inside the programs. They do not return any variables a values, only as arguments. All operations they perform are floating--point. {\tt pzer(po)}. Initializes {\tt po} to 0. {\tt pcopy(po1,po2)}. Copies {\tt po1} into {\tt po2}. {\tt pnorm(po1)}. Returns a {\tt double} with a floating--point approximation to $\lnorm{{\tt po1}}_1$. {\tt psub(po1,po2,po3)}. Puts in {\tt po3} the algebraic difference of {\tt po1} and {\tt po2}. {\tt pprod(po1,po2,po3)}. Puts in {\tt po3} the algebraic product of {\tt po1} and {\tt po2} truncated to {\tt DEGREE}. {\tt pinte(p01,po2)}. Puts in {\tt po2} the algebraic integral of {\tt po1}, truncated to {\tt DEGREE}. {\tt psca(po1,x,po2)}. Puts in {\tt po2} the product of {\tt po1} by the {\tt double x}. {\tt pscale(po1,x,po2)}. Puts in {\tt po2} the scaled polynomial {\tt po1$({\tt x}\cdot t)$}. {\tt myprpower(po1,x,po2)}. Takes {\tt po1} to the power {\tt x} and puts the result in {\tt po2}. Last, but not least, we also need functions that give the {\it decimal} expansion of rationals bounds for representable numbers. This is required, for instance, to be able to state Lemma~3.21 in the form we did, rather than in a form where the bounds claimed are given in the harder to visualize hexadecimal form. The construction of such functions, while not trivial, is not too hard and we omit the details. \vskip3em \title{7.3 Aperiodicity Programs.} \parskip=10truept The following is a brief itemized explanation of the computer programs included at the end of this paper. \vskip1em Throughout the programs, we will use the external variables \ {\medtype\tt\obeylines\parskip0pt extern GRS Y, YRS, U; extern INTERVL Le, UL, De, UD, C1, W, RC, BC, ALPHA; extern RSERIES HINF, H0; } The variable {\tt ALPHA} will contain consist of bounds for $\oh(\sqrt{73}-7)$ computed once and for all at the beginning of each program. The variables {\tt De} and {\tt UD} correspond to $\delta$ and $u(\delta)$ of Section~4, and ${\tt Le}$ and ${\tt UL}$ correspond to variables $L$ and $u(L)$ also in Section~4, and they are introduced in functions {\tt h\_at\_0()} and {\tt hinf()} respectively. The rest will be explained below. {\tt lipreg()}. Implements Algorithm 3.1, returning the Lipschitz norm if we can show that it is less than 1, and returning 1 otherwise. {\tt vtffx()}. Implements Algorithm 3.3. Algorithm 3.2, which is needed for the execution of the former, is implemented explicitly inside the function. {\tt supervtffx()}. Implements Algorithm 3.5. Note that in this function, as well as in {\tt vtffx()}, the values and derivatives of the solution of the ODE are returned as arguments, while the {\tt double} that the function returns as value is an approximation to $\norm{ p-\tilde Tp}$, which, as pointed out before, will be used in deciding how much to increase or decrease the next choice of $r_i$. {\tt vtffxi()}. Implements Algorithm 3.2. Gives neighborhoods of type 2. This function (and {\tt vtffxi2()} below) returns a neighborhood valid for all centers in the interval {\tt xin}. As a consequence, no {\tt .center} of type {\tt BND} can be naturally specified in the {\tt RSERIES} it returns; we assigned the value 0 (a better choice would be {\tt NaN}). This means that we cannot manipulate the outcome of this particular function with any general--purpose function which would attempt to make use of the structure member {\tt .center}. All such manipulations should be done explicitly taking into account that the centers are contained in {\tt xin}.\footnote{$^1$}{ \medtype One could attempt to define a new variable type in which centers are of type {\tt INTERVL}. Since this is the only place in which we use this special choice, we decided not to do it in this particular situation. \par } {\tt tfypoly()}. See below. {\tt vtffxi2()}. Implements Algorithm 3.2. Gives neighborhoods of type $\infty$. The power matching scheme is done in {\tt tfypoly()} {\tt lip0()}. Implements Algorithm 3.6, returning the Lipschitz norm if we can show that it is less than 1, and returning 1 otherwise. {\tt vtff0()}. Implements Algorithm 3.8. Again, Algorithm 3.7 is built in. {\tt y\_at\_0()}. Implements Algorithm 3.7. {\tt tfw()}. Implements Algorithm 3.15. In our description of this Algorithm above, the $x_i$ and $r_i$ are given. From the logical point of view, this is true, but from the computational point of view, the $x_i$ and $r_i$ are produced within {\tt tfw()}. The successive rigorous bounds we find for $w_0$ are printed in hexadecimal form as they are obtained. The reason for this is that it takes a long time to run each iteration. In this way, one has rigorous bounds for $w_0$ even if the function does not finish (due to computer shut down, or impatience on our part). {\tt tff(w)}. This function implements Algorithm~3.16, where the {\tt INTERVL w} has as endpoints a rigorous upper and lower bound for $w_0$. As in {\tt tfw()}, the choice of the $x_i$ and $r_i$ is done inside the function. The values $y_i$ and $y_i'$ are returned as a single variable of type {\tt GRS}. {\tt tfrs(y)}. Implements Algorithm~3.20. The argument ${\tt y}$, which is of type {\tt GRS}, is the output of {\tt tff(w)}. {\tt Omega(u)}. The argument {\tt u}, a {\tt GRS} variable, contains bounds for $u_{\rm TF}(x)=x\cdot y_{\rm TF}(x)$. The function then returns an interval $\ccint r^{\rm dn},{r^{\rm dn}}$ which is guaranteed to contain $r_c$. Recall that $r_c$ is uniquely defined by the identity $u'(r_c)=0$. Thus, we first look in a heuristic manner for $r^{\rm dn}$ and $r^{\rm up}$, and conclude that they are valid bounds after checking that $u'(r^{\rm dn})\ge 0$ and $u'(r^{\rm up})\le 0$, which we can easily do using the general purpose interval arithmetic package, namely, function {\tt grsdereval()}. The heuristic construction of the interval is done via a bisection method, slightly modified so that the interval produced is optimal, in the sense that any representable $r>r^{\rm dn}$, the bounds we obtain for $u'(r)$ would not be strictly positive (similarly for $r^{\rm up}$). {\tt tfprint()}. This is a bookeeping function. It prints the output of {\tt tfw()}, bounds for $w_0$ and $-b_1$, the output of {\tt tff()} (Algorithm~3.16), {\tt tfrs()} (Algorithm~3.18), together with a {\tt GRS} variable containing $u_{\rm TF}$, the bounds for $r_c$ produced in {\tt Omega()}, and bounds for $\Omega_c^2$. The bounds for $u_{\rm TF}$, and $\Omega_c^2$ are easily obtained via the general purpose functions {\tt grstimesx()} and {\tt grseval()}. The print out is done in hexadecimal form, so that it can be printed on a file and read rigorously for later use. {\tt tfread()}. This function simply reads the output of {\tt tfprint()}. {\tt r0(w)}. Given an {\tt INTERVL w}, this function produces an interval $\ccint a,b$ containing all solutions $r\le r_c$ to the equation $u(r)=w^2$, for all values $w$ contained in {\tt w}. The bounds are, first, obtained heuristically using bisection (as in {\tt Omega()}), and seen to be correct by checking that an interval containing $u(b)$ is entirely to the right of (i.e. larger than or equal to) {\tt w}$^2$, and an interval containing $u(a)$ is entirely to the left of {\tt w}$^2$. {\tt r1(w)}. Same as before, except that the solutions we are looking for are $u(r)=w^2$ for $r\ge r_c$. Both functions produce optimal intervals, in the sense described in {\tt Omega()}. This function only returns a true bound when {\tt w}$\ge10^{-20}$. This is perfectly fine, since it is only invoked for {\tt w}$\ge \Omega_\epsilon$, and we check that $\Omega_\epsilon>10^{-20}$ (if fact, $\Omega_\epsilon\sim 0.01$). {\tt vtfinf()}. Implements Algorithm~3.13, for representable values of $b$={\tt -a0} and $R$={\tt t}. As usual, Algorithm~3.12 is implemented inside. {\tt tfc1()}. We use our bounds for $w_0$, which {\tt tff()} transforms for bounds for $y_{\rm TF}$, and, for a representable {\tt ctest} we return 1 if we can guarantee that $b_1\le -${\tt ctest}, $-1$ if $b_1\ge -${\tt ctest} and 0 if we cannot guarantee any inequality. {\tt getc1()}. Organizes {\tt tfc1()} to implement Algorithm~3.19. The bounds for $-b_1$ are stored in the external variable {\tt C1}. As in {\tt tfw()}, instead of returning an interval value for our bounds at the end, this function prints the successive rigorous bounds it obtains in hexadecimal form. {\tt refiney()}. This function implements Algorithm~3.18, with the extra obvious feature that instead of obtaining $y_i^\ast$ and ${y'}_i^\ast$ alone, it takes care of comparing them with the old bounds we had, given by function {\tt tff()} and stored in {\tt Y}, and takes the intersection of them. This requires the $x_i$ to be the same as before, which poses no problem, of course. {\tt refine\_numbers()}. Similar to {\tt tfprint()}, except that, once bounds for $b_1$ are computed, it takes care of using them to improve the bounds for $y_{\rm TF}$ before printing them out. {\tt tfw2(w)}. According to Algorithm~3.19, once new bounds for $b_1$ are obtained, and the corresponding bounds for $y_{\rm TF}$ are obtained, one can attempt to improve the bounds for $w_0$. This function takes care of this, by returning 1 if, using the scheme in Algorithm~3.19, we can show that $w_0\ge{\tt w}$, $-1$ if $w_0\le{\tt w}$, and 0 if we cannot claim any inequality. {\tt mygetw()}. This function simply organizes the previous one. {\tt refineY()}. This function implements Algorithm~3.16 again. The difference with {\tt tff()} is only a programming one, since the bounds this function computes are assumed to be refinements of previous ones. Thus, the $x_i$ need not be recomputed. {\tt rstfu2(x,r,y)}. This function implements a trivial variant of Algorithm~3.2, in the type $\infty$ case, except that instead of returning bounds for $y_{\rm TF}$ alone (which are returned in the pointer variable {\tt y}), it returns $r\cdot y_{\rm TF}(r)$ also. As pointed out before, multiplication by $r$, which is implemented as a general purpose routine {\tt rstimesx()}, is not available here, since {\tt vtffxi2()} returns a {\tt RSERIES} without a meaningful {\tt .center} structure member, needed in {\tt rstimesx()}. {\tt yinf(t,m)}. This function implements Algorithm~3.12. The {\tt int} variable {\tt m} represents the order of the expansion we want. {\tt expandY()}. Given the variable {\tt Y} of type {\tt RSERIES} containing bounds for $y_{\rm TF}$ and $y'_{\rm TF}$ at certains points $x_i$, this function returns another {\tt RSERIES} with the same bounds at the same $x_i$, plus the trivial bounds $y_{\rm TF}\in\ccint 0,1$, $y'_{\rm TF}\in \ccint -2,0$ at other points $x_i$, chosen heuristically inside it. This is justified since if at the stage we use this function we already know that $w_0\le 2$, which we do because by the time we use this function we would have already run {\tt mygetw()}. Obviously, this bound can be replaced by any other we know to be true by the time we run this function, and it will probably have no effect on the final answer, since the information {\tt expandY()} produces will most probably never be used until {\tt refineY()} has already improved it to a quite sharp bound. After doing this, it also destroys {\tt Y} by freeing the memory allocated to it. Note that this function has a purely administrative role. \vskip1em The functions below refer to the algorithms presented in Section~4. In the explanation to follow, we use the notation introduced there. {\tt secder0(w,a)}. Computes $I_2$ in Section~4, for the thin interval {\tt w} and $a={\tt a}$. If ${\tt w}\le \bar\Omega_2$, it simply invokes {\tt secder0\_sp()} below. Note that $\bar\Omega_2$ is implicitly defined by the first {\tt if} statement in this function. {\tt secder1(w,a)}. Same as before, but for $I_3$ this time. {\tt dermatrix()}. Computes the numbers $a_{k,i}$ and $b_{k,i}$ involved in the computation of $I_1$ in Section~4. They are stored in the polynomial part of {\tt RSERIES} variables. {\tt secdermat(w,a1,a2,der1,der2,i)}. Uses the numbers $a_{k,i}$ and $b_{k,i}$ (given in {\tt a1}, {\tt a2}, {\tt der1} and {\tt der2} respectively) to compute $\tilde J_{i}({\tt w})$. The choice of the $t_i$ is made using {\tt grsintpt()}. {\tt secderdir()}. This function computes $J_i$ directly, i.e. computes bounds for the functions $f_i$ in Section~4 involved in the computation of $I_1$, without use of the numbers $a_{k,i}$ and $b_{k,i}$. This function is intended to compute these $f_i$ for representable arguments. {\tt super\_secderdir()}. This function does the same as the previous, but for interval values of the argument. In other words, for the thin interval under consideration $\ccint z_1,{z_2}$, this function uses the previous one to compute the $f_i(z_1)$ and $f_i(z_2)$, and then sets $J_i=f_i(z_1)\cup_I f_i(z_2)$. Recall that this is justified due to the monotonicity of the $f_i$. {\tt secder\_help()}. This function uses the previous one to compute bounds for $I_1$. It selects the $i_0$ and $i_1$, computes the $\tilde J_i$ either directly or using the $a_{k,i}$ and $b_{k,i}$, and adds them up together. {\tt alim()}. This function selects heuristically the number $a$ (as a function of $\Omega$) involved in the break up of $I$ into the $I_i$ ($i=1,2,3$) in (4.1). {\tt blim()}. Same as before, but for $b$. {\tt secder2()}. This function organizes the previous ones to produce bounds for $-F''$ in a thin interval $\Omega$. {\tt supersecder2()}. Given an interval $\Omega$, it runs the previous function to check whether $-F''$ is strictly positive. If we can check that it is, it reports a success and returns the bounds. If it is not, it subdivides the interval and tries each half recursively. Note that the fact that this function eventually finishes implies that $-F''$ is strictly positive on the original interval. {\tt supersupersecderdn(w,vup,dup)}. (At this stage, the reader will probably notice our lack of imagination in picking names for all the functions involved in this proof.) Given a fat interval {\tt w}, and numbers $a_{2,i}$ (stored in {\tt vup}), and $b_{2,i}$ (stored in {\tt dup}), corresponding to the upper endpoint of {\tt w}, this function computes the missing $a_{1,i}$ and $b_{1,i}$ corresponding to the lower endpoint of {\tt w}; then, it breaks {\tt w} into thin subintervals using the heuristic variable {\tt step}, and then invokes the previous function to check that $-F''$ is strictly positive in each thin subinterval. Once this is done, we know that $-F''$ is strictly positive all over {\tt w}. Before returning, this function replaces the arguments {\tt vup} and {\tt dup} by the values of the $a_{1,i}$ and $b_{1,i}$ corresponding to the lower end of {\tt w}. The reason for this will be explained in the next function. {\tt secderdn(r,step)}. Given representable {\tt r} and {\tt step}, this function computes the $a_i$ and $b_i$ corresponding to $r$, constructs the fat interval $w_1=\ccint {\tt r}-{\tt step},{\tt r}$, and gives them to the previous function (note that these are exactly the arguments it needs) to do its job. When {\tt supersupersecderdn()} is finished, it returns to us the $a_i$ and $b_i$ corresponding to the lower end of $w_1$; then, we construct the new fat interval $w_2=\ccint {\tt r}-2 \cdot{\tt step},{{\tt r}-{\tt step}}$, and we give it to {\tt supersupersecderdn()} again. Note that the $a_i$ and $b_i$ that we need now are exactly the ones returned to us by {\tt supersupersecderdn()}. And so on. Note that in the construction of the $w_i$ we used expressions of the type ${\tt r}-i\cdot{\tt step}$; there is no need to make these computations rigorous, as long we make sure that the lower endpoint of each interval is exactly the upper endpoint of the next, which is trivial to arrange. We do not include any stopping criterion for this function, rather, we instruct it to print in exact hexadecimal form each fat interval on which we can guarantee that $-F''$ is strictly positive. The reason for this is that it takes a very long time to do each fat interval; thus, we prefer to let several computers run (say six of them) on complementary ranges, and stop them as they redundantly start to get into each other's territory. {\tt supersupersecderup()}. Same as {\tt supersupersecderdn()}, but designed to go up, rather than down. {\tt secderup()}. ``Up'' version of {\tt secderdn()}. {\tt hinf()}. Implements Algorithm~4.7. All sub--algorithms are explicitly implemented inside as needed. ${\tt Le}$ is chosen here. {\tt h\_at\_0()}. Implements Algorithm~4.3. Also, sub--algorithms are implemented inside. ${\tt De}$ is chosen inside also. {\tt printh()}. Runs the two previous functions, and types the output in hexadecimal form for later use. {\tt tfint1(alpha,x)}. This function computes bounds for $$\int_1^{1+x}(t-1)^{-\frac 1,2}t^{\alpha}\,dt$$ for $x<1$. It does it by Taylor--expanding the integrand around 1. {\tt tfint2(alpha,a,b)}. Computes rough bounds for $$\int_a^b(t-1)^{-\frac 1,2}t^{\alpha}\,dt$$ by bounding $t^{\alpha}$ and integrating $(t-1)^{-\frac 1,2}$. {\tt tfint3(alpha,a,b)}. Computes rough bounds for $$\int_a^b(t-1)^{-\frac 1,2}t^{\alpha}\,dt$$ by bounding $(t-1)^{-\frac 1,2}$ and integrating $t^{\alpha}$. {\tt tfint4(alpha,a,b)}. Computes precise bounds for $$\int_a^b(t-1)^{-\frac 1,2}t^{\alpha}\,dt$$ by Taylor--expanding the integrand. This function is to be used when we require precision and are willing to give up speed. The two previous ones are intended for a fast, rather inaccurate answer. {\tt tfintf1(alpha,x)}. This function computes rough bounds for $$\int_1^{1+x}(t-1)^{-\frac 1,2}t^{\alpha}\,dt$$ for all {\tt x}. It does it by using {\tt tfint1()} around 1, and using {\tt tfint2()} (or {\tt tfint3()}) in several other small intervals away from 0. {\tt tfintf2(alpha,x)}. As the previous function, but using {\tt tfint4()} instead for precise, slow bounds. {\tt secder0\_sp(w)}. Computes $I_2$ when $\Omega_\epsilon \le{\tt w}\le\bar\Omega_2$, as explained in Section~4. It checks that ${\tt w}^2\le u(\delta)$: otherwise, it aborts the program. Thus, if the program eventually ends without abortions, we are guaranteed that $\bar\Omega_2^2\le u(\delta)$. {\tt secder1\_sp()}. Computes $I_3$ when $\Omega_\epsilon \le\Omega\le\bar\Omega_3$. The same comments as {\tt secder0\_sp()} apply. {\tt secder0\_speps()}. Computes $T_1(\Omega)$, as in Section~4. {\tt secder1\_speps()}. Computes $T_2(\Omega)$, as in Section~4. As before, it also checks that ${\tt w}^2\le u(L)$: otherwise, it aborts the program. Thus, if the program eventually ends without abortions, we are guaranteed that $\bar\Omega_3^2\le u(L)$. {\tt secder\_eps()}. Computes $T_3$. \vskip1em The next functions are related to Section~5. We also use the same notation used there. {\tt rtpoly()}. See below. {\tt tfwz(t,x)}. Implements Algorithm~5.3. The neighborhood ${\cal U}_0$ is computed using {\tt vtffxi2()}, where $R={\tt x}$. The value {\tt t} is $T$ in the statement of the algorithm, which we check it is less than or equal to $\tilde T$. The power--matching scheme is done in {\tt rtpoly()}. {\tt sdinv(w0,w)}. As in Algorithm~5.4, computes bounds for $-F''$ in the interval ${\tt w0}$, using {\tt w}, the output of {\tt tfwz()}. {\tt super\_sdinv(a,ww)}. Given an interval {\tt a} (which will be all of Zone II), and {\tt ww}, the output of {\tt tfwz()}, this function subdivides {\tt a} recursively until it checks, using the previous function, that $-F''$ is strictly positive in each subinterval of {\tt a}. When this function exits, we know that $-F''$ is strictly positive all over {\tt a}. \vskip2em In order to display how the previous functions can be used to prove Theorem~1.1, we conclude the present discussion with a list of the final programs used in proving our theorem. In doing this, we omit the trivial, but lengthy, statements such as those dealing with variable declarations. The following obtains ---from scratch--- bounds for $w_0$, which are printed in exact hexadecimal form. \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt W.dn = (double) 1; W.up = (double) 2; tfw(W,0.0); } Note that it looks as if in the previous program we are assuming the apparently trivial bounds $\ccint 1,2$ for $w_0$ before we start. In fact, we are not, since these initial bounds are used only to make heuristic choices where to look. The only thing to bear in mind, is that the choices we will be using will be in $\ccint 1,2$. Therefore, once we exit the program, we only have to check that there is at least one of those choices for which we were able to conclude that it bounds $w_0$ from above, and that there is one of those choices that bounds $w_0$ from below. Once we know this, the final bounds obtained will be true bounds for $w_0$. Using the bounds for $w_0$ obtained before, we can now obtain bounds for $y_{\rm TF}$ and use these to obtain bounds for $b_1$. \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt W=readivalio(); printivalio(W); Y=tff(W); Y=expandY(); printgrsio(Y); fflush(stdout); C1.up = (double) 14; C1.dn = (double) 13; getc1();} In the previous program, note that without the statement {\tt expandY()}, the points at which we have bounds for {\tt Y} may not be very many, and when trying to solve the ODE backwards, starting at the largest $x_i$ stored inside {\tt Y}, we may run into trouble, since, as pointed out in Section~3, we need large $x_i$ to be able to solve the ODE around $\infty$. Note also that the bounds stored in {\tt Y} are printed out in exact hexadecimal form, since we will be using them in all programs to follow. Next, we organize the output of the previous program so that it contains, first, the bounds for $w_0$, next, the bounds for $-b_1$, and then the bounds for $y_{\rm TF}$ contained in {\tt Y}, all in exact hexadecimal form. Then, this output can be used as input for the following program, which will use the new bounds for $-b_1$ to obtain improved bounds for $y_{\rm TF}$, stored in {\tt Y}. It will also compute the corresponding {\tt YRS} and {\tt U}. \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt W=readivalio(); C1 = readivalio(); Y=readgrsio(); refine\_numbers();} The next program refines our bounds for $w_0$, $b_1$ and $y_{\rm TF}$, as described in Algorithm~3.19. The output is printed out with same format as usual in exact hexadecimal form. \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt tfread(); mygetw(); refineY(); getc1(); refine\_numbers();} A comment concerning the previous program. Since the bounds that the previous procedures yield are quite sharp, the computer may have to solve ODE's with initial values close to the critical ones that cause the solutions to vanish, but only very slowly. As a result, when trying to check bounds for $w_0$ with {\tt mygetw()}, some choices of {\tt wtest} may yield a failure of some of the ODE--solving algorithms, which will cause the previous program to be aborted. The thing to do in this case is to take whatever bounds were successfully obtained by {\tt mygetw()}, use them to replace the old bounds for $w_0$ written in some file, and restart the previous program without using {\tt mygetw()}. To achieve even greater accuracy, one may also rerun the previous program (after replacing the bounds for $w_0$ with the new ones) with a different choice of the heuristic parameter {\tt t} in {\tt mygetw()}. These comments extend also to $b_1$ and {\tt getc1()}, although those occurrences are very unlikely in this case. Once we are happy with all the bounds for the Thomas--Fermi data, we run the following program, which will write in exact hexadecimal form the neighborhoods for $h(t)$ in Algorithm~4.7, for $h(t)$ in Algorithm~4.3, and bounds for $L$, $u(L)$, $\delta$ and $u(\delta)$. \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt tfread(); printh();} \vskip1em The program to follow will check that $-F''$ is strictly positive for $\Omega_\epsilon\le\Omega\le\Omega_c$. \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt tfread(); readh(); i1 = ratpower(BC,1,2); i1.dn = 0.6956; /*{\it This amounts to setting {\rm Zone II}$=\ccint 0{.}6956,{\Omega_c}$}*/ res = tfwz(0.0605,0.462); super\_sdinv(i1,res); /*{\it At this point we know that $-F''>0$ on \rm Zone II}*/ secderdn(i1.dn,1.e-4); } This program can be complemented with programs of the type \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt tfread(); readh(); secderup(x,t); } or \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt tfread(); readh(); secderdn(x,t); } for {\tt double} values of {\tt x} and {\tt t}, which can run on separate computers. Note that these three last programs will tell us in exact hexadecimal form which fat intervals $W$ are guaranteed to satisfy $-F''>0$ on $W$, but will never stop trying to get $W$ closer and closer to 0. The thing to do is, as long as we see that we have checked all intervals inside $\ccint \sim 10^{-2}, {\Omega_c}$, halt the program, set $\Omega_\epsilon$ equal to the lower end of the last interval checked, and run the following last program: \ {\medtype\tt\obeylines \parindent=1truecm \parskip=0pt tfread(); readh(); i1 = readivalio(); b1 = minb(uabs(UL),uabs(UD)); b1 = ldivb(b1,btwo); printbd(b1); if(lsb(b1,ucvtib(square(i1))))$\{$ printf("error"); abort(); $\}$ i3 = secder1\_speps(i1); i4 = secder0\_speps(i1); i2 = secder\_eps(); i2 = plus(i2,neg(plus(i3,i4))); i3 = poweri(i1,plus(imone,ALPHA)); i3 = mult(i3,divi(HINF.p.p[1],itwo)); i3 = mult(i3, poweri(divi(cvtbi(HINF.center),cvtinti(12)),ALPHA)); i2 = plus(i2,i3); if(lseqb(lcvtib(i2),b0ero))$\{$ printf("error"); abort(); $\}$ else printf("PROVED"); } This program checks that $T_1(\Omega)+T_2(\Omega)+T_3>0$ for all $\Omega\in{\tt i1}$, after checking that $\Omega_\epsilon={\tt i1.up}$ satisfies (4.11c) and (4.22). As a result, any $\Omega_\epsilon\in{\tt i1}$ would finish the proof. In our case, {\tt i1.dn}$=${\tt i1.up}$=$lower end of the last thin interval for which we successfully run {\tt supersupersecderdn()}. \bibliography \medtype \BBArnoldBB \BBEKWBB \BBEckmannWittwerBB \BBFeffermanLlaveBB \BBFeffSecaBB \BBFeffSecbBB \BBFeffSeccBB \BBFeffSecdBB \BBFeffSeceBB \BBFeffSecfBB \BBFeffSecgBB \BBKaucherMirankerBB \BBHKSWBB \BBHilleBB \BBLanfordLlaveBB \BBLlaveBB \BBLohnerBB \BBMooreBB \BBRanaBB \BBthesisBB \BBSecoBB \BBSiedentopWeikardLBBB % @(#)tgrindmac.tex 1.4 (LBL) 3/30/85 % Macros for TeX "tgrind" (a TeX equivalent of 4bsd "vgrind"). % % Copyright (C) 1985 by Van Jacobson, Lawrence Berkeley Laboratory. % This program may be freely used and copied but may not be sold % without the author's written permission. This notice must remain % in any copy or derivative. % % Please send improvements, bug fixes, comments, etc., to % van@lbl-rtsg.arpa % ...ucbvax!lbl-csam!van % % Modifications. % -------------- % 10Feb85, vj Written. % 23Mar85, rf Substitute ambx10 for amb10 % 29Mar85, Chris Torek: Use tt font for all characters in strings. % Print decent quotes in comments rather than use tt % font quotes. Show filename (to terminal & log file) % at start of each new file. % 30Mar85, vj Fixed bug in tabbing. \font\sevenrm=cmr7 % font for right margin line numbers \font\twelvebf=cmbx10 scaled \magstep1 % font for page headers \font\forteenrm=cmr10 scaled \magstep2 % font for right margin proc names % tfontedpr outputs a "\Head{Hdr text}" if you give it the "-h" flag. % We remember the text in "\Header" so it can be included in the % head line. \def\Head#1{\def\Header{#1}} \def\Header{\null} % We get a "\File{Filename},{Last Mod Time},{Last Mod Date}" at the start of % each new file. We remember this stuff for inclusion in the page head & foot. % We reset the page number & current line number to 0 and output a null % mark to let the output routine know we're starting a new file. % We set up the \headline & \footline token lists inside the File macro to % save remembering the filename & mod time with yet other macros. \def\File#1,#2,#3{\vfill\eject\mark{\empty} \global\linecount=0\linenext=9\pageno=1\message{#1} \headline={\twelvebf\Header\hfil \edef\a{\topmark}\edef\b{\botmark}\edef\c{\firstmark} \ifx\c\empty\botmark\else \ifx\a\empty\botmark\else \ifx\b\empty\topmark\else \ifx\a\b\topmark\else\topmark--\botmark\fi \fi\fi\fi(#1)} \footline={\it{}#2 #3\hfil{}Page \folio{} of #1}} % There's a "\Proc{Proc Name}" at the start of each procedure. If % the language definition allows nested procedures (e.g., pascal), there % will be a "\ProcCont{Proc Name}" at the end of each inner procedure. % (In this case, "proc name" is the name of the outer procedure. I.e., % ProcCont marks the continuation of "proc name"). \def\Proc#1{\global\def\Procname{#1}\global\setbox\procbox=\hbox{\forteenrm #1}} \def\ProcCont#1{\global\def\Procname{#1} \global\setbox\procbox=\hbox{\forteenrm$\ldots$#1}} \newbox\procbox \def\Procname{\null} % Each formfeed in the input is replaced by a "\NewPage" macro. If % you really want a page break here, define this as "\vfill\eject". \def\NewPage{\filbreak\bigskip} % Each line of the program text is enclosed by a "\L{...}". We turn % each line into an hbox of size hsize. If we saw a procedure name somewhere % in the line (i.e., "procbox" is not null), we right justify "procbox" % on the line. Every 10 lines we output a small, right justified line number. \def\L#1{\filbreak\hbox to \hsize{\CF\strut\global\advance\linecount by1 #1\hss\ifvoid\procbox\linebox\else\box\procbox\mark{\Procname}\fi}} \newcount\linecount \linecount=0 \newcount\linenext \linenext=9 \def\linebox{\ifnum\linecount>\linenext\global\advance\linenext by10 \hbox{\sevenrm\the\linecount}\fi} % The following weirdness is to deal with tabs. "Pieces" of a line % between tabs are output as "\LB{...}". E.g., a line with a tab at % column 16 would be output as "\LB{xxx}\Tab{16}\LB{yyy}". (Actually, to % reduce the number of characters in the .tex file the \Tab macro % supplies the 2nd & subsequent \LB's.) We accumulate the LB stuff in an % hbox. When we see a Tab, we grab this hbox (using "\lastbox") and turn % it into a box that extends to the tab position. We stash this box in % "\linesofar" & use "\everyhbox" to get \linesofar concatenated onto the % front of the next piece of the line. (There must be a better way of % doing tabs [cf., the Plain.tex tab macros] but I'm not not enough of a % TeX wizard to come up with it. Suggestions would be appreciated.) \def\LB{\CF\hbox} \newbox\linesofar\setbox\linesofar=\null \everyhbox={\box\linesofar} \newdimen\TBwid \def\Tab#1{\setbox\tbox=\lastbox\TBwid=1\wd\tbox\advance\TBwid by 1\ts \ifdim\TBwid>#1\ts \setbox\linesofar=\hbox{\box\tbox\space}\else \setbox\linesofar=\hbox to #1\ts{\box\tbox\hfil}\fi\LB} % A normal space is too thin for code listings. We make spaces & tabs % be in "\ts" units (which are the width of a "0" in the current font). \newdimen\ts \newbox\tbox \setbox\tbox=\hbox{0} \ts=1\wd\tbox \setbox\tbox=\hbox{\hskip 1\ts} \def\space{\hskip 1\ts\relax} % Font changing stuff for keywords, comments & strings. We put keywords % in boldface, comments in text-italic & strings in typewriter. Since % we're usually changing the font inside of a \LB macro, we remember the % current font in \CF & stick a \CF at the start of each new box. % Also, the characters " and ' behave differently in comments than in % code, and others behave differently in strings than in code. \newif\ifcomment\newif\ifstring \let\CF=\rm \def\K#1{{\bf #1}} % Keyword \def\C{\it\global\let\CF=\it\global\commenttrue\relax} % Comment Start \def\CE{\rm\global\let\CF=\rm\global\commentfalse\relax}% Comment End \def\S{\tt\global\let\CF=\tt\global\stringtrue\relax} % String Start \def\SE{\rm\global\let\CF=\rm\global\stringfalse\relax} % String End % Special characters. \def\{{\ifmmode\lbrace\else\ifstring{\char'173}\else$\lbrace$\fi\fi} \def\}{\ifmmode\rbrace\else\ifstring{\char'175}\else$\rbrace$\fi\fi} \def\!{\ifmmode\backslash\else\ifstring{\char'134}\else$\backslash$\fi\fi} \def\|{\ifmmode|\else\ifstring{\char'174}\else$|$\fi\fi} \def\<{\ifmmode<\else\ifstring<\else$<$\fi\fi} \def\>{\ifmmode>\else\ifstring>\else$>$\fi\fi} \def\/{\ifmmode/\else\ifstring/\else$/$\fi\fi} \def\-{\ifmmode-\else\ifstring-\else$-$\fi\fi} \def\_{\ifstring{\char'137}\else\underbar{\ }\fi} \def\&{{\char'046}} \def\#{{\char'043}} \def\%{{\char'045}} \def\~{{\char'176}} \def\"{\ifcomment''\else{\tt\char'042}\fi} \def\'{\ifcomment'\else{\tt\char'047}\fi} \def\^{{\char'136}} \def\${{\rm\char'044}} \raggedright\obeyspaces\let =\space% \File{},{18:34},{Apr 17 1993} \L{\LB{\K{\#include} \}} \L{\LB{\K{\#include} \}} \L{\LB{\K{\#include} \S{}\"extern.inc\"\SE{}}} \L{\LB{INTERVL r0(), r1();}} \L{\LB{RSERIES yinf(), y\_at\_0(), vtffxi();}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder0}secder0(w,a)}} \L{\LB{INTERVL w, a;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL yw0, yw1, x, ith;}} \L{\LB{}\Tab{8}{INTERVL secder0\_sp(), coo, coo12, sol;}} \L{\LB{}\Tab{8}{BND b;}} \L{\LB{}\Tab{8}{RSERIES y, u, y2;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}\Tab{8}{coo = U.f[1].p.p[0];}} \L{\LB{}\Tab{8}{\K{if}(lsb(ucvtib(square(w)),lcvtib(coo)))\K{return}(secder0\_sp(w));}} \L{\LB{}} \L{\LB{}\Tab{8}{ith = divi(cvtinti(\-3),cvtinti(2));}} \L{\LB{}\Tab{8}{x = r0(w);}} \L{\LB{}\Tab{8}{yw0 = grseval(YRS,x);}} \L{\LB{}\Tab{8}{yw1 = grsdereval(YRS,x);}} \L{\LB{}\Tab{8}{y = vtffxi( x, a, yw0, yw1);}} \L{\LB{}} \L{\LB{}\Tab{8}{u = rscopy(y);}} \L{\LB{}\Tab{8}{\K{for}(i = y.p.deg; i\>=1; \-\-i)}} \L{\LB{}\Tab{16}{u.p.p[i] = plus(mult(y.p.p[i],x),mult(y.p.p[i\-1], cvtbi(y.r)));}} \L{\LB{}\Tab{8}{u.p.p[0] = mult(y.p.p[0] ,x);}} \L{\LB{}\Tab{8}{u.g = umultb(y.g,uplusb(ucvtib(x),y.r));}} \L{\LB{}\Tab{8}{u.h = umultb(y.h,uplusb(ucvtib(x),y.r));}} \L{\LB{}\Tab{8}{u.h = uplusb(u.h,umultb(uabs(y.p.p[y.p.deg]),y.r));}} \L{\LB{}\Tab{8}{u.p.p[1] = intersect(u.p.p[1],mult(grsdereval(U,x),cvtbi(u.r)));}} \L{\LB{}} \L{\LB{}\Tab{8}{y2 = rs(y.p.deg\-1,y.center,y.r);}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= y2.p.deg; ++i) y2.p.p[i] = u.p.p[i+1];}} \L{\LB{}\Tab{8}{y2.g = u.g, y2.h = u.h;}} \L{\LB{}} \L{\LB{}\Tab{8}{y2 = rspowerf(y2,divi(ith,itwo));}} \L{\LB{}\Tab{8}{y2 = rsmultf(y2,y2);}} \L{\LB{}\Tab{8}{y2 = rsmultf(y2,y);}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = izero;}} \L{\LB{}\Tab{8}{coo = divi(iabs(plus(a,neg(x))),cvtdi(y.r.b));}} \L{\LB{}\Tab{8}{\K{if}(coo.up \> 1) coo.up = (\K{double}) 1;}} \L{\LB{}\Tab{8}{coo12 = iexp(divi(ilog(coo),imtwo));}} \L{\LB{}\Tab{8}{\K{if}(coo12.dn \< 1) coo12.dn = (\K{double}) 1;}} \L{\LB{}\Tab{8}{\K{for}(i = y2.p.deg; i \>= 0; i\-\- )\{}} \L{\LB{}\Tab{16}{sol = mult(sol,coo);}} \L{\LB{}\Tab{16}{sol = plus(sol,divi(y2.p.p[i],plus(cvtinti(i),neg(ihalf))));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = mult(sol, coo12);}} \L{\LB{}\Tab{8}{b = uplusb(umultb(y2.g,btwo),udivb(y2.h,}} \L{\LB{}\Tab{8}{ lplusb(cvtintb(y2.p.deg+1),negb(bhalf))));}} \L{\LB{}\Tab{8}{sol = mult(ienlarge(sol,b), cvtdi(y2.r.b));}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{freep(u.p), freep(y2.p);}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder1}secder1(w,a)}} \L{\LB{INTERVL w, a;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL yw0, yw1, x, ith;}} \L{\LB{}\Tab{8}{INTERVL secder1\_sp(), coo, coo12, sol;}} \L{\LB{}\Tab{8}{BND b;}} \L{\LB{}\Tab{8}{RSERIES y, u, y2;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}\Tab{8}{coo = U.f[U.n\-1].p.p[0];}} \L{\LB{}\Tab{8}{\K{if}(lsb(ucvtib(square(w)),lcvtib(coo)))\K{return}(secder1\_sp(w));}} \L{\LB{}} \L{\LB{}\Tab{8}{ith = divi(cvtinti(\-3),cvtinti(2));}} \L{\LB{}\Tab{8}{x = r1(w);}} \L{\LB{}\Tab{8}{yw0 = grseval(YRS,x);}} \L{\LB{}\Tab{8}{yw1 = grsdereval(YRS,x);}} \L{\LB{}\Tab{8}{y = vtffxi( x, a, yw0, yw1);}} \L{\LB{}} \L{\LB{}\Tab{8}{u = rscopy(y);}} \L{\LB{}\Tab{8}{\K{for}(i = y.p.deg; i\>=1; \-\-i)}} \L{\LB{}\Tab{16}{u.p.p[i] = plus(mult(y.p.p[i],x),mult(y.p.p[i\-1], cvtbi(y.r)));}} \L{\LB{}\Tab{8}{u.p.p[0] = mult(y.p.p[0] ,x);}} \L{\LB{}\Tab{8}{u.g = umultb(y.g,uplusb(ucvtib(x),y.r));}} \L{\LB{}\Tab{8}{u.h = umultb(y.h,uplusb(ucvtib(x),y.r));}} \L{\LB{}\Tab{8}{u.h = uplusb(u.h,umultb(uabs(y.p.p[y.p.deg]),y.r));}} \L{\LB{}\Tab{8}{u.p.p[1] = intersect(u.p.p[1],mult(grsdereval(U,x),cvtbi(u.r)));}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{y2 = rs(y.p.deg\-1,y.center,y.r);}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= y2.p.deg; ++i) y2.p.p[i] = neg(u.p.p[i+1]);}} \L{\LB{}\Tab{8}{y2.g = u.g, y2.h = u.h;}} \L{\LB{}} \L{\LB{}\Tab{8}{y2 = rspowerf(y2,divi(ith, itwo));}} \L{\LB{}\Tab{8}{y2 = rsmultf(y2, y2);}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{y2 = rsmultf(y2,y);}} \L{\LB{}\Tab{8}{sol = izero;}} \L{\LB{}\Tab{8}{coo = divi(iabs(plus(a,neg(x))),cvtdi(y.r.b));}} \L{\LB{}\Tab{8}{\K{if}(coo.up \> 1) coo.up = (\K{double}) 1;}} \L{\LB{}\Tab{8}{coo12 = iexp(divi(ilog(coo),imtwo));}} \L{\LB{}\Tab{8}{\K{if}(coo12.dn \< 1) coo12.dn = (\K{double}) 1;}} \L{\LB{}\Tab{8}{\K{for}(i = y2.p.deg; i \>= 0; i\-\- )\{}} \L{\LB{}\Tab{16}{sol = mult(sol,coo);}} \L{\LB{}\Tab{16}{\K{if}(i \% 2 )}} \L{\LB{}\Tab{24}{sol = plus(sol,divi(y2.p.p[i],plus(cvtinti(\-i),ihalf)));}} \L{\LB{}\Tab{16}{\K{else}}} \L{\LB{}\Tab{24}{sol = plus(sol,divi(y2.p.p[i],plus(cvtinti(i),neg(ihalf))));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = mult(sol, coo12);}} \L{\LB{}\Tab{8}{b = uplusb(umultb(y2.g,btwo),udivb(y2.h,}} \L{\LB{}\Tab{8}{ lplusb(cvtintb(y2.p.deg+1),negb(bhalf))));}} \L{\LB{}\Tab{8}{sol = mult(ienlarge(sol,b), cvtdi(y2.r.b));}} \L{\LB{}} \L{\LB{}\Tab{8}{freep(u.p), freep(y2.p);}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{\Proc{dermatrix}dermatrix(w, sec, thi)}} \L{\LB{INTERVL w;}} \L{\LB{RSERIES *sec, *thi;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL mw2, ifh;}} \L{\LB{}\Tab{8}{INTERVL t1, t2;}} \L{\LB{}\Tab{8}{RSERIES rsw1, rsw;}} \L{\LB{}\Tab{8}{\K{int} i, i0, i1;}} \L{\LB{}} \L{\LB{}\Tab{8}{ifh = divi(cvtinti(\-5),cvtinti(2));}} \L{\LB{}\Tab{8}{mw2 = neg(square(w));}} \L{\LB{}} \L{\LB{}\Tab{8}{i0 = grsloc(U,r0(w));}} \L{\LB{}\Tab{8}{i1 = grsloc(U,r1(w));}} \L{\LB{}} \L{\LB{}\Tab{8}{i = i0+5;}} \L{\LB{}\Tab{8}{t1 = cvtdi(grsintpt(U,i));}} \L{\LB{}\Tab{8}{\K{while}(i \<= i1\-5)\{}} \L{\LB{}\Tab{16}{++i;}} \L{\LB{}\Tab{16}{\K{if}(i== 60 *(i\/60 ))}} \L{\LB{}\Tab{24}{printf(\S{}\"\%d points done\!n\"\SE{}, i);}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{16}{rsw = rscopy(U.f[i]);}} \L{\LB{}\Tab{16}{rsw.p.p[0] = plus(rsw.p.p[0],mw2);}} \L{\LB{}\Tab{16}{t2 = cvtdi(grsintpt(U,i));}} \L{\LB{}\Tab{16}{rsw1 = rspower(rsw,divi(ifh,ifour));}} \L{\LB{}\Tab{16}{rsw1 = rsmultf(rsw1,rsw1);}} \L{\LB{}\Tab{16}{rsw1 = rsmultf(rsmult(rsw1,YRS.f[i]),rsw1);}} \L{\LB{}\Tab{16}{thi\-\>p.p[i] = mult(mult(ithree,w),rsdint(rsw1,t2,t1));}} \L{\LB{}\Tab{16}{rsw1 = rsmultf(rsw1,rsw);}} \L{\LB{}\Tab{16}{sec\-\>p.p[i] = rsdintf(rsw1,t2,t1);}} \L{\LB{}\Tab{16}{t1 = t2;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secdermat}secdermat(w, a1, a2, der1, der2, i)}} \L{\LB{INTERVL w;}} \L{\LB{RSERIES a1, a2;}} \L{\LB{RSERIES der1, der2;}} \L{\LB{\K{int} i;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL i1, v1, v2, de1, de2, sol;}} \L{\LB{}} \L{\LB{}\Tab{8}{v1 = a1.p.p[i];}} \L{\LB{}\Tab{8}{v2 = a2.p.p[i];}} \L{\LB{}\Tab{8}{de1 = der1.p.p[i];}} \L{\LB{}\Tab{8}{de2 = der2.p.p[i];}} \L{\LB{}} \L{\LB{}\Tab{8}{i1 = plus(v1,neg(v2));}} \L{\LB{}\Tab{8}{i1 = divi(i1,plus(cvtbi(a1.center),neg(cvtbi(a2.center))));}} \L{\LB{}\Tab{8}{i1 = mult(i1,plus(w,neg(cvtbi(a1.center))));}} \L{\LB{}\Tab{8}{i1 = plus(i1,v1);}} \L{\LB{}\Tab{8}{sol.up = i1.up;}} \L{\LB{}} \L{\LB{}\Tab{8}{i1 = mult(de1,plus(w,neg(cvtbi(a1.center))));}} \L{\LB{}\Tab{8}{i1 = plus(i1,v1);}} \L{\LB{}\Tab{8}{sol.dn = i1.dn;}} \L{\LB{}} \L{\LB{}\Tab{8}{i1 = mult(de2,plus(w,neg(cvtbi(a2.center))));}} \L{\LB{}\Tab{8}{i1 = plus(i1,v2);}} \L{\LB{}\Tab{8}{sol.dn = maxm(sol.dn,i1.dn);}} \L{\LB{}\Tab{8}{\K{if}(sol.up \< sol.dn)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"SECDERMAT: negative interval !!!\!n\"\SE{});}} \L{\LB{}\Tab{16}{printival(sol);}} \L{\LB{}\Tab{16}{printf(\S{}\"w:\"\SE{});}} \L{\LB{}\Tab{16}{printival(w);}} \L{\LB{}\Tab{16}{printf(\S{}\"\!ncenter1:\"\SE{});}} \L{\LB{}\Tab{16}{printbd(a1.center);}} \L{\LB{}\Tab{16}{printf(\S{}\"v1:\"\SE{});}} \L{\LB{}\Tab{16}{printival(v1);}} \L{\LB{}\Tab{16}{printf(\S{}\"d1:\"\SE{});}} \L{\LB{}\Tab{16}{printival(de1);}} \L{\LB{}\Tab{16}{printf(\S{}\"\!ncenter2:\"\SE{});}} \L{\LB{}\Tab{16}{printbd(a2.center);}} \L{\LB{}\Tab{16}{printf(\S{}\"v2:\"\SE{});}} \L{\LB{}\Tab{16}{printival(v2);}} \L{\LB{}\Tab{16}{printf(\S{}\"d2:\"\SE{});}} \L{\LB{}\Tab{16}{printival(de2);}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secderdir}secderdir(mw2, t1, t2, i)}} \L{\LB{INTERVL mw2, t1, t2;}} \L{\LB{\K{int} i;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES rsw;}} \L{\LB{}} \L{\LB{}\Tab{8}{rsw = rscopy(U.f[i]);}} \L{\LB{}\Tab{8}{rsw.p.p[0] = plus(rsw.p.p[0],mw2);}} \L{\LB{}\Tab{8}{rsw = rspowerf(rsw,ration(\-3,4));}} \L{\LB{}\Tab{8}{rsw = rsmultf(rsw,rsw);}} \L{\LB{}\Tab{8}{rsw = rsmultf(rsw,rscopy(YRS.f[i]));}} \L{\LB{}\Tab{8}{\K{return}(rsdintf(rsw,t2,t1));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{super\_secderdir}super\_secderdir(mw2, t1,t2,i)}} \L{\LB{INTERVL t1, t2;}} \L{\LB{INTERVL mw2;}} \L{\LB{\K{int} i;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{return}(iunion(secderdir(cvtdi(mw2.up),t1,t2,i),}} \L{\LB{}\Tab{8}{ secderdir(cvtdi(mw2.dn),t1,t2,i)));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder\_help}secder\_help(w, a, b, v1, v2, de1, de2)}} \L{\LB{RSERIES v1, v2, de1, de2;}} \L{\LB{INTERVL w, a, b;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL mw2;}} \L{\LB{}\Tab{8}{INTERVL sol, t1, t2;}} \L{\LB{}\Tab{8}{\K{int} i, i0, i1;}} \L{\LB{}} \L{\LB{}\Tab{8}{mw2 = neg(square(w));}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = izero;}} \L{\LB{}\Tab{8}{i0 = grsloc(U,a);}} \L{\LB{}\Tab{8}{\K{if}(U.f[i0].center.b \< a.dn)++i0;}} \L{\LB{}\Tab{8}{i1 = grsloc(U,b);}} \L{\LB{}\Tab{8}{\K{if}(U.f[i1].center.b \> b.up)\-\-i1;}} \L{\LB{}} \L{\LB{}\Tab{8}{t1 = a;}} \L{\LB{}\Tab{8}{\K{for}(i= i0; i \<= i1\-1; ++i)\{}} \L{\LB{}\Tab{16}{t2 = cvtdi(grsintpt(U,i));}} \L{\LB{}\Tab{16}{\K{if}(}} \L{\LB{}\Tab{24}{i == i0 \|\| }} \L{\LB{}\Tab{16}{ v1.p.p[i].up == (\K{double}) 0 \|\| }} \L{\LB{}\Tab{16}{ v2.p.p[i].up == (\K{double}) 0}} \L{\LB{}\Tab{16}{ )\{}} \L{\LB{}\Tab{24}{sol = plus(sol, super\_secderdir(mw2,t1, t2,i));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{\K{else} sol = plus(sol,secdermat(w,v1, v2, de1, de2,i));}} \L{\LB{}\Tab{16}{t1 = t2;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{sol = plus(sol, super\_secderdir(mw2,t1, b,i));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{\K{double}}} \L{\LB{\Proc{alim}alim(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{int} j, i;}} \L{\LB{}\Tab{8}{\K{double} rat, diff;}} \L{\LB{}\Tab{8}{RSERIES rsw;}} \L{\LB{}\Tab{8}{INTERVL x, w2;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{rat = 3.0;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.136 \&\& w.up \< .140) rat = 2.0;}} \L{\LB{}\Tab{8}{x = U.f[1].p.p[0];}} \L{\LB{}\Tab{8}{\K{if}(lsb(ucvtib(square(w)),lcvtib(x)))\{}} \L{\LB{}\Tab{16}{\K{if}(De.up == De.dn) \K{return}(De.up);}} \L{\LB{}\Tab{16}{\K{else} printf(\S{}\"De error\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{w2 = square(w);}} \L{\LB{}} \L{\LB{}\Tab{8}{i = 0;}} \L{\LB{}\Tab{8}{\K{while}(U.f[i].p.p[0].dn \< w2.up)\{}} \L{\LB{}\Tab{16}{++i;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{diff = 1.0;}} \L{\LB{}\Tab{8}{\-\-i;}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{while}(diff \> 0.0)\{}} \L{\LB{}\Tab{16}{++i;}} \L{\LB{}\Tab{16}{rsw = U.f[i];}} \L{\LB{}\Tab{16}{diff = (w2.dn \- rsw.p.p[0].dn)\/rat;}} \L{\LB{}\Tab{16}{\K{for}(j=1; j \<= rsw.p.deg; ++j)}} \L{\LB{}\Tab{24}{diff += fabs(rsw.p.p[j].up);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\-\-i;}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(grsintpt(U,i));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{\K{double}}} \L{\LB{\Proc{blim}blim(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{int} j, i;}} \L{\LB{}\Tab{8}{\K{double} rat, diff;}} \L{\LB{}\Tab{8}{RSERIES rsw;}} \L{\LB{}\Tab{8}{INTERVL w2;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{w2 = U.f[U.n\-1].p.p[0];}} \L{\LB{}\Tab{8}{\K{if}(lsb(ucvtib(square(w)),lcvtib(w2)))\{}} \L{\LB{}\Tab{16}{\K{if}(Le.up == Le.dn) \K{return}(Le.up);}} \L{\LB{}\Tab{16}{\K{else} printf(\S{}\"Le error\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(w.up \< .09)\{}} \L{\LB{}\Tab{16}{w2 = r1(w);}} \L{\LB{}\Tab{16}{\K{return}(w2.dn\-16.0);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{w2 = square(w);}} \L{\LB{}} \L{\LB{}\Tab{8}{i = U.n;}} \L{\LB{}\Tab{8}{\K{while}(U.f[i].p.p[0].dn \< w2.up)\-\-i;}} \L{\LB{}} \L{\LB{}\Tab{8}{diff = 1.0;}} \L{\LB{}\Tab{8}{++i;}} \L{\LB{}\Tab{8}{rat = 2.1;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.695) rat = 1.5;}} \L{\LB{}\Tab{8}{\K{if}(w.up \< 0.2388 \&\& w.dn \> .1) rat = 1.8;}} \L{\LB{}\Tab{8}{\K{if}(w.up \< 0.1668 \&\& w.dn \> .1) rat = 1.6;}} \L{\LB{}\Tab{8}{\C{}\/*else if(w.dn \> 0.136 \&\& w.up \< .140) rat = 1.8;*\/\CE{}}} \L{\LB{}\Tab{8}{\K{while}(diff \> 0.0)\{}} \L{\LB{}\Tab{16}{\-\-i;}} \L{\LB{}\Tab{16}{rsw = U.f[i];}} \L{\LB{}\Tab{16}{diff = (w2.dn \- rsw.p.p[0].dn)\/rat;}} \L{\LB{}\Tab{16}{\K{for}(j=1; j \<= rsw.p.deg; ++j)}} \L{\LB{}\Tab{24}{diff += fabs(rsw.p.p[j].up);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(grsintpt(U,i));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder2}secder2(w, v1, v2, de1, de2)}} \L{\LB{RSERIES v1, v2, de1, de2;}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}} \L{\LB{}\Tab{8}{INTERVL sol3, sol1, sol2;}} \L{\LB{}\Tab{8}{\K{double} i1, i2;}} \L{\LB{}} \L{\LB{}\Tab{8}{i1 = alim(w);}} \L{\LB{}\Tab{8}{i2 = blim(w);}} \L{\LB{}} \L{\LB{}\Tab{8}{sol1 = secder1(w,cvtdi(i2));}} \L{\LB{}} \L{\LB{}\Tab{8}{sol2 = secder\_help(w,cvtdi(i1),cvtdi(i2), v1, v2, de1, de2);}} \L{\LB{}\Tab{8}{sol3 = secder0(w,cvtdi(i1));}} \L{\LB{}\Tab{8}{\K{return}(plus(plus(sol1,sol2),sol3));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{supersecder2}supersecder2(w, v1, v2, de1, de2)}} \L{\LB{RSERIES v1, v2, de1, de2;}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL sol, w1, w2;}} \L{\LB{}\Tab{8}{sol = secder2(w, v1, v2, de1, de2);}} \L{\LB{}\Tab{8}{\K{if}(sol.dn \<= (\K{double}) 0)\{}} \L{\LB{}\Tab{16}{w1.up = w.up;}} \L{\LB{}\Tab{16}{w1.dn = 0.5*(w.up+w.dn);}} \L{\LB{}\Tab{16}{w2.up = w1.dn;}} \L{\LB{}\Tab{16}{w2.dn = w.dn;}} \L{\LB{}\Tab{16}{\K{return}(iunion(supersecder2(w1, v1, v2, de1, de2),}} \L{\LB{}\Tab{16}{ supersecder2(w2, v1, v2, de1, de2)));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{supersupersecderdn}supersupersecderdn(w, vup, dup)}} \L{\LB{INTERVL w;}} \L{\LB{RSERIES *vup, *dup;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL w1, sol;}} \L{\LB{}\Tab{8}{RSERIES v2, de2;}} \L{\LB{}\Tab{8}{\K{double} step = 5.e\-5;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.672)step = 8.e\-6;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.688)step = 3.e\-6;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.693)step = 1.5e\-6;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.694)step = 4.e\-7;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.6956)step = 2.e\-7;}} \L{\LB{}} \L{\LB{}\Tab{8}{v2 = rs(U.n,cvtdb(w.dn),b0ero);}} \L{\LB{}\Tab{8}{de2 = rs(U.n,cvtdb(w.dn),b0ero);}} \L{\LB{}\Tab{8}{dermatrix(cvtdi(w.dn),\&v2, \&de2);}} \L{\LB{}\Tab{8}{w1.up = w.up;}} \L{\LB{}\Tab{8}{w1.dn = w1.up \- step;}} \L{\LB{}\Tab{8}{sol = supersecder2(w1, *vup, v2, *dup, de2);}} \L{\LB{}\Tab{8}{w1.up = w1.dn;}} \L{\LB{}\Tab{8}{w1.dn \-= step;}} \L{\LB{}\Tab{8}{\K{while}(w1.dn \> w.dn + step\/5.0)\{}} \L{\LB{}\Tab{16}{sol = iunion(sol,supersecder2(w1, *vup, v2, *dup, de2));}} \L{\LB{}\Tab{16}{w1.up = w1.dn;}} \L{\LB{}\Tab{16}{w1.dn \-= step;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{w1.dn = w.dn;}} \L{\LB{}\Tab{8}{sol = iunion(sol, supersecder2(w1, *vup, v2, *dup, de2));}} \L{\LB{}\Tab{8}{freep(vup\-\>p), freep(dup\-\>p);}} \L{\LB{}\Tab{8}{*vup = v2;}} \L{\LB{}\Tab{8}{*dup = de2;}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{\Proc{secderdn}secderdn(r, step)}} \L{\LB{\K{double} r, step;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES v, de;}} \L{\LB{}\Tab{8}{INTERVL w;}} \L{\LB{}} \L{\LB{}\Tab{8}{v = rs(U.n,cvtdb(r),b0ero);}} \L{\LB{}\Tab{8}{de = rs(U.n,cvtdb(r),b0ero);}} \L{\LB{}\Tab{8}{dermatrix(cvtdi(r),\&v, \&de);}} \L{\LB{}\Tab{8}{w.dn = r;}} \L{\LB{}\Tab{8}{\K{while}(w.dn \> 0)\{}} \L{\LB{}\Tab{16}{w.up = w.dn;}} \L{\LB{}\Tab{16}{w.dn \-= step;}} \L{\LB{}\Tab{16}{printival(supersupersecderdn(w, \&v, \&de));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{supersupersecderup}supersupersecderup(w, vup, dup)}} \L{\LB{INTERVL w;}} \L{\LB{RSERIES *vup, *dup;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL w1, sol;}} \L{\LB{}\Tab{8}{RSERIES v2, de2;}} \L{\LB{}\Tab{8}{\K{double} step = 0.00005;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.672)step = 8.e\-6;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.688)step = 3.e\-6;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.693)step = 1.5e\-6;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.694)step = 4.e\-7;}} \L{\LB{}\Tab{8}{\K{if}(w.dn \> 0.6956)step = 2.e\-7;}} \L{\LB{}} \L{\LB{}\Tab{8}{v2 = rs(U.n,cvtdb(w.up),b0ero);}} \L{\LB{}\Tab{8}{de2 = rs(U.n,cvtdb(w.up),b0ero);}} \L{\LB{}\Tab{8}{dermatrix(cvtdi(w.up),\&v2, \&de2);}} \L{\LB{}\Tab{8}{w1.dn = w.dn;}} \L{\LB{}\Tab{8}{w1.up = w1.dn + step;}} \L{\LB{}\Tab{8}{sol = supersecder2(w1, *vup, v2, *dup, de2);}} \L{\LB{}\Tab{8}{w1.dn = w1.up;}} \L{\LB{}\Tab{8}{w1.up += step;}} \L{\LB{}\Tab{8}{\K{while}(w1.up \< w.up \- step\/5.0)\{}} \L{\LB{}\Tab{16}{sol = iunion(sol,supersecder2(w1, *vup, v2, *dup, de2));}} \L{\LB{}\Tab{16}{w1.dn = w1.up;}} \L{\LB{}\Tab{16}{w1.up += step;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{w1.up = w.up;}} \L{\LB{}\Tab{8}{sol = iunion(sol, supersecder2(w1, *vup, v2, *dup, de2));}} \L{\LB{}\Tab{8}{freep(vup\-\>p), freep(dup\-\>p);}} \L{\LB{}\Tab{8}{*vup = v2;}} \L{\LB{}\Tab{8}{*dup = de2;}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\Proc{secderup}secderup(r, step)}} \L{\LB{\K{double} r, step;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES v, de;}} \L{\LB{}\Tab{8}{INTERVL w;}} \L{\LB{}} \L{\LB{}\Tab{8}{v = rs(U.n,cvtdb(r),b0ero);}} \L{\LB{}\Tab{8}{de = rs(U.n,cvtdb(r),b0ero);}} \L{\LB{}\Tab{8}{dermatrix(cvtdi(r),\&v, \&de);}} \L{\LB{}\Tab{8}{w.up = r;}} \L{\LB{}\Tab{8}{\K{while}(w.up \> 0)\{}} \L{\LB{}\Tab{16}{w.dn = w.up;}} \L{\LB{}\Tab{16}{w.up += step;}} \L{\LB{}\Tab{16}{printival(supersupersecderup(w, \&v, \&de));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{POLY}} \L{\LB{\Proc{rtpoly}rtpoly(u)}} \L{\LB{POLY u;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{POLY res, res2;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}\Tab{8}{res = make\_poly(u.deg\-2);}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= res.deg; ++i) res.p[i] = neg(u.p[i+2]);}} \L{\LB{}\Tab{8}{res = polypowerf(res,ihalf);}} \L{\LB{}\Tab{8}{res2 = make\_poly(res.deg+1);}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= res.deg; ++i) res2.p[i+1] = res.p[i];}} \L{\LB{}\Tab{8}{freep(res);}} \L{\LB{}\Tab{8}{\K{return}(polyinv(res2));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{RSERIES}} \L{\LB{\Proc{tfwz}tfwz(t, x)}} \L{\LB{\K{double} t, x;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL i2, a0l, a4, i1, rmr, a0, a1, a2, a3;}} \L{\LB{}\Tab{8}{BND m, tt, h;}} \L{\LB{}\Tab{8}{RSERIES u, y, rstfu2(), r, rp;}} \L{\LB{}\Tab{8}{\K{int} oldeg;}} \L{\LB{}} \L{\LB{}\Tab{8}{\C{}\/**** t = 0.0605, and x = 0.462 will do the job for w=0.6956 ****\/\CE{}}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{oldeg = DEGREE;}} \L{\LB{}\Tab{8}{DEGREE = SIZE \-1;}} \L{\LB{}\Tab{8}{r.k = rp.k = 0;}} \L{\LB{}\Tab{8}{r.h = b0ero;}} \L{\LB{}\Tab{8}{r.g = b0ero;}} \L{\LB{}\Tab{8}{rp.h = b0ero;}} \L{\LB{}\Tab{8}{rp.g = b0ero;}} \L{\LB{}} \L{\LB{}\Tab{8}{u = rstfu2(RC,x, \&y);}} \L{\LB{}\Tab{8}{rmr = plus(RC,cvtbi(negb(u.r)));}} \L{\LB{}\Tab{8}{a0l = ienlarge(y.p.p[0],bl1nrsf(rsplusc(y,neg(y.p.p[0]))));}} \L{\LB{}\Tab{8}{a0 = cvtbi(bl1nrs(y));}} \L{\LB{}\Tab{8}{a2 = divi(ratpower(a0,3,2),ratpower(rmr,1,2));}} \L{\LB{}\Tab{8}{a1 = plus(divi(BC,square(RC)),mult(a2,cvtbi(u.r)));}} \L{\LB{}} \L{\LB{}\Tab{8}{a3 = mult(a1,mult(ration(3,2),ratpower(a0,1,2)));}} \L{\LB{}\Tab{8}{a3 = plus(a3,divi(ratpower(a0,3,2),mult(itwo,rmr)));}} \L{\LB{}\Tab{8}{a3 = divi(a3,ratpower(rmr,1,2));}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{a4 = divi(ratpower(a0,3,2),rmr);}} \L{\LB{}\Tab{8}{a4 = plus(a4, mult(mult(itwo,a1),poweri(a0,ihalf)));}} \L{\LB{}\Tab{8}{a4 = divi(a4,rmr);}} \L{\LB{}\Tab{8}{a4 = plus(a4,divi(square(a1),poweri(a0l,ihalf)));}} \L{\LB{}\Tab{8}{a4 = divi(a4,poweri(rmr,ihalf));}} \L{\LB{}\Tab{8}{a4 = plus(a4, mult(itwo,divi(square(a0),rmr)));}} \L{\LB{}\Tab{8}{a4 = mult(a4, ration(3,4));}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{m = uabs(plus(mult(ifour,a3),mult(a4,plus(RC,cvtbi(u.r)))));}} \L{\LB{}} \L{\LB{}\Tab{8}{r.p = rtpoly(polyscale(u.p,inv(cvtbi(u.r))));}} \L{\LB{}\Tab{8}{r.r = cvtdb(t);}} \L{\LB{}} \L{\LB{}\Tab{8}{tt = labsi(u.p.p[2]);}} \L{\LB{}\Tab{8}{i1 = neg(divi(u.p.p[2],square(cvtbi(u.r))));}} \L{\LB{}\Tab{8}{i2 = u.p.p[2];}} \L{\LB{}\Tab{8}{u.p.p[0] = u.p.p[1] = u.p.p[2] = izero;}} \L{\LB{}\Tab{8}{h = bl1nrs(u);}} \L{\LB{}\Tab{8}{tt = lplusb(tt,negb(h));}} \L{\LB{}\Tab{8}{\K{if}(lseqb(tt,b0ero))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"TFWZ: BAD !!!\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{tt = lcvtib(ratpower(cvtbi(tt),1,2));}} \L{\LB{}} \L{\LB{}\Tab{8}{h = udivb(h,lmultb(u.r,u.r));}} \L{\LB{}\Tab{8}{rp.h = ucvtib(poweri(cvtbi(uplusb(uabs(i1),h)),ihalf));}} \L{\LB{}\Tab{8}{i2 = plus(iabs(mult(i2,itwo)),neg(iabs(mult(ithree,u.p.p[3]))));}} \L{\LB{}\Tab{8}{i2 = divi(i2,square(cvtbi(u.r)));}} \L{\LB{}\Tab{8}{i2 = ienlarge(i2,umultb(udivb(m,cvtintb(6)),upowerb(u.r,2)));}} \L{\LB{}\Tab{8}{rp.h = umultb(btwo,udivb(rp.h,lcvtib(i2)));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(lseqb(rp.h,b0ero))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"TFWZ: radius in the expansion for U is too large !!!\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{r.h = udivb(cvtdb(t),tt);}} \L{\LB{}\Tab{8}{\K{if}(lseqb(bone,r.h))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"TFWZ: too large t !!!\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{r.h = udivb(upowerb(r.h,r.p.deg),lplusb(bone,negb(r.h)));}} \L{\LB{}} \L{\LB{}\Tab{8}{rp.h = umultb(rp.h,r.h);}} \L{\LB{}\Tab{8}{r.h = udivb(rp.h,cvtintb(r.p.deg+1));}} \L{\LB{}\Tab{8}{r.h = umultb(r.h, cvtdb(t));}} \L{\LB{}} \L{\LB{}\Tab{8}{rp.p = polyder(r.p);}} \L{\LB{}\Tab{8}{rp.p = polyscalef(rp.p,cvtbi(r.r));}} \L{\LB{}\Tab{8}{r.p = polyscalef(r.p,cvtbi(r.r));}} \L{\LB{}\Tab{8}{r.p.p[0] = RC;}} \L{\LB{}\Tab{8}{rp.r = r.r;}} \L{\LB{}\Tab{8}{r.center = b0ero;}} \L{\LB{}\Tab{8}{rp.center = b0ero;}} \L{\LB{}} \L{\LB{}\Tab{8}{r = rsmultf(rp,rspowerf(r,imone));}} \L{\LB{}\Tab{8}{DEGREE = oldeg;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(r);}\Tab{24}{}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL }} \L{\LB{\Proc{sdinv}sdinv(w0, w)}} \L{\LB{INTERVL w0;}} \L{\LB{RSERIES w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}\Tab{8}{INTERVL a[30], sol2, sol, g;}} \L{\LB{}\Tab{8}{BND b;}} \L{\LB{}\Tab{8}{\K{double} t;}} \L{\LB{}\Tab{8}{\K{int} m;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{t = w.r.b;}} \L{\LB{}\Tab{8}{a[0] = ione;}} \L{\LB{}\Tab{8}{a[1] = ihalf;}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= 29; ++i)}} \L{\LB{}\Tab{16}{a[i] = divi(ichoose(cvtinti(2*i),i),power(cvtinti(2),2*i));}} \L{\LB{}} \L{\LB{}\Tab{8}{g = plus(BC,neg(square(w0)));}} \L{\LB{}\Tab{8}{g = divi(g,square(cvtdi(t)));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(grteqb(ucvtib(g),bone))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"SDINV: Omega value tries to escape domain of convergence !!!. \!n\"\SE{});}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(w.p.deg \% 2) m = (w.p.deg\-1)\/2;}} \L{\LB{}\Tab{8}{\K{else} m = (w.p.deg)\/2;}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = mult(w.p.p[2*m],a[m]);}} \L{\LB{}\Tab{8}{\K{for}(i=m\-1; i \>= 0; \-\-i)}} \L{\LB{}\Tab{16}{sol = plus(mult(sol,g),mult(w.p.p[2*i],}} \L{\LB{}\Tab{16}{ a[i]));}} \L{\LB{}} \L{\LB{}\Tab{8}{sol2 = mult(mult(cvtinti(m),w.p.p[2*m]),a[m]);}} \L{\LB{}\Tab{8}{\K{for}(i=m\-1; i \>= 1; \-\-i)}} \L{\LB{}\Tab{16}{sol2 = plus(mult(sol2,g),}} \L{\LB{}\Tab{16}{ mult(mult(cvtinti(i),w.p.p[2*i]),a[i]));}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = plus(sol,mult(sol2,neg(mult(itwo,square(divi(w0,cvtbi(w.r)))))));}} \L{\LB{}} \L{\LB{}\Tab{8}{b = uabs(inv(ilog(g)));}} \L{\LB{}\Tab{8}{\K{if}(grteqb(cvtintb(m+1),b))\{}} \L{\LB{}\Tab{16}{b = umultb(cvtintb(2*(m+1)),ucvtib(divi(BC,square(cvtbi(w.r)))));}} \L{\LB{}\Tab{16}{b = umultb(uplusb(uabs(g),b),}} \L{\LB{}\Tab{16}{ upowerb(uabs(g),m));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{else}\{}} \L{\LB{}\Tab{16}{b = umultb(btwo,ucvtib(divi(BC,square(cvtbi(w.r)))));}} \L{\LB{}\Tab{16}{b = udivb(b,labsi(mult(mult(iexp(ione),g),ilog(g))));}} \L{\LB{}\Tab{16}{b = uplusb(upowerb(uabs(g),m+1),b);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{b = umultb(w.h, umultb(uabs(a[m+1]), b));}} \L{\LB{}\Tab{8}{\K{return}(ienlarge(sol,b));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{\Proc{super\_sdinv}super\_sdinv(a, ww)}} \L{\LB{INTERVL a;}} \L{\LB{RSERIES ww;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL sol;}} \L{\LB{}\Tab{8}{RSERIES w;}} \L{\LB{}} \L{\LB{}\Tab{8}{w = rstrunc(ww,53);}} \L{\LB{}\Tab{8}{sol = sdinv(a, w);}} \L{\LB{}\Tab{8}{\K{if}(sol.dn \> (\K{double}) 0)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"SUCCESS for \"\SE{});}} \L{\LB{}\Tab{16}{printival(a);}} \L{\LB{}\Tab{16}{printivalio(a);}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{else} \{}} \L{\LB{}\Tab{16}{printf(\S{}\"trying...\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{sol.up = a.up;}} \L{\LB{}\Tab{16}{sol.dn = .5*(a.dn+a.up);}} \L{\LB{}\Tab{16}{super\_sdinv(sol,w);}} \L{\LB{}\Tab{16}{a.up = sol.dn;}} \L{\LB{}\Tab{16}{super\_sdinv(a,w);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{RSERIES}} \L{\LB{\Proc{hinf}hinf()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp, uP, upp, r, rp, rpp, h, rm2, rma, *rpow;}} \L{\LB{}\Tab{8}{RSERIES rw, rm1;}} \L{\LB{}\Tab{8}{INTERVL i1;}} \L{\LB{}\Tab{8}{BND err, unr;}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{double} t;}} \L{\LB{}\Tab{8}{\K{int} m;}} \L{\LB{}\Tab{8}{\K{int} i, j, k;}} \L{\LB{}\Tab{8}{\K{unsigned} size;}} \L{\LB{}} \L{\LB{}\Tab{8}{t = U.f[U.n\-60].center.b;}} \L{\LB{}\Tab{8}{m = 10;}} \L{\LB{}} \L{\LB{}\Tab{8}{Le = cvtdi(295.0);}} \L{\LB{}\Tab{8}{UL = grseval(U,Le);}} \L{\LB{}} \L{\LB{}\Tab{8}{y = yinf(t, m);}} \L{\LB{}\Tab{8}{y.center = cvtdb(t);}} \L{\LB{}\Tab{8}{y.r = b0ero;}} \L{\LB{}\Tab{8}{ypp = rspower(y, ration(3,2));}} \L{\LB{}\Tab{8}{yp = rs(y.p.deg, y.center, y.r);}} \L{\LB{}\Tab{8}{uP = rs(y.p.deg, y.center, y.r);}} \L{\LB{}\Tab{8}{upp = rs(y.p.deg, y.center, y.r);}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(i=1; i \<= ypp.p.deg; ++i)\{}} \L{\LB{}\Tab{16}{yp.p.p[i] = mult(divi(ypp.p.p[i],plus(ifour,}} \L{\LB{}\Tab{16}{ mult(cvtinti(i),ALPHA))),ifour);}} \L{\LB{}\Tab{16}{uP.p.p[i] = divi(plus(y.p.p[i],mult(imthree,yp.p.p[i])),}} \L{\LB{}\Tab{16}{ imtwo);}} \L{\LB{}\Tab{16}{upp.p.p[i] = plus(mult(ypp.p.p[i],itwo),neg(yp.p.p[i]));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{yp.g = umultb(ypp.g, udivb(bfour, lplusb(bfour,lcvtib(}} \L{\LB{}\Tab{8}{ mult(itwo,ALPHA)))));}} \L{\LB{}\Tab{8}{yp.h = umultb(ypp.h, udivb(bfour, lplusb(bfour,lmultb(cvtintb(}} \L{\LB{}\Tab{8}{ ypp.p.deg+1),lcvtib(ALPHA)))));}} \L{\LB{}} \L{\LB{}\Tab{8}{uP.g = udivb(uplusb(y.g,umultb(yp.g,bthree)),btwo);}} \L{\LB{}\Tab{8}{uP.h = udivb(uplusb(y.h,umultb(yp.h,bthree)),btwo);}} \L{\LB{}} \L{\LB{}\Tab{8}{upp.g = uplusb(yp.g,umultb(ypp.g,btwo));}} \L{\LB{}\Tab{8}{upp.h = uplusb(yp.h,umultb(ypp.h,btwo));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(grtb(bl1nrs(uP),bhalf))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"HINF: Derivative is not bounded below by 1\/2\!n \"\SE{});}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(grtb(bl1nrs(upp),bone))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"HINF: u is not convex\!n\"\SE{});}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{r = rs(y.p.deg, y.center, y.r);}} \L{\LB{}} \L{\LB{}\Tab{8}{r.p.p[0] = ione;}} \L{\LB{}\Tab{8}{r.p.p[1] = divi(y.p.p[1],itwo);}} \L{\LB{}\Tab{8}{r.p.deg = 1;}} \L{\LB{}\Tab{8}{\K{while} (r.p.deg \< y.p.deg )\{}} \L{\LB{}\Tab{16}{rma = rspower(r,neg(ALPHA));}} \L{\LB{}\Tab{16}{rpow = (RSERIES *)calloc(size=r.p.deg+1,\K{sizeof}(RSERIES));}} \L{\LB{}\Tab{16}{rsmatpower(rma, rpow);}} \L{\LB{}\Tab{16}{rw = rs(r.p.deg+1,r.center,r.r);}} \L{\LB{}\Tab{16}{\K{for}(j=1; j \<= r.p.deg+1; ++j)\{}} \L{\LB{}\Tab{24}{err= b0ero;}} \L{\LB{}\Tab{24}{\K{for}(k=1; k \<= j; ++k)\{}} \L{\LB{}\Tab{32}{\K{if}(k \<= r.p.deg)}} \L{\LB{}\Tab{40}{i1 = rpow[k].p.p[j\-k];}} \L{\LB{}\Tab{32}{\K{else} }} \L{\LB{}\Tab{40}{i1 = ione;}} \L{\LB{}\Tab{32}{rw.p.p[j] = plus(mult(i1,y.p.p[k]),rw.p.p[j]);}} \L{\LB{}\Tab{32}{\K{if}(k \> 1) err = maxb(err, uabs(i1));}} \L{\LB{}\Tab{24}{\}}} \L{\LB{}\Tab{24}{rw.p.p[j] = ienlarge(rw.p.p[j],umultb(err,y.g));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{16}{rw.p.p[0] = ione;}} \L{\LB{}\Tab{16}{r.p.deg++;}} \L{\LB{}\Tab{16}{rm2 = rspower(r, imtwo);}} \L{\LB{}\Tab{16}{\K{for}(j=0; j \<= r.p.deg; ++j)\{}} \L{\LB{}\Tab{24}{r.p.p[r.p.deg] = plus(r.p.p[r.p.deg],}} \L{\LB{}\Tab{24}{ mult(rm2.p.p[j], rw.p.p[r.p.deg\-j]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{r.p.p[r.p.deg] = divi(r.p.p[r.p.deg], itwo);}} \L{\LB{}} \L{\LB{}\Tab{16}{\K{for}(j=0; j \<= r.p.deg\-1; ++j) freep(rpow[j].p);}} \L{\LB{}\Tab{16}{freep(rm2.p), freep(rma.p), free((\K{char} *)rpow), rpow = NULL;}} \L{\LB{}\Tab{16}{freep(rw.p);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{rma = rspower(r,neg(ALPHA));}} \L{\LB{}\Tab{8}{rm2 = rspower(r,imone);}} \L{\LB{}\Tab{8}{rm2 = rsmultf(rm2,rm2);}} \L{\LB{}\Tab{8}{rpow = (RSERIES *)calloc(size=r.p.deg+1,\K{sizeof}(RSERIES));}} \L{\LB{}\Tab{8}{rsmatpower(rma, rpow);}} \L{\LB{}\Tab{8}{rw = rs(r.p.deg,r.center,r.r);}} \L{\LB{}} \L{\LB{}\Tab{8}{rw.p.p[0] = ione;}} \L{\LB{}\Tab{8}{err = b0ero;}} \L{\LB{}\Tab{8}{\K{for}(j=1; j \<= r.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{\K{for}(k=r.p.deg\-j+1; k \<= r.p.deg; ++k)}} \L{\LB{}\Tab{24}{rpow[j].h = uplusb(rpow[j].h,uabs(rpow[j].p.p[k]));}} \L{\LB{}\Tab{16}{err = maxb(err, rpow[j].h);}} \L{\LB{}\Tab{16}{\K{for}(k = r.p.deg; k \>= j; \-\-k)}} \L{\LB{}\Tab{24}{rpow[j].p.p[k] = rpow[j].p.p[k\-j];}} \L{\LB{}\Tab{16}{\K{for}(k = 0; k \< j; ++k)}} \L{\LB{}\Tab{24}{rpow[j].p.p[k] = izero;}} \L{\LB{}\Tab{16}{rw = rsplusf(rw, rsca(rpow[j],y.p.p[j]));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(err,y.g));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(j=2; j \<= rw.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{err = b0ero;}} \L{\LB{}\Tab{16}{\K{for}(k=2; k \<= j; ++k)\{}} \L{\LB{}\Tab{24}{err = maxb(err,uabs(rpow[k].p.p[j]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{rw.p.p[j] = ienlarge(rw.p.p[j],umultb(y.g,err));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{r.p.p[0] = izero;}} \L{\LB{}\Tab{8}{unr = bl1nrs(r);}} \L{\LB{}\Tab{8}{\K{if} (unr.b \>= (\K{double}) 1)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"HINF: error no. 1\!n\"\SE{});}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{unr = lplusb(bone,negb(unr));}} \L{\LB{}\Tab{8}{unr = uabs(poweri(cvtbi(unr),mult(cvtinti(\-r.p.deg\-1),ALPHA)));}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h, umultb(y.g, unr));}} \L{\LB{}\Tab{8}{r.p.p[0] = ione;}} \L{\LB{}} \L{\LB{}\Tab{8}{rw = rsmult(rw,rm2);}} \L{\LB{}\Tab{8}{r.h = rw.h;}} \L{\LB{}\Tab{8}{freep(rw.p);}} \L{\LB{}} \L{\LB{}\Tab{8}{freep(rm2.p), freep(rma.p);}} \L{\LB{}\Tab{8}{\K{for}(j=0; j \<= r.p.deg; ++j)freep(rpow[j].p);}} \L{\LB{}} \L{\LB{}\Tab{8}{rma = rspower(r,neg(ALPHA));}} \L{\LB{}\Tab{8}{rm2 = rsmult(r,r);}} \L{\LB{}\Tab{8}{rm2 = rsmultf(rm2,rscopy(r));}} \L{\LB{}\Tab{8}{rsmatpower(rma, rpow);}} \L{\LB{}\Tab{8}{rw = rs(r.p.deg,r.center,r.r);}} \L{\LB{}\Tab{8}{rpp = rs(r.p.deg,r.center,r.r);}} \L{\LB{}} \L{\LB{}\Tab{8}{rw.p.p[0] = ione;}} \L{\LB{}\Tab{8}{err = b0ero;}} \L{\LB{}\Tab{8}{\K{for}(j=1; j \<= r.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{rpp.h = rpow[j].h;}} \L{\LB{}\Tab{16}{\K{for}(k=r.p.deg\-j+1; k \<= r.p.deg; ++k)\{}} \L{\LB{}\Tab{24}{rpp.h = uplusb(rpp.h,uabs(rpow[j].p.p[k]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{err = maxb(err, rpp.h);}} \L{\LB{}} \L{\LB{}\Tab{16}{\K{for}(k = r.p.deg; k \>= j; \-\-k)}} \L{\LB{}\Tab{24}{rpp.p.p[k] = rpow[j].p.p[k\-j];}} \L{\LB{}\Tab{16}{\K{for}(k = 0; k \< j; ++k)}} \L{\LB{}\Tab{24}{rpp.p.p[k] = izero;}} \L{\LB{}\Tab{16}{rw = rsplusf(rw,}} \L{\LB{}\Tab{16}{ rsca(rpp,uP.p.p[j]));}} \L{\LB{}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(err,uP.g));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(j=2; j \<= rw.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{err = b0ero;}} \L{\LB{}\Tab{16}{\K{for}(k=2; k \<= j; ++k)\{}} \L{\LB{}\Tab{24}{err = maxb(err,uabs(rpow[k].p.p[j\-k]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{rw.p.p[j] = ienlarge(rw.p.p[j],umultb(uP.g,err));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{r.p.p[0] = izero;}} \L{\LB{}\Tab{8}{unr = bl1nrs(r);}} \L{\LB{}\Tab{8}{\K{if} (unr.b \>= (\K{double}) 1)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"HINF: error no. 3\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{unr = lplusb(bone,negb(unr));}} \L{\LB{}} \L{\LB{}\Tab{8}{err = labsi(divi(cvtinti(12),poweri(UL,ihalf)));}} \L{\LB{}\Tab{8}{\K{if}(lsb(lmultb(err,unr),cvtdb(t)))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"HINF: wrong choice of L\!n\"\SE{});}} \L{\LB{}\Tab{16}{printf(\S{}\"err = \"\SE{});}} \L{\LB{}\Tab{16}{printbd(err);}} \L{\LB{}\Tab{16}{printf(\S{}\"unr = \"\SE{});}} \L{\LB{}\Tab{16}{printbd(unr);}} \L{\LB{}\Tab{16}{printf(\S{}\" UL = \"\SE{});}} \L{\LB{}\Tab{16}{printival(UL);}} \L{\LB{}\Tab{16}{printf(\S{}\"M = \%e\"\SE{},t);}} \L{\LB{}\Tab{16}{printrs(r);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{unr = uabs(poweri(cvtbi(unr), mult(cvtinti(\-r.p.deg\-1),ALPHA)));}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(unr,uplusb(uP.h, uP.g)));}} \L{\LB{}\Tab{8}{rw.g = b0ero;}} \L{\LB{}\Tab{8}{r.p.p[0] = ione;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{rp = rsmultf(rspowerf(rw,imone),rm2);}} \L{\LB{}\Tab{8}{rp.p.p[0] = ione;}} \L{\LB{}\Tab{8}{freep(rpp.p), freep(rma.p);}} \L{\LB{}} \L{\LB{}\Tab{8}{rm1 = rspower(r, imone);}} \L{\LB{}\Tab{8}{rm2 = rsmult(rm1,rm1);}} \L{\LB{}\Tab{8}{rm2 = rsmultf(rm2,rm2);}} \L{\LB{}} \L{\LB{}\Tab{8}{rw = rs(r.p.deg, r.center, r.r);}} \L{\LB{}\Tab{8}{rw.p.p[0] = ione;}} \L{\LB{}\Tab{8}{err = b0ero;}} \L{\LB{}\Tab{8}{\K{for}(j=1; j \<= r.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{\K{for}(k=r.p.deg\-j+1; k \<= r.p.deg; ++k)}} \L{\LB{}\Tab{24}{rpow[j].h = uplusb(rpow[j].h,uabs(rpow[j].p.p[k]));}} \L{\LB{}\Tab{16}{err = maxb(err,rpow[j].h);}} \L{\LB{}\Tab{16}{\K{for}(k = r.p.deg; k \>= j; \-\-k)}} \L{\LB{}\Tab{24}{rpow[j].p.p[k] = rpow[j].p.p[k\-j];}} \L{\LB{}\Tab{16}{\K{for}(k = 0; k \< j; ++k)}} \L{\LB{}\Tab{24}{rpow[j].p.p[k] = izero;}} \L{\LB{}\Tab{16}{rw = rsplusf(rw,}} \L{\LB{}\Tab{16}{ rsca(rpow[j],upp.p.p[j]));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(err,upp.g));}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(j=2; j \<= rw.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{err = b0ero;}} \L{\LB{}\Tab{16}{\K{for}(k=2; k \<= j; ++k)\{}} \L{\LB{}\Tab{24}{err = maxb(err,uabs(rpow[k].p.p[j]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{rw.p.p[j] = ienlarge(rw.p.p[j],umultb(upp.g,err));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(unr,uplusb(upp.h, upp.g)));}} \L{\LB{}\Tab{8}{rw.g = b0ero;}} \L{\LB{}} \L{\LB{}\Tab{8}{rpp = rsmultf(rm2,rw);}} \L{\LB{}\Tab{8}{rpp = rsmultf(rsmult(rp,rp),rpp);}} \L{\LB{}\Tab{8}{rpp = rsmultf(rpp, rscopy(rp));}} \L{\LB{}\Tab{8}{rpp.p.p[0] = ione;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{rw = rsmult(rp,rm1);}} \L{\LB{}\Tab{8}{h = rsplusf(rsca(rw,itwo),}} \L{\LB{}\Tab{8}{ rsplusf(rscaf(rsmult(rpp,rm1),imthree),rsmult(rw,rw)));}} \L{\LB{}} \L{\LB{}\Tab{8}{freep(y.p), freep(yp.p), freep(ypp.p);}} \L{\LB{}\Tab{8}{freep(uP.p), freep(upp.p);}} \L{\LB{}\Tab{8}{freep(r.p), freep(rp.p), freep(rpp.p);}} \L{\LB{}\Tab{8}{freep(rm1.p), freep(rw.p);}} \L{\LB{}\Tab{8}{\K{for}(j=0; j \<= r.p.deg; ++j) freep(rpow[j].p);}} \L{\LB{}\Tab{8}{free((\K{char} *)rpow), rpow = NULL;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(h);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{RSERIES}} \L{\LB{\Proc{h\_at\_0}h\_at\_0()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp, uP, upp, r, rp, rpp, h, rma, *rpow;}} \L{\LB{}\Tab{8}{RSERIES rw, rm1;}} \L{\LB{}\Tab{8}{INTERVL sc, i1;}} \L{\LB{}\Tab{8}{BND err, unr;}} \L{\LB{}\Tab{8}{\K{int} i, j, k;}} \L{\LB{}\Tab{8}{\K{unsigned} size;}} \L{\LB{}\Tab{8}{\K{double} t;}} \L{\LB{}\Tab{8}{\K{int} m = 20;}} \L{\LB{}} \L{\LB{}\Tab{8}{De = cvtdi(0.0099);}} \L{\LB{}\Tab{8}{UD = grseval(U,De);}} \L{\LB{}} \L{\LB{}\Tab{8}{i = 0;}} \L{\LB{}\Tab{8}{t = 0.012;}} \L{\LB{}} \L{\LB{}\Tab{8}{y = y\_at\_0(t, m);}} \L{\LB{}\Tab{8}{ypp = rspower(y, ration(3,2));}} \L{\LB{}\Tab{8}{yp = rs(y.p.deg+1, y.center, y.r);}} \L{\LB{}\Tab{8}{uP = rs(y.p.deg, y.center, y.r);}} \L{\LB{}\Tab{8}{yp.p.p[0] = neg(W);}} \L{\LB{}\Tab{8}{ypp.p.p[0] = ione;}} \L{\LB{}} \L{\LB{}\Tab{8}{sc = poweri(cvtdi(t),ihalf);}} \L{\LB{}\Tab{8}{\K{for}(i=1; i \<= yp.p.deg; ++i)}} \L{\LB{}\Tab{16}{yp.p.p[i] = mult(divi(mult(ypp.p.p[i\-1],itwo),cvtinti(i)),sc);}} \L{\LB{}\Tab{8}{yp.g = umultb(ypp.g, btwo);}} \L{\LB{}\Tab{8}{yp.g = umultb(yp.g, ucvtib(sc));}} \L{\LB{}\Tab{8}{yp.g = udivb(yp.g,bthree);}} \L{\LB{}\Tab{8}{yp.h = udivb(ypp.h, ldivb(cvtintb(yp.p.deg+1),btwo));}} \L{\LB{}\Tab{8}{yp.h = umultb(yp.h, ucvtib(sc));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(i=2; i \<= y.p.deg; ++i)}} \L{\LB{}\Tab{16}{uP.p.p[i] = plus(y.p.p[i],mult(yp.p.p[i\-2],cvtdi(t)));}} \L{\LB{}\Tab{8}{uP.g = uplusb(y.g,umultb(yp.g,cvtdb(t)));}} \L{\LB{}\Tab{8}{\K{for}(i=y.p.deg\-1; i \<= yp.p.deg; ++i)}} \L{\LB{}\Tab{16}{uP.h = uplusb(uabs(yp.p.p[i]),uP.h);}} \L{\LB{}\Tab{8}{uP.h = uplusb(y.h,umultb(uplusb(yp.h,uP.h),cvtdb(t)));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(grtb(bl1nrs(uP),bhalf))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"H\_AT\_0: Derivative too small !!!\!n\"\SE{});}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{uP.p.p[0] = ione;}} \L{\LB{}} \L{\LB{}\Tab{8}{upp = rsca(yp,itwo);}} \L{\LB{}\Tab{8}{\K{for}(i=1; i \<= upp.p.deg; ++i)}} \L{\LB{}\Tab{16}{upp.p.p[i] = plus(upp.p.p[i], mult(ypp.p.p[i\-1],sc));}} \L{\LB{}\Tab{8}{upp.g = uplusb(upp.g,umultb(ypp.g,ucvtib(sc)));}} \L{\LB{}\Tab{8}{upp.h = uplusb(upp.h,umultb(ypp.h,ucvtib(sc)));}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{r = rs(y.p.deg, y.center, y.r);}} \L{\LB{}} \L{\LB{}\Tab{8}{r.p.p[0] = ione;}} \L{\LB{}\Tab{8}{r.p.p[2] = neg(y.p.p[2]);}} \L{\LB{}\Tab{8}{r.p.deg = 2;}} \L{\LB{}\Tab{8}{\K{while} (r.p.deg \< y.p.deg )\{}} \L{\LB{}\Tab{16}{rma = rspower(r,ihalf);}} \L{\LB{}\Tab{16}{rpow = (RSERIES *)calloc(size=r.p.deg+1,\K{sizeof}(RSERIES));}} \L{\LB{}\Tab{16}{\K{for}(j=0; j \<= r.p.deg; ++j) rpow[j] = rs(r.p.deg,r.center,r.r);}} \L{\LB{}\Tab{16}{rsmatpower(rma, rpow);}} \L{\LB{}\Tab{16}{rw = rs(r.p.deg+1,r.center,r.r);}} \L{\LB{}\Tab{16}{\K{for}(j=1; j \<= r.p.deg+1; ++j)\{}} \L{\LB{}\Tab{24}{err= b0ero;}} \L{\LB{}\Tab{24}{\K{for}(k=1; k \<= j; ++k)\{}} \L{\LB{}} \L{\LB{}\Tab{32}{\K{if}(k \<= r.p.deg)}} \L{\LB{}\Tab{40}{i1 = rpow[k].p.p[j\-k];}} \L{\LB{}\Tab{32}{\K{else} }} \L{\LB{}\Tab{40}{i1 = ione;}} \L{\LB{}} \L{\LB{}\Tab{32}{rw.p.p[j] = plus(mult(i1,y.p.p[k]),rw.p.p[j]);}} \L{\LB{}\Tab{32}{\K{if}(k \> 1) err = maxb(err, uabs(i1));}} \L{\LB{}\Tab{24}{\}}} \L{\LB{}\Tab{24}{rw.p.p[j] = ienlarge(rw.p.p[j],umultb(err,y.g));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{16}{rw.p.p[0] = ione;}} \L{\LB{}\Tab{16}{r.p.deg++;}} \L{\LB{}\Tab{16}{\K{for}(j=0; j \< r.p.deg; ++j)\{}} \L{\LB{}\Tab{24}{r.p.p[r.p.deg] = plus(r.p.p[r.p.deg],}} \L{\LB{}\Tab{24}{ mult(r.p.p[j], rw.p.p[r.p.deg\-j]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{r.p.p[r.p.deg] = neg(r.p.p[r.p.deg]);}} \L{\LB{}} \L{\LB{}\Tab{16}{\K{for}(j=0; j \<= r.p.deg\-1; ++j) freep(rpow[j].p);}} \L{\LB{}\Tab{16}{freep(rma.p), free((\K{char} *)rpow), rpow = NULL;}} \L{\LB{}\Tab{16}{freep(rw.p);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{rma = rspower(r,ihalf);}} \L{\LB{}\Tab{8}{rpow = (RSERIES *)calloc(size=r.p.deg+1,\K{sizeof}(RSERIES));}} \L{\LB{}\Tab{8}{\K{for}(j=0; j \<= r.p.deg; ++j) rpow[j] = rs(r.p.deg,r.center,r.r);}} \L{\LB{}\Tab{8}{rsmatpower(rma, rpow);}} \L{\LB{}\Tab{8}{rw = rs(r.p.deg,r.center,r.r);}} \L{\LB{}} \L{\LB{}\Tab{8}{rw.p.p[0] = ione;}} \L{\LB{}\Tab{8}{err = b0ero;}} \L{\LB{}\Tab{8}{\K{for}(j=1; j \<= r.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{\K{for}(k=r.p.deg\-j+1; k \<= r.p.deg; ++k)}} \L{\LB{}\Tab{24}{rpow[j].h = uplusb(rpow[j].h,uabs(rpow[j].p.p[k]));}} \L{\LB{}\Tab{16}{err = maxb(err,rpow[j].h);}} \L{\LB{}} \L{\LB{}\Tab{16}{\K{for}(k = r.p.deg; k \>= j; \-\-k)}} \L{\LB{}\Tab{24}{rpow[j].p.p[k] = rpow[j].p.p[k\-j];}} \L{\LB{}\Tab{16}{\K{for}(k = 0; k \< j; ++k)}} \L{\LB{}\Tab{24}{rpow[j].p.p[k] = izero;}} \L{\LB{}\Tab{16}{rw = rsplusf(rw,rsca(rpow[j],y.p.p[j]));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(err,y.g));}} \L{\LB{}\Tab{8}{\K{for}(j=2; j \<= rw.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{err = b0ero;}} \L{\LB{}\Tab{16}{\K{for}(k=2; k \<= j; ++k)\{}} \L{\LB{}\Tab{24}{err = maxb(err,uabs(rpow[k].p.p[j]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{rw.p.p[j] = ienlarge(rw.p.p[j],umultb(y.g,err));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{unr = bl1nrs(r);}} \L{\LB{}\Tab{8}{unr = uabs(poweri(cvtbi(unr),divi(cvtinti(r.p.deg+1),itwo)));}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h, umultb(y.g, unr));}} \L{\LB{}} \L{\LB{}\Tab{8}{rw = rsmultf(rw,rscopy(r));}} \L{\LB{}} \L{\LB{}\Tab{8}{r.h = umultb(rw.h,btwo);}} \L{\LB{}\Tab{8}{freep(rw.p);}} \L{\LB{}} \L{\LB{}\Tab{8}{freep(rma.p), free((\K{char} *)rpow), rpow = NULL;}} \L{\LB{}} \L{\LB{}\Tab{8}{rma = rspower(r,ihalf);}} \L{\LB{}\Tab{8}{rpow = (RSERIES *)calloc(size=r.p.deg+1,\K{sizeof}(RSERIES));}} \L{\LB{}\Tab{8}{\K{for}(j=0; j \<= r.p.deg; ++j) rpow[j] = rs(r.p.deg,r.center,r.r);}} \L{\LB{}\Tab{8}{rsmatpower(rma, rpow);}} \L{\LB{}\Tab{8}{rw = rs(r.p.deg,r.center,r.r);}} \L{\LB{}\Tab{8}{rpp = rs(r.p.deg,r.center,r.r);}} \L{\LB{}} \L{\LB{}\Tab{8}{rw.p.p[0] = ione;}} \L{\LB{}\Tab{8}{err = b0ero;}} \L{\LB{}\Tab{8}{\K{for}(j=1; j \<= r.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{rpp.h = rpow[j].h;}} \L{\LB{}\Tab{16}{\K{for}(k=r.p.deg\-j+1; k \<= r.p.deg; ++k)}} \L{\LB{}\Tab{24}{rpp.h = uplusb(rpp.h,uabs(rpow[j].p.p[k]));}} \L{\LB{}\Tab{16}{err = maxb(err,rpp.h);}} \L{\LB{}\Tab{16}{\K{for}(k = r.p.deg; k \>= j; \-\-k)}} \L{\LB{}\Tab{24}{rpp.p.p[k] = rpow[j].p.p[k\-j];}} \L{\LB{}\Tab{16}{\K{for}(k = 0; k \< j; ++k)}} \L{\LB{}\Tab{24}{rpp.p.p[k] = izero;}} \L{\LB{}\Tab{16}{rw = rsplusf(rw,}} \L{\LB{}\Tab{16}{ rsca(rpp,uP.p.p[j]));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{rw.h = uplusb(umultb(err, uP.g), rw.h);}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(j=2; j \<= rw.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{err = b0ero;}} \L{\LB{}\Tab{16}{\K{for}(k=2; k \<= j; ++k)\{}} \L{\LB{}\Tab{24}{err = maxb(err,uabs(rpow[k].p.p[j\-k]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{rw.p.p[j] = ienlarge(rw.p.p[j],umultb(uP.g,err));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{unr = bl1nrs(r);}} \L{\LB{}\Tab{8}{err = umultb(ucvtib(UD),unr);}} \L{\LB{}\Tab{8}{\K{if}(grtb(err,cvtdb(t)))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"H\_at\_0: wrong choice of D\!n\"\SE{});}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{unr = uabs(poweri(cvtbi(unr), divi(cvtinti(r.p.deg+1),itwo)));}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(unr,uplusb(uP.h, uP.g)));}} \L{\LB{}\Tab{8}{rw.g = b0ero;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{rp = rspowerf(rw,imone);}} \L{\LB{}\Tab{8}{rp.p.p[0] = ione;}} \L{\LB{}\Tab{8}{freep(rpp.p), freep(rma.p);}} \L{\LB{}} \L{\LB{}\Tab{8}{rm1 = rspower(r, imone);}} \L{\LB{}} \L{\LB{}\Tab{8}{rw = rs(r.p.deg, r.center, r.r);}} \L{\LB{}\Tab{8}{rw.p.p[0] = upp.p.p[0];}} \L{\LB{}\Tab{8}{err = b0ero;}} \L{\LB{}\Tab{8}{\K{for}(j=1; j \<= r.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{\K{for}(k=r.p.deg\-j+1; k \<= r.p.deg; ++k)}} \L{\LB{}\Tab{24}{rpow[j].h = uplusb(rpow[j].h,uabs(rpow[j].p.p[k]));}} \L{\LB{}\Tab{16}{err = maxb(err,rpow[j].h);}} \L{\LB{}\Tab{16}{\K{for}(k = r.p.deg; k \>= j; \-\-k)}} \L{\LB{}\Tab{24}{rpow[j].p.p[k] = rpow[j].p.p[k\-j];}} \L{\LB{}\Tab{16}{\K{for}(k = 0; k \< j; ++k)}} \L{\LB{}\Tab{24}{rpow[j].p.p[k] = izero;}} \L{\LB{}\Tab{16}{rw = rsplusf(rw,rsca(rpow[j],upp.p.p[j]));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(err,upp.g));}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(j=3; j \<= rw.p.deg; ++j)\{}} \L{\LB{}\Tab{16}{err = b0ero;}} \L{\LB{}\Tab{16}{\K{for}(k=3; k \<= j; ++k)\{}} \L{\LB{}\Tab{24}{err = maxb(err,uabs(rpow[k].p.p[j]));}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{rw.p.p[j] = ienlarge(rw.p.p[j],umultb(upp.g,err));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{rw.h = uplusb(rw.h,umultb(unr,uplusb(upp.h,upp.g)));}} \L{\LB{}\Tab{8}{rw.g = b0ero;}} \L{\LB{}} \L{\LB{}\Tab{8}{rpp = rsmultf(rsmult(rp,rp),rw);}} \L{\LB{}\Tab{8}{rpp = rsmultf(rpp, rsca(rp,imone));}} \L{\LB{}\Tab{8}{rpp.p.p[0] = mult(itwo,W);}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{rw = rsmult(rp,rm1);}} \L{\LB{}\Tab{8}{h = rsminusf(rsmult(rw,rw), rw);}} \L{\LB{}} \L{\LB{}\Tab{8}{h.p.deg = h.p.deg \- 2;}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= h.p.deg; ++i)h.p.p[i] = divi(h.p.p[i+2],cvtdi(t));}} \L{\LB{}\Tab{8}{h.h = udivb(h.h,cvtdb(t));}} \L{\LB{}} \L{\LB{}\Tab{8}{h = rsminusf(rsmult(rpp,rm1), h);}} \L{\LB{}} \L{\LB{}\Tab{8}{freep(y.p), freep(yp.p), freep(ypp.p);}} \L{\LB{}\Tab{8}{freep(uP.p), freep(upp.p);}} \L{\LB{}\Tab{8}{freep(r.p), freep(rp.p), freep(rpp.p);}} \L{\LB{}\Tab{8}{freep(rm1.p);}} \L{\LB{}\Tab{8}{\K{for}(j=0; j \<= r.p.deg; ++j) freep(rpow[j].p);}} \L{\LB{}\Tab{8}{free((\K{char} *)rpow), rpow = NULL;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(rstruncf(h,9));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{\K{void}}} \L{\LB{\Proc{printh}printh()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES h;}} \L{\LB{}\Tab{8}{printrsio(hinf());}} \L{\LB{}\Tab{8}{h = h\_at\_0();}} \L{\LB{}\Tab{8}{printrsio(h);}} \L{\LB{}\Tab{8}{printivalio(Le);}} \L{\LB{}\Tab{8}{printivalio(UL);}} \L{\LB{}\Tab{8}{printivalio(De);}} \L{\LB{}\Tab{8}{printivalio(UD);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\K{void}}} \L{\LB{\Proc{readh}readh()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{HINF = readrsio();}} \L{\LB{}\Tab{8}{H0 = readrsio();}} \L{\LB{}\Tab{8}{Le = readivalio();}} \L{\LB{}\Tab{8}{UL= readivalio();}} \L{\LB{}\Tab{8}{De = readivalio();}} \L{\LB{}\Tab{8}{UD = readivalio();}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{tfint1}tfint1(alpha,x)}} \L{\LB{INTERVL alpha, x;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL sol;}} \L{\LB{}\Tab{8}{\K{int} m, i;}} \L{\LB{}\Tab{8}{BND err;}} \L{\LB{}} \L{\LB{}\Tab{8}{m = 100;}} \L{\LB{}\Tab{8}{\K{if}( x.up \>= one \|\| x.dn \< zero )\{}} \L{\LB{}\Tab{16}{printf(\S{}\"TFINT1: error\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = izero;}} \L{\LB{}\Tab{8}{err = udivb(btwo,cvtintb(2*m+3));}} \L{\LB{}\Tab{8}{\K{for}(i=m; i \>= 1; i\-\-)\{}} \L{\LB{}\Tab{16}{sol = mult(plus(sol,inv(plus(cvtinti(i),ihalf))),}} \L{\LB{}\Tab{16}{ divi(mult(x,plus(alpha,cvtinti(\-i+1))),cvtinti(i)));}} \L{\LB{}\Tab{16}{err = udivb(}} \L{\LB{}\Tab{16}{ umultb(umultb(uabs(x),err),uplusb(uabs(alpha),}} \L{\LB{}\Tab{16}{ cvtintb(i\-1))),}} \L{\LB{}\Tab{16}{ cvtintb(i));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = mult(plus(sol, itwo), poweri(x,ihalf));}} \L{\LB{}\Tab{8}{err = umultb(err,udivb(umultb(uplusb(cvtintb(m),negb(lcvtib(alpha))),}} \L{\LB{}\Tab{8}{ uabs(ratpower(x,3,2))),cvtintb(m+1)));}} \L{\LB{}\Tab{8}{\K{return}(ienlarge(sol, err));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{tfint2}tfint2(alpha, a, b)}} \L{\LB{INTERVL a, b, alpha;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL sol;}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = poweri(iunion(a,b),alpha);}} \L{\LB{}\Tab{8}{sol = mult(sol, mult(itwo,plus(b,neg(a))));}} \L{\LB{}\Tab{8}{sol = divi(sol,}} \L{\LB{}\Tab{8}{ plus(poweri(plus(b,imone),ihalf),poweri(plus(a,imone),ihalf)));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{tfint3}tfint3(alpha, a, b)}} \L{\LB{INTERVL a, b, alpha;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL sol;}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = mult(poweri(plus(iunion(a,b),imone),ihalf),plus(alpha,ione));}} \L{\LB{}\Tab{8}{sol = divi(plus(poweri(b,plus(alpha,ione)),}} \L{\LB{}\Tab{8}{ neg(poweri(a,plus(alpha,ione)))),sol);}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{tfint4}tfint4(alpha, a, b)}} \L{\LB{INTERVL a, alpha, b;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES rw1, rw2;}} \L{\LB{}\Tab{8}{BND x, r;}} \L{\LB{}} \L{\LB{}\Tab{8}{x.b = (a.dn+b.up)\/2.0;}} \L{\LB{}\Tab{8}{r = maxb(uplusb(x,negb(lcvtib(a))),uplusb(negb(x),ucvtib(b)));}} \L{\LB{}\Tab{8}{rw1 = rslinpower(ione,imone,neg(ihalf),8,x,r);}} \L{\LB{}\Tab{8}{rw2 = rslinpower(ione,izero,alpha,8,x,r);}} \L{\LB{}\Tab{8}{\K{return}(rsdintf(rsmultf(rw1,rw2),b,a));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{tfintf1}tfintf1(alpha,x)}} \L{\LB{INTERVL alpha, x;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL sol, a, b;}} \L{\LB{}\Tab{8}{\K{double} step;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(x.up \< 1.8)\K{return}(tfint1(alpha,plus(x,imone)));}} \L{\LB{}} \L{\LB{}\Tab{8}{b.up = b.dn = 1.8;}} \L{\LB{}\Tab{8}{sol = tfint1(alpha, plus(b,imone));}} \L{\LB{}} \L{\LB{}\Tab{8}{step = x.up\/100.0;}} \L{\LB{}\Tab{8}{\K{while}(b.up \< x.dn)\{}} \L{\LB{}\Tab{16}{a = b;}} \L{\LB{}\Tab{16}{b.up = b.dn = b.dn + step;}} \L{\LB{}\Tab{16}{a = tfint2(alpha,a,b);}} \L{\LB{}\Tab{16}{sol = plus(sol,a);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(plus(sol,tfint2(alpha,b,x)));}} \L{\LB{\}}} \L{\LB{INTERVL}} \L{\LB{\Proc{tfintf2}tfintf2(alpha,x)}} \L{\LB{INTERVL alpha, x;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL sol, a, b;}} \L{\LB{}\Tab{8}{\K{double} step;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(x.up \< 1.8)\K{return}(tfint1(alpha,plus(x,imone)));}} \L{\LB{}} \L{\LB{}\Tab{8}{b.up = b.dn = 1.8;}} \L{\LB{}\Tab{8}{sol = tfint1(alpha, plus(b,imone));}} \L{\LB{}} \L{\LB{}\Tab{8}{step = x.up\/100.0;}} \L{\LB{}\Tab{8}{\K{while}(b.up \< x.dn)\{}} \L{\LB{}\Tab{16}{a = b;}} \L{\LB{}\Tab{16}{b.up = b.dn = b.dn + step;}} \L{\LB{}\Tab{16}{a = tfint4(alpha,a,b);}} \L{\LB{}\Tab{16}{sol = plus(sol,a);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(plus(sol,tfint2(alpha,b,x)));}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder0\_sp}secder0\_sp(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL a, sol, uLwm2, m, upL, ta;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}\Tab{8}{m = cvtbi(H0.r);}} \L{\LB{}\Tab{8}{upL = grsdereval(U,De);}} \L{\LB{}\Tab{8}{uLwm2 = divi(UD,square(w));}} \L{\LB{}\Tab{8}{\K{if}(uLwm2.dn \< one)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"SECDER0\_sp: Omega is too large\!n\"\SE{});}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{ta = poweri(m,neg(ihalf));}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = izero;}} \L{\LB{}} \L{\LB{}\Tab{8}{a = divi(cvtinti(H0.p.deg+1),itwo);}} \L{\LB{}\Tab{8}{sol = izero;}} \L{\LB{}\Tab{8}{sol = ienlarge(sol,umultb(uabs(mult(w,ta)),}} \L{\LB{}\Tab{8}{ umultb(H0.h,uabs(tfintf1(a,uLwm2)))));}} \L{\LB{}\Tab{8}{\K{for}(i=H0.p.deg; i \>= 0; \-\-i)\{}} \L{\LB{}\Tab{16}{a = divi(cvtinti(i),itwo);}} \L{\LB{}\Tab{16}{\K{if}(i \> 1) a = tfintf1(a,uLwm2);}} \L{\LB{}\Tab{16}{\K{else} a = tfintf2(a,uLwm2);}} \L{\LB{}} \L{\LB{}\Tab{16}{sol = mult(plus(mult(H0.p.p[i],a),mult(sol,ta)),w);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = plus(divi(mult(imtwo,UD),mult(mult(De,upL),poweri(}} \L{\LB{}\Tab{8}{ plus(UD,neg(square(w))),ihalf))), mult(sol,itwo));}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder1\_sp}secder1\_sp(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL wa, a, sol, a2, uLwm2, m, upL, ta;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}\Tab{8}{m = cvtbi(HINF.center);}} \L{\LB{}\Tab{8}{upL = grsdereval(U,Le);}} \L{\LB{}\Tab{8}{uLwm2 = divi(UL,square(w));}} \L{\LB{}\Tab{8}{\K{if}(uLwm2.dn \< one)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"SECDER1\_sp: Omega is too large\!n\"\SE{});}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{a2 = divi(ALPHA,itwo);}} \L{\LB{}\Tab{8}{wa = poweri(w,ALPHA);}} \L{\LB{}\Tab{8}{ta = mult(wa, poweri(divi(m,cvtinti(12)),ALPHA));}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{a = plus(mult(cvtinti(HINF.p.deg+1),a2),imone);}} \L{\LB{}\Tab{8}{sol = izero;}} \L{\LB{}\Tab{8}{sol = ienlarge(sol, }} \L{\LB{}\Tab{8}{ umultb(uabs(ta),umultb(HINF.h,uabs(tfintf1(a,uLwm2)))));}} \L{\LB{}\Tab{8}{\K{for}(i=HINF.p.deg; i \>= 1; \-\-i)\{}} \L{\LB{}\Tab{16}{a = plus(mult(cvtinti(i),a2),imone);}} \L{\LB{}\Tab{16}{\K{if}( i \> 1) a = tfintf1(a,uLwm2);}} \L{\LB{}\Tab{16}{\K{else} a = tfintf2(a,uLwm2);}} \L{\LB{}\Tab{16}{sol = mult(plus(sol,mult(HINF.p.p[i],a)),ta);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{sol = divi(divi(sol,itwo),w);}} \L{\LB{}\Tab{8}{sol = plus(divi(mult(itwo,UL),mult(mult(Le,upL),poweri(}} \L{\LB{}\Tab{8}{ plus(UL,neg(square(w))),ihalf))), sol);}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder0\_speps}secder0\_speps(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL r, sol;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{r = poweri(divi(UD,cvtbi(H0.r)),ihalf);}} \L{\LB{}\Tab{8}{sol = cvtbi(H0.h);}} \L{\LB{}\Tab{8}{\K{for}(i=H0.p.deg; i \>= 0; \-\-i)\{}} \L{\LB{}\Tab{16}{sol = mult(sol,r);}} \L{\LB{}\Tab{16}{\K{if}(H0.p.p[i].dn \< (\K{double}) 0)}} \L{\LB{}\Tab{24}{sol = plus(sol,iabs(H0.p.p[i]));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{sol = mult(sol,poweri(UD,ihalf));}} \L{\LB{}\Tab{8}{sol = mult(sol,cvtinti(4));}} \L{\LB{}\Tab{8}{sol = plus(divi(mult(itwo,UD),mult(mult(De,grsdereval(U,De)),poweri(}} \L{\LB{}\Tab{8}{ plus(UD,neg(square(w))),ihalf))),sol);}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder1\_speps}secder1\_speps(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL r, sol, i1;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}\Tab{8}{r = mult(poweri(UL,ihalf),divi(cvtbi(HINF.center),cvtinti(12)));}} \L{\LB{}\Tab{8}{r = poweri(r,ALPHA);}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = mult(cvtbi(HINF.h),r);}} \L{\LB{}\Tab{8}{\K{if}(HINF.p.p[2].dn \< (\K{double}) 0)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"SECDER1\_SPEPS: second term negative.\!n\"\SE{});}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{for}(i=HINF.p.deg; i \>= 2; \-\-i)\{}} \L{\LB{}\Tab{16}{\K{if}(HINF.p.p[i].dn \< (\K{double}) 0)}} \L{\LB{}\Tab{24}{sol = plus(sol,iabs(HINF.p.p[i]));}} \L{\LB{}\Tab{16}{sol = mult(sol,r);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{sol = mult(sol,r);}} \L{\LB{}\Tab{8}{sol = divi(sol, poweri(UL,ihalf));}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{i1 = mult(Le,iabs(grsdereval(U,Le)));}} \L{\LB{}\Tab{8}{i1 = mult(i1,poweri(plus(UL,neg(square(w))),ihalf));}} \L{\LB{}\Tab{8}{sol = plus(divi(mult(itwo,UL), i1), sol);}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{secder\_eps}secder\_eps()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{GRS yw;}} \L{\LB{}} \L{\LB{}\Tab{8}{yw = grstimesx(grstimesx(grstimesx(YRS)));}} \L{\LB{}\Tab{8}{yw = grspowerf(yw,neg(ihalf));}} \L{\LB{}\Tab{8}{\K{return}(grsdintf(yw,De,Le));}} \L{\LB{\}}} \L{\LB{}} \vfill\eject \File{},{18:35},{Apr 17 1993} \L{\LB{\K{\#include} \}} \L{\LB{\K{\#include} \}} \L{\LB{}} \L{\LB{BND}} \L{\LB{\Proc{lipreg}lipreg(u0,u1,x0,r,a)}} \L{\LB{INTERVL u0, u1, x0;}} \L{\LB{BND r, a;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{BND g1, sol, g0, c1, c2;}} \L{\LB{}} \L{\LB{}\Tab{8}{g0 = umultb(r,uabs(u1));}} \L{\LB{}\Tab{8}{g1 = udivb(g0,lcvtib(u0));}} \L{\LB{}\Tab{8}{g0 = udivb(uplusb(g0,a),lcvtib(u0));}} \L{\LB{}\Tab{8}{c1 = umultb(frak22(neg(ihalf),udivb(r,lcvtib(x0))),ucvtib(ratpower(x0,}} \L{\LB{}\Tab{8}{ \-1,2)));}} \L{\LB{}\Tab{8}{c2 = umultb(ucvtib(ratpower(u0,1,2)),}} \L{\LB{}\Tab{8}{ frac22(ration(3,2),g0));}} \L{\LB{}} \L{\LB{}\Tab{8}{c1 = udivb(umultb(usquareb(r),c1),btwo);}} \L{\LB{}\Tab{8}{sol = umultb(c1,c2);}} \L{\LB{}\Tab{8}{c2 = umultb(ucvtib(ratpower(u0,3,2)),}} \L{\LB{}\Tab{8}{ frak22(ration(3,2),g1));}} \L{\LB{}} \L{\LB{}\Tab{8}{c1 = umultb(c1, c2);}} \L{\LB{}\Tab{8}{c2 = lmultb(a,lplusb(bone,negb(sol)));}} \L{\LB{}\Tab{8}{\K{if} (lseqb(c1,c2)) \K{return}(sol);}} \L{\LB{}\Tab{8}{\K{else}\{ }} \L{\LB{}\Tab{16}{\K{return}(bone);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{BND}} \L{\LB{\Proc{lip0}lip0(w,r,a)}} \L{\LB{INTERVL w;}} \L{\LB{BND r, a;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{BND g1, sol, g0, c1, c2;}} \L{\LB{}} \L{\LB{}\Tab{8}{g1 = umultb(r,uabs(w));}} \L{\LB{}\Tab{8}{g0 = uplusb(g1,a);}} \L{\LB{}\Tab{8}{sol = frac22(ration(3,2),g0);}} \L{\LB{}\Tab{8}{c2 = frak22(ration(3,2),g1);}} \L{\LB{}\Tab{8}{c1 = udivb(umultb(bfour,ucvtib(ratpower(cvtbi(r),3,2))),cvtintb(3));}} \L{\LB{}} \L{\LB{}\Tab{8}{sol = umultb(sol, c1);}} \L{\LB{}} \L{\LB{}\Tab{8}{c1 = umultb(c1,c2);}} \L{\LB{}\Tab{8}{c2 = lmultb(a,lplusb(bone,negb(sol)));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if} (lseqb(c1,c2)) \K{return}(sol);}} \L{\LB{}\Tab{8}{\K{else}\{ }} \L{\LB{}\Tab{16}{\K{return}(bone);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{\K{double}}} \L{\LB{\Proc{vtffx}vtffx(xin, xout, u0, u1,y0, y1)}} \L{\LB{\K{double} xin, u0, u1;}} \L{\LB{INTERVL xout, *y0, *y1;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp, rsw;}} \L{\LB{}\Tab{8}{INTERVL ithreehalfs, r;}} \L{\LB{}\Tab{8}{BND bn, eps;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}\Tab{8}{\K{double} p[SIZE], pw[SIZE];}} \L{\LB{}} \L{\LB{}\Tab{8}{pzer(p), pzer(pw);}} \L{\LB{}\Tab{8}{p[0] = u0, p[1] = u1*fabs(.5*(xout.dn+xout.up)\-xin);}} \L{\LB{}} \L{\LB{}\Tab{8}{pw[0] = xin; }} \L{\LB{}\Tab{8}{pw[1] = fabs(.5*(xout.up+xout.dn)\-xin);}} \L{\LB{}\Tab{8}{myprpower(pw, \-0.5, pw);}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(i=1; i \<= DEGREE; ++i)\{}} \L{\LB{}\Tab{16}{myprpower(p,1.5,p);}} \L{\LB{}\Tab{16}{pprod(p,pw,p);}} \L{\LB{}\Tab{16}{pinte(p,p);}} \L{\LB{}\Tab{16}{pinte(p,p);}} \L{\LB{}\Tab{16}{psca(p,(.5*(xout.up+xout.dn)\-xin)*(.5*(xout.up+xout.dn)\-xin),p);}} \L{\LB{}\Tab{16}{p[0] = u0, p[1] = u1*fabs(.5*(xout.up+xout.dn)\-xin);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{r = iabs(plus(xout,cvtdi(\-xin)));}} \L{\LB{}} \L{\LB{}\Tab{8}{y = rs(DEGREE,cvtdb(xin),cvtdb(r.up));}} \L{\LB{}\Tab{8}{\K{for}(i=2; i\<= DEGREE; ++i) y.p.p[i] = cvtdi(p[i]);}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = bl1nrs(y);}} \L{\LB{}\Tab{8}{bn = lipreg(cvtdi(u0), cvtdi(u1), cvtdi(xin), ucvtib(r), eps);}} \L{\LB{}\Tab{8}{\K{while}(grteqb(bn,bone))\{}} \L{\LB{}\Tab{16}{eps = maxb(eps, cvtdb(eps.b * 1.1));}} \L{\LB{}\Tab{16}{bn = lipreg(cvtdi(u0), cvtdi(u1), cvtdi(xin), ucvtib(r), eps);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{y.p.p[0] = cvtdi(u0);}} \L{\LB{}\Tab{8}{y.p.p[1] = mult(cvtdi(r.up),cvtdi(u1));}} \L{\LB{}} \L{\LB{}\Tab{8}{rsw = rs(DEGREE,cvtdb(xin),cvtdb(r.up));}} \L{\LB{}\Tab{8}{rsw.p.p[0] = cvtdi(xin);}} \L{\LB{}\Tab{8}{rsw.p.p[1] = cvtdi(r.up);}} \L{\LB{}\Tab{8}{rsw = rspowerf(rsw,neg(ihalf));}} \L{\LB{}} \L{\LB{}\Tab{8}{ithreehalfs = divi(ithree,itwo);}} \L{\LB{}\Tab{8}{yp = rspower(y,ithreehalfs);}} \L{\LB{}\Tab{8}{ypp = rsintegf(rsintegf(rsmultf(yp,rscopy(rsw))));}} \L{\LB{}} \L{\LB{}\Tab{8}{yp = rscopy(y);}} \L{\LB{}\Tab{8}{yp.p.p[0] = izero;}} \L{\LB{}\Tab{8}{yp.p.p[1] = izero;}} \L{\LB{}\Tab{8}{eps = bl1nrsf(rsminusf(yp,ypp));}} \L{\LB{}} \L{\LB{}\Tab{8}{y.g = udivb(eps,lplusb(bone, negb(bn)));}} \L{\LB{}\Tab{8}{y.k = 2;}} \L{\LB{}\Tab{8}{rsw.k = 2;}} \L{\LB{}\Tab{8}{*y0 = rseval(y,xout);}} \L{\LB{}} \L{\LB{}\Tab{8}{*y1 = rsevalf(rsintegf(rsmultf(rspower(y,ithreehalfs),rsw)), xout);}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{*y1 = plus(*y1,cvtdi(u1));}} \L{\LB{}\Tab{8}{freep(y.p);}} \L{\LB{}\Tab{8}{\K{return}(eps.b);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{\K{double}}} \L{\LB{\Proc{supervtffx}supervtffx(xin, xout, u0, u1,y0, y1)}} \L{\LB{\K{double} xin;}} \L{\LB{INTERVL xout, u0, u1;}} \L{\LB{INTERVL *y0, *y1;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL yw0, yw1;}} \L{\LB{}\Tab{8}{\K{double} eps;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(xin \<= xout.dn)\{}} \L{\LB{}\Tab{16}{eps = vtffx(xin, xout, u0.up, u1.up, \&yw0, \&yw1);}} \L{\LB{}\Tab{16}{y0\-\>up = yw0.up, y1\-\>up = yw1.up;}} \L{\LB{}\Tab{16}{eps += vtffx(xin, xout, u0.dn, u1.dn, \&yw0, \&yw1);}} \L{\LB{}\Tab{16}{y0\-\>dn = yw0.dn, y1\-\>dn = yw1.dn;}} \L{\LB{}\Tab{16}{\K{return}(eps);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{else} \K{if}(xin \>= xout.up)\{}} \L{\LB{}\Tab{16}{eps = vtffx(xin, xout, u0.dn, u1.up, \&yw0, \&yw1);}} \L{\LB{}\Tab{16}{y0\-\>dn = yw0.dn, y1\-\>up = yw1.up;}} \L{\LB{}\Tab{16}{eps += vtffx(xin, xout, u0.up, u1.dn, \&yw0, \&yw1);}} \L{\LB{}\Tab{16}{y0\-\>up = yw0.up, y1\-\>dn = yw1.dn;}} \L{\LB{}\Tab{16}{\K{return}(eps);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{else}\{}} \L{\LB{}\Tab{16}{printf(\S{}\"SUPERVTFFX: case not considered\!n\!n\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{RSERIES}} \L{\LB{\Proc{vtffxi}vtffxi(xin, xout, u0, u1)}} \L{\LB{INTERVL xin, xout, u0, u1;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp, rsw;}} \L{\LB{}\Tab{8}{INTERVL ithreehalfs, r;}} \L{\LB{}\Tab{8}{BND bn, eps;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}\Tab{8}{\K{double} p[SIZE], pw[SIZE];}} \L{\LB{}} \L{\LB{}\Tab{8}{pzer(p), pzer(pw);}} \L{\LB{}\Tab{8}{p[0] = .5*(u0.dn+u0.up);}} \L{\LB{}\Tab{8}{p[1] = .5*(u1.up+u1.dn)*fabs(.5*(xout.up+xout.dn)}} \L{\LB{}\Tab{8}{ \-0.5*(xin.up+xin.dn));}} \L{\LB{}} \L{\LB{}\Tab{8}{pw[0] = 0.5*(xin.up+xin.dn);}} \L{\LB{}\Tab{8}{pw[1] = fabs(.5*(xout.up+xout.dn)\-0.5*(xin.up+xin.dn));}} \L{\LB{}\Tab{8}{myprpower(pw, \-0.5, pw);}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(i=1; i \<= DEGREE; ++i)\{}} \L{\LB{}\Tab{16}{myprpower(p,1.5,p);}} \L{\LB{}\Tab{16}{pprod(p,pw,p);}} \L{\LB{}\Tab{16}{pinte(p,p);}} \L{\LB{}\Tab{16}{pinte(p,p);}} \L{\LB{}\Tab{16}{psca(p,(.5*(xout.up+xout.dn)\-0.5*(xin.up+xin.dn))}} \L{\LB{}\Tab{16}{ *(.5*(xout.up+xout.dn)\-0.5*(xin.up+xin.dn)),p);}} \L{\LB{}\Tab{16}{p[0] = .5*(u0.dn+u0.up);}} \L{\LB{}\Tab{16}{p[1] = .5*(u1.up+u1.dn)*fabs(.5*(xout.up+xout.dn)}} \L{\LB{}\Tab{16}{ \-0.5*(xin.up+xin.dn));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{r = iabs(plus(xout,neg(xin)));}} \L{\LB{}} \L{\LB{}\Tab{8}{y = rs(DEGREE,b0ero,cvtdb(r.up));}} \L{\LB{}\Tab{8}{y.p.p[0] = u0;}} \L{\LB{}\Tab{8}{y.p.p[1] = mult(cvtdi(r.up),u1);}} \L{\LB{}\Tab{8}{\K{for}(i=2; i\<= DEGREE; ++i) y.p.p[i] = cvtdi(p[i]);}} \L{\LB{}} \L{\LB{}\Tab{8}{rsw = rs(DEGREE,b0ero,cvtdb(r.up));}} \L{\LB{}\Tab{8}{rsw.p.p[0] = xin;}} \L{\LB{}\Tab{8}{rsw.p.p[1] = cvtdi(r.up);}} \L{\LB{}\Tab{8}{rsw = rspowerf(rsw,neg(ihalf));}} \L{\LB{}} \L{\LB{}\Tab{8}{ithreehalfs = divi(ithree,itwo);}} \L{\LB{}\Tab{8}{yp = rspower(y,ithreehalfs);}} \L{\LB{}\Tab{8}{ypp = rsintegf(rsintegf(rsmultf(yp,rsw)));}} \L{\LB{}} \L{\LB{}\Tab{8}{yp = rscopy(y);}} \L{\LB{}\Tab{8}{yp.p.p[0] = izero;}} \L{\LB{}\Tab{8}{yp.p.p[1] = izero;}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = bl1nrs(yp);}} \L{\LB{}\Tab{8}{bn = lipreg(u0, u1, xin, ucvtib(r), eps);}} \L{\LB{}\Tab{8}{\K{while}(grteqb(bn,bone))\{}} \L{\LB{}\Tab{16}{eps = maxb(eps,cvtdb(eps.b * 1.1));}} \L{\LB{}\Tab{16}{bn = lipreg(u0, u1, xin, ucvtib(r), eps);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = bl1nrsf(rsminusf(yp,ypp));}} \L{\LB{}\Tab{8}{y.g = udivb(eps,lplusb(bone, negb(bn)));}} \L{\LB{}\Tab{8}{y.k = 2;}} \L{\LB{}\Tab{8}{\K{return}(y);}} \L{\LB{\}}} \L{\LB{POLY}} \L{\LB{\Proc{tfypoly}tfypoly(u0, u1, r, i)}} \L{\LB{INTERVL u0, u1, r;}} \L{\LB{\K{int} i;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{POLY poly, res;}} \L{\LB{}\Tab{8}{POLY rsw;}} \L{\LB{}\Tab{8}{\K{int} j, k;}} \L{\LB{}\Tab{8}{poly = make\_poly(i);}} \L{\LB{}\Tab{8}{poly.p[0] = u0, poly.p[1] = u1;}} \L{\LB{}} \L{\LB{}\Tab{8}{rsw = make\_poly(i);}} \L{\LB{}\Tab{8}{rsw.p[0] = r;}} \L{\LB{}\Tab{8}{rsw.p[1] = ione;}} \L{\LB{}\Tab{8}{rsw = polypowerf(rsw,neg(ihalf));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(k=2; k \<= i; ++k)\{}} \L{\LB{}\Tab{16}{res = make\_poly(k\-1);}} \L{\LB{}\Tab{16}{\K{for}(j=0; j\<=k\-2; ++j) res.p[j] = poly.p[j];}} \L{\LB{}\Tab{16}{res = polypowerf(res, ration(3,2));}} \L{\LB{}\Tab{16}{res.p[k\-2] = coeffmult(res,rsw,k\-2);}} \L{\LB{}\Tab{16}{poly.p[k] = divi(res.p[k\-2], cvtinti(k*(k\-1)));}} \L{\LB{}\Tab{16}{freep(res);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{freep(rsw);}} \L{\LB{}\Tab{8}{\K{return}(poly);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{RSERIES}} \L{\LB{\Proc{vtffxi2}vtffxi2(xin, xout, u0, u1)}} \L{\LB{INTERVL xin, xout, u0, u1;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp, rsw;}} \L{\LB{}\Tab{8}{INTERVL ithreehalfs, r;}} \L{\LB{}\Tab{8}{BND bn, eps;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}\Tab{8}{POLY poly;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{r = iabs(plus(xout,neg(xin)));}} \L{\LB{}\Tab{8}{poly = polyscalef(tfypoly(u0, u1, xin, DEGREE), cvtdi(r.up));}} \L{\LB{}} \L{\LB{}\Tab{8}{y = rs(DEGREE,b0ero,cvtdb(r.up));}} \L{\LB{}\Tab{8}{y.p.p[0] = u0;}} \L{\LB{}\Tab{8}{y.p.p[1] = mult(cvtdi(r.up),u1);}} \L{\LB{}\Tab{8}{\K{for}(i=2; i\<= DEGREE; ++i)}} \L{\LB{}\Tab{16}{y.p.p[i] = cvtdi(.5*(poly.p[i].up+poly.p[i].dn));}} \L{\LB{}} \L{\LB{}\Tab{8}{rsw = rs(DEGREE,b0ero,cvtdb(r.up));}} \L{\LB{}\Tab{8}{rsw.p.p[0] = xin;}} \L{\LB{}\Tab{8}{rsw.p.p[1] = cvtdi(r.up);}} \L{\LB{}\Tab{8}{rsw = rspowerf(rsw,imhalf);}} \L{\LB{}} \L{\LB{}\Tab{8}{ithreehalfs = divi(ithree,itwo);}} \L{\LB{}\Tab{8}{yp = rspower(y,ithreehalfs);}} \L{\LB{}\Tab{8}{ypp = rsintegf(rsintegf(rsmultf(yp,rsw)));}} \L{\LB{}} \L{\LB{}\Tab{8}{yp = rscopy(y);}} \L{\LB{}\Tab{8}{yp.p.p[0] = izero;}} \L{\LB{}\Tab{8}{yp.p.p[1] = izero;}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = bl1nrs(yp);}} \L{\LB{}\Tab{8}{bn = lipreg(u0, u1, xin, ucvtib(r), eps);}} \L{\LB{}\Tab{8}{\K{while}(grteqb(bn,bone))\{}} \L{\LB{}\Tab{16}{eps = maxb(eps,cvtdb(eps.b * 1.1));}} \L{\LB{}\Tab{16}{bn = lipreg(u0, u1, xin, ucvtib(r), eps);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = bl1nrsf(rsminusf(yp,ypp));}} \L{\LB{}\Tab{8}{y.h = udivb(eps,lplusb(bone, negb(bn)));}} \L{\LB{}\Tab{8}{\K{for}(i=2; i\<= DEGREE; ++i)}} \L{\LB{}\Tab{16}{y.p.p[i] = iunion(y.p.p[i], poly.p[i]);}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(y);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\K{void}}} \L{\LB{\Proc{vtff0}vtff0(w,t,y0, y1)}} \L{\LB{\K{double} w;}} \L{\LB{BND t;}} \L{\LB{INTERVL *y0, *y1;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp;}} \L{\LB{}\Tab{8}{INTERVL sc;}} \L{\LB{}\Tab{8}{INTERVL ithreehalfs;}} \L{\LB{}\Tab{8}{BND eps, b0;}} \L{\LB{}\Tab{8}{\K{double} p[SIZE];}} \L{\LB{}\Tab{8}{\K{int} i, j;}} \L{\LB{}} \L{\LB{}\Tab{8}{pzer(p);}} \L{\LB{}\Tab{8}{p[0] = 1.0, p[2] = \-w;}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= DEGREE; ++i)\{}} \L{\LB{}\Tab{16}{myprpower(p,1.5,p);}} \L{\LB{}\Tab{16}{\K{for}(j=DEGREE; j \>= 3; \-\-j)}} \L{\LB{}\Tab{24}{p[j] = 4.0*p[j\-3]\/(j*(j\-2));}} \L{\LB{}\Tab{16}{p[0] = 1.0, p[1]=zero, p[2]= \-w;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{pscale(p,sqrt(t.b),p);}} \L{\LB{}\Tab{8}{y = rs(DEGREE,b0ero,t);}} \L{\LB{}\Tab{8}{\K{for}(i=3; i\<= DEGREE; ++i) y.p.p[i] = cvtdi(p[i]);}} \L{\LB{}\Tab{8}{b0 = bl1nrs(y);}} \L{\LB{}\Tab{8}{y.p.p[0] = ione;}} \L{\LB{}\Tab{8}{y.p.p[2] = mult(cvtdi(\-w),cvtbi(t));}} \L{\LB{}} \L{\LB{}\Tab{8}{ithreehalfs = divi(ithree,itwo);}} \L{\LB{}\Tab{8}{yp = rspower(y,ithreehalfs);}} \L{\LB{}\Tab{8}{ypp = rs(DEGREE+3, b0ero, t);}} \L{\LB{}\Tab{8}{\K{for} (i=0; i \<= DEGREE; ++i) ypp.p.p[i+3]= divi(yp.p.p[i],}} \L{\LB{}\Tab{8}{ divi(cvtinti((i+1)*(i+3)),cvtinti(4)));}} \L{\LB{}\Tab{8}{ypp.h = umultb(yp.h,}} \L{\LB{}\Tab{8}{ udivb(cvtintb(4),cvtintb((DEGREE+2)*(DEGREE+4))));}} \L{\LB{}\Tab{8}{sc = iexp(mult(divi(cvtinti(3),cvtinti(2)),ilog(cvtbi(t))));}} \L{\LB{}\Tab{8}{ypp = rscaf(ypp, sc);}} \L{\LB{}\Tab{8}{ypp.p.p[0] = y.p.p[0];}} \L{\LB{}\Tab{8}{ypp.p.p[1] = y.p.p[1];}} \L{\LB{}\Tab{8}{ypp.p.p[2] = y.p.p[2];}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = btwo;}} \L{\LB{}\Tab{8}{\K{while}(grteqb(eps,bone))\{}} \L{\LB{}\Tab{16}{eps = lip0(cvtdi(w),t,b0);}} \L{\LB{}\Tab{16}{b0 = maxb(b0,cvtdb(b0.b * 1.01));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{b0 = eps;}} \L{\LB{}\Tab{8}{eps = bl1nrsf(rsminus(y,ypp));}} \L{\LB{}\Tab{8}{y.g = umultb(eps, lplusb(bone,negb(b0)));}} \L{\LB{}\Tab{8}{*y0 = izero, *y1 = izero;}} \L{\LB{}\Tab{8}{\K{for}(i=0; i\<=y.p.deg; ++i)\{}} \L{\LB{}\Tab{16}{*y0 = plus(*y0,y.p.p[i]);}} \L{\LB{}\Tab{16}{*y1 = plus(*y1,divi(yp.p.p[i],divi(cvtinti(i+1),itwo)));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{*y0 = ienlarge(*y0,uplusb(y.g,y.h));}} \L{\LB{}\Tab{8}{ypp = rspower(y,ithreehalfs);}} \L{\LB{}\Tab{8}{sc = iexp(mult(ihalf,ilog(cvtbi(t))));}} \L{\LB{}} \L{\LB{}\Tab{8}{ypp.g = udivb(ypp.g,btwo);}} \L{\LB{}\Tab{8}{ypp.h = udivb(ypp.h,cvtintb(2*(ypp.p.deg+1)));}} \L{\LB{}\Tab{8}{*y1 = mult(ienlarge(*y1,uplusb(ypp.g,ypp.h)), sc);}} \L{\LB{}\Tab{8}{*y1 = plus(*y1,neg(cvtdi(w)));}} \L{\LB{}\Tab{8}{freep(y.p), freep(yp.p), freep(ypp.p);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{RSERIES}} \L{\LB{\Proc{y\_at\_0}y\_at\_0(tb,m)}} \L{\LB{\K{double} tb;}} \L{\LB{\K{int} m;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp;}} \L{\LB{}\Tab{8}{INTERVL sc;}} \L{\LB{}\Tab{8}{INTERVL ithreehalfs;}} \L{\LB{}\Tab{8}{BND t, eps, b0;}} \L{\LB{}\Tab{8}{\K{double} w, p[SIZE];}} \L{\LB{}\Tab{8}{\K{int} olddeg, i, j;}} \L{\LB{}} \L{\LB{}\Tab{8}{t = cvtdb(tb);}} \L{\LB{}\Tab{8}{sc = iexp(mult(divi(ithree,itwo ),ilog(cvtbi(t))));}} \L{\LB{}\Tab{8}{olddeg = DEGREE;}} \L{\LB{}\Tab{8}{DEGREE = m;}} \L{\LB{}\Tab{8}{\K{if} (m \>= SIZE)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"Y\_AT\_0: DEGREE \%d is not possible\!n\"\SE{},m);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{pzer(p);}} \L{\LB{}\Tab{8}{w = .5*(W.up+W.dn);}} \L{\LB{}\Tab{8}{p[0] = 1.0, p[2] = \-w*tb;}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= DEGREE; ++i)\{}} \L{\LB{}\Tab{16}{myprpower(p,1.5,p);}} \L{\LB{}\Tab{16}{\K{for}(j=DEGREE; j \>= 3; \-\-j)}} \L{\LB{}\Tab{24}{p[j] = 2.0*(sc.up+sc.dn)*p[j\-3]\/(j*(j\-2));}} \L{\LB{}\Tab{16}{p[0] = 1.0, p[1]=zero, p[2]= \-w*tb;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{y = rs(DEGREE,b0ero,t);}} \L{\LB{}\Tab{8}{\K{for}(i=3; i\<= DEGREE; ++i) y.p.p[i] = cvtdi(p[i]);}} \L{\LB{}\Tab{8}{b0 = bl1nrs(y);}} \L{\LB{}\Tab{8}{y.p.p[0] = ione;}} \L{\LB{}\Tab{8}{y.p.p[2] = neg(mult(W,cvtbi(t)));}} \L{\LB{}} \L{\LB{}\Tab{8}{ithreehalfs = divi(ithree,itwo);}} \L{\LB{}\Tab{8}{yp = rspower(y,ithreehalfs);}} \L{\LB{}\Tab{8}{ypp = rs(DEGREE+3, b0ero, t);}} \L{\LB{}\Tab{8}{\K{for} (i=0; i \<= DEGREE; ++i) ypp.p.p[i+3]= divi(yp.p.p[i],}} \L{\LB{}\Tab{8}{ divi(cvtinti((i+1)*(i+3)),ifour));}} \L{\LB{}\Tab{8}{ypp.h = umultb(yp.h,udivb(cvtintb(4),cvtintb((DEGREE+2)*(DEGREE+4))));}} \L{\LB{}\Tab{8}{ypp = rscaf(ypp, sc);}} \L{\LB{}} \L{\LB{}\Tab{8}{yp = rscopy(y);}} \L{\LB{}\Tab{8}{ypp.p.p[0] = y.p.p[0] = izero;}} \L{\LB{}\Tab{8}{ypp.p.p[1] = y.p.p[1] = izero;}} \L{\LB{}\Tab{8}{ypp.p.p[2] = y.p.p[2] = izero;}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = bl1nrsf(rsminusf(y,ypp));}} \L{\LB{}\Tab{8}{\K{while}(grteqb(lip0(W,t,b0),bone)) b0 = maxb(b0, cvtdb(b0.b*1.1));}} \L{\LB{}\Tab{8}{yp.g = udivb(eps, dengeob(lip0(W,t,b0)));}} \L{\LB{}\Tab{8}{DEGREE = olddeg;}} \L{\LB{}\Tab{8}{\K{return}(yp);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{tfw}tfw(w, tol)}} \L{\LB{INTERVL w;}} \L{\LB{\K{double} tol;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{BND b;}} \L{\LB{}\Tab{8}{INTERVL i1, i2, i3, i4, r;}} \L{\LB{}\Tab{8}{\K{double} eps, wtest, x;}} \L{\LB{}} \L{\LB{}\Tab{8}{b.b = 0.008;}} \L{\LB{}\Tab{8}{\K{while}(w.up\-w.dn\>tol)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"\!n\!nBOUNDS FOR w:\!n\"\SE{});}} \L{\LB{}\Tab{16}{printival(w);}} \L{\LB{}\Tab{16}{printivalio(w);}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{16}{wtest = .5*(w.up+w.dn);}} \L{\LB{}\Tab{16}{vtff0(wtest, b, \&i1, \&i2);}} \L{\LB{}\Tab{16}{r.up = b.b;}} \L{\LB{}\Tab{16}{x = 0.0008;}} \L{\LB{}\Tab{16}{\K{while}(i2.dn\<= 0 \&\& r.up \< 250.0)\{}} \L{\LB{}\Tab{24}{r.dn = r.up, r.up = r.dn+x;}} \L{\LB{}\Tab{24}{\K{if}(r.dn \< 2.104025275 }} \L{\LB{}\Tab{24}{ \&\& r.up \> 2.104025275)}} \L{\LB{}\Tab{32}{r.up = 2.104025275;}} \L{\LB{}\Tab{24}{eps = supervtffx(r.dn, cvtdi(r.up), }} \L{\LB{}\Tab{24}{ i1, i2, \&i1, \&i2);}} \L{\LB{}} \L{\LB{}\Tab{24}{i3 = divi(cvtinti(5),cvtinti(2));}} \L{\LB{}\Tab{24}{i3 = iexp(mult(i3,ilog(i1)));}} \L{\LB{}\Tab{24}{i3 = divi(i3,iexp(mult(ihalf,ilog(cvtdi(r.up)))));}} \L{\LB{}\Tab{24}{i4 = square(i2);}} \L{\LB{}\Tab{24}{i3 = mult(itwo,i3);}} \L{\LB{}\Tab{24}{\K{if}(i3.up \<= i4.dn) i2.dn = 1.0;}} \L{\LB{}} \L{\LB{}\Tab{24}{\K{if}(eps}\Tab{32}{\> 5.e\-17) x *= 0.7;}} \L{\LB{}\Tab{24}{\K{if}(eps}\Tab{32}{\> 5.e\-16) x *= 0.5;}} \L{\LB{}\Tab{24}{\K{if}(eps}\Tab{32}{\< 1.e\-17) x *= 1.2;}} \L{\LB{}\Tab{24}{x = minm(x,0.3);}} \L{\LB{}\Tab{24}{\K{if}(r.up \> 10.0) x = minm(x,0.5);}} \L{\LB{}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{\K{if}(r.up \>= 250.0)\{}} \L{\LB{}\Tab{24}{\K{return}(w);}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{\K{else} \K{if}(i3.up \<= i4.dn) w.up = wtest;}} \L{\LB{}\Tab{16}{\K{else} w.dn = wtest;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{return}(w);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{GRS}} \L{\LB{\Proc{tff}tff(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{GRS y;}} \L{\LB{}\Tab{8}{BND b;}} \L{\LB{}\Tab{8}{\K{unsigned} size;}} \L{\LB{}\Tab{8}{INTERVL i1[1000], i2[1000], i3, i4 , i5[1000], i6[1000];}} \L{\LB{}\Tab{8}{\K{double} eps, x, r[1000];}} \L{\LB{}\Tab{8}{\K{int} count = 0;}} \L{\LB{}\Tab{8}{\K{int} count2 = 0;}} \L{\LB{}} \L{\LB{}\Tab{8}{b.b = 0.008;}} \L{\LB{}\Tab{8}{vtff0(w.up, b, \&i1[0], \&i2[0]);}} \L{\LB{}\Tab{8}{vtff0(w.dn, b, \&i5[0], \&i6[0]);}} \L{\LB{}\Tab{8}{r[count] = b.b;}} \L{\LB{}\Tab{8}{x = 0.0008;}} \L{\LB{}\Tab{8}{i3.up = 1.0, i4.dn = zero;}} \L{\LB{}\Tab{8}{\K{while}(i3.up \> i4.dn \|\| i6[count].dn \< 0)\{}} \L{\LB{}\Tab{16}{\K{if}(i3.up \<= i4.dn \|\| i6[count].dn \>= 0) ++count2;}} \L{\LB{}\Tab{16}{r[count+1] = r[count]+x;}} \L{\LB{}\Tab{16}{\K{if}(r[count] \< 2.10402528 \&\& r[count+1] \> 2.10402528)}} \L{\LB{}\Tab{24}{r[count+1]=maxm(r[count],2.10402528);}} \L{\LB{}\Tab{16}{\K{if}(i3.up \> i4.dn )}} \L{\LB{}\Tab{24}{eps = supervtffx(r[count], cvtdi(r[count+1]), }} \L{\LB{}\Tab{24}{ i1[count], i2[count], \&i1[count+1], \&i2[count+1]);}} \L{\LB{}\Tab{16}{\K{if}(i6[count].dn \< 0)}} \L{\LB{}\Tab{24}{eps += supervtffx(r[count], cvtdi(r[count+1]), }} \L{\LB{}\Tab{24}{ i5[count], i6[count], \&i5[count+1], \&i6[count+1]);}} \L{\LB{}} \L{\LB{}\Tab{16}{++count;}} \L{\LB{}\Tab{16}{\K{if}(i3.up \> i4.dn )\{}} \L{\LB{}\Tab{24}{i3 = divi(cvtinti(5),cvtinti(2));}} \L{\LB{}\Tab{24}{i3 = iexp(mult(i3,ilog(i1[count])));}} \L{\LB{}\Tab{24}{i3 = divi(i3,iexp(mult(ihalf,ilog(cvtdi(r[count])))));}} \L{\LB{}\Tab{24}{i4 = square(i2[count]);}} \L{\LB{}\Tab{24}{i3 = mult(itwo,i3);}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}} \L{\LB{}\Tab{16}{\K{if}(eps}\Tab{24}{\> 1.e\-16) x *= 0.7;}} \L{\LB{}\Tab{16}{\K{if}(eps}\Tab{24}{\> 1.e\-15) x *= 0.5;}} \L{\LB{}\Tab{16}{\K{if}(eps}\Tab{24}{\< 2.e\-17) x *= 1.2;}} \L{\LB{}\Tab{16}{x = minm(x,4.0);}} \L{\LB{}\Tab{16}{\K{if}(r[count] \> 200.0) x = minm(x,1.0);}} \L{\LB{}\Tab{16}{\K{if}(r[count] \< 10.0) x = minm(x,0.05);}} \L{\LB{}\Tab{16}{\K{if}(r[count] \> 250.0) x = minm(x,0.3);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{count \-= count2;}} \L{\LB{}\Tab{8}{y.n = count+1;}} \L{\LB{}\Tab{8}{y.f = (RSERIES *)calloc(size=count+2,\K{sizeof}(RSERIES));}} \L{\LB{}\Tab{8}{y.f[0] = rs(1,b0ero,bone);}} \L{\LB{}\Tab{8}{y.f[0].p.p[0] = ione;}} \L{\LB{}\Tab{8}{y.f[0].p.p[1] = w;}} \L{\LB{}\Tab{8}{\K{for}(count=1; count \<= y.n; ++count)\{}} \L{\LB{}\Tab{16}{y.f[count] = rs(1,cvtdb(r[count\-1]),bone);}} \L{\LB{}\Tab{16}{y.f[count].p.p[0].up = i5[count\-1].up;}} \L{\LB{}\Tab{16}{y.f[count].p.p[0].dn = i1[count\-1].dn;}} \L{\LB{}\Tab{16}{y.f[count].p.p[1].up = i6[count\-1].up;}} \L{\LB{}\Tab{16}{y.f[count].p.p[1].dn = i2[count\-1].dn;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{return}(y);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{GRS}} \L{\LB{\Proc{tfrs}tfrs(y)}} \L{\LB{GRS y;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}\Tab{8}{GRS sol;}} \L{\LB{}\Tab{8}{\K{unsigned} size;}} \L{\LB{}\Tab{8}{\C{}\/*char *calloc();*\/\CE{}}} \L{\LB{}\Tab{8}{INTERVL cvtdi(), x1, x2;}} \L{\LB{}} \L{\LB{}\Tab{8}{sol.f = (RSERIES *)calloc(size=y.n\-1,\K{sizeof}(RSERIES));}} \L{\LB{}\Tab{8}{sol.n = y.n\-2;}} \L{\LB{}\Tab{8}{\K{for} (i=0; i \<= sol.n; ++i)\{}} \L{\LB{}\Tab{16}{x1 = y.f[i+1].p.p[0];}} \L{\LB{}\Tab{16}{x2 = y.f[i+1].p.p[1];}} \L{\LB{}\Tab{16}{sol.f[i] = vtffxi(cvtdi(y.f[i+1].center.b),}} \L{\LB{}\Tab{16}{ cvtdi(y.f[i+2].center.b),x1,x2);}} \L{\LB{}\Tab{16}{sol.f[i].center = y.f[i+1].center;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(sol);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{Omega}Omega(u)}} \L{\LB{GRS u;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{int} d;}} \L{\LB{}\Tab{8}{INTERVL ievders(), sol, derw;}} \L{\LB{}\Tab{8}{\K{double} otest, otest1, otest2;}} \L{\LB{}} \L{\LB{}\Tab{8}{sol.dn = (\K{double}) 2;}} \L{\LB{}\Tab{8}{sol.up = (\K{double}) 3;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(;;)\{}} \L{\LB{}\Tab{16}{otest = .5*(sol.up+sol.dn);}} \L{\LB{}\Tab{16}{\K{if}(otest == sol.up \|\| otest == sol.dn)}} \L{\LB{}\Tab{24}{\K{return}(sol);}} \L{\LB{}\Tab{16}{derw = grsdereval(u,cvtdi(otest));}} \L{\LB{}\Tab{16}{\K{if}(derw.up \<= zero) sol.up = otest;}} \L{\LB{}\Tab{16}{\K{else} \K{if}(derw.dn \>= zero) sol.dn = otest;}} \L{\LB{}\Tab{16}{\K{else} }} \L{\LB{}\Tab{16}{\{}} \L{\LB{}\Tab{24}{otest1 = otest;}} \L{\LB{}\Tab{24}{d = 1;}} \L{\LB{}\Tab{24}{\K{while}(d )\{}} \L{\LB{}\Tab{32}{otest2 = .5*(otest+sol.up);}} \L{\LB{}\Tab{32}{\K{if}(otest2 == otest \|\| otest2 == sol.up)d = 0;}} \L{\LB{}\Tab{32}{derw = grsdereval(u,cvtdi(otest2));}} \L{\LB{}\Tab{32}{\K{if}(derw.up \<= zero ) sol.up = otest2;}} \L{\LB{}\Tab{32}{\K{else} otest = otest2; }} \L{\LB{}\Tab{24}{\}}} \L{\LB{}} \L{\LB{}\Tab{24}{otest = otest1;}} \L{\LB{}\Tab{24}{d = 1;}} \L{\LB{}\Tab{24}{\K{while}(d )\{}} \L{\LB{}\Tab{32}{otest2 = .5*(otest+sol.dn);}} \L{\LB{}\Tab{32}{\K{if}(otest2 == otest \|\| otest2 == sol.dn)d = 0;}} \L{\LB{}\Tab{32}{derw = grsdereval(u,cvtdi(otest2));}} \L{\LB{}\Tab{32}{\K{if}(derw.up \>= zero ) sol.dn = otest2;}} \L{\LB{}\Tab{32}{\K{else} otest = otest2; }} \L{\LB{}\Tab{24}{\}}} \L{\LB{}} \L{\LB{}\Tab{24}{\K{return}(sol);}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\K{void} }} \L{\LB{\Proc{tfprint}tfprint()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{GRS grstimesx();}} \L{\LB{}\Tab{8}{INTERVL grseval();}} \L{\LB{}} \L{\LB{}\Tab{8}{printivalio(W);}} \L{\LB{}\Tab{8}{printivalio(C1);}} \L{\LB{}\Tab{8}{Y = tff(W);}} \L{\LB{}\Tab{8}{printgrsio(Y);}} \L{\LB{}\Tab{8}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{8}{YRS = tfrs(Y);}} \L{\LB{}\Tab{8}{printgrsio(YRS);}} \L{\LB{}\Tab{8}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{8}{U = grstimesx(YRS);}} \L{\LB{}\Tab{8}{printgrsio(U);}} \L{\LB{}\Tab{8}{printivalio(RC = Omega(U));}} \L{\LB{}\Tab{8}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{8}{printivalio(BC = grseval(U,RC));}} \L{\LB{}\Tab{8}{fflush(stdout);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\K{void} }} \L{\LB{\Proc{tfread}tfread()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{W = readivalio();}} \L{\LB{}\Tab{8}{C1 = readivalio();}} \L{\LB{}\Tab{8}{Y = readgrsio();}} \L{\LB{}\Tab{8}{YRS = readgrsio();}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= YRS.n; ++i)YRS.f[i].k = 2;}} \L{\LB{}\Tab{8}{U = readgrsio();}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= U.n; ++i)U.f[i].k = 2;}} \L{\LB{}\Tab{8}{RC = readivalio();}} \L{\LB{}\Tab{8}{BC = readivalio();}} \L{\LB{}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{r0}r0(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL r, i1, cvtdi(), grseval(), w2, square();}} \L{\LB{}\Tab{8}{\K{double} rtest, rtest1, rtest2;}} \L{\LB{}\Tab{8}{\K{int} d;}} \L{\LB{}} \L{\LB{}\Tab{8}{w2 = square(w);}} \L{\LB{}\Tab{8}{r = cvtbi(uplusb(U.f[0].center,negb(U.f[0].r)));}} \L{\LB{}\Tab{8}{i1 = rseval(U.f[0],r);}} \L{\LB{}\Tab{8}{\K{if}(i1.up \>= w2.dn)\{}} \L{\LB{}\Tab{16}{r.dn = (\K{double}) 0;}} \L{\LB{}\Tab{16}{d = \-1;}} \L{\LB{}\Tab{16}{\K{while}(i1.dn \< w2.up)\{}} \L{\LB{}\Tab{24}{++d;}} \L{\LB{}\Tab{24}{i1 = U.f[d].p.p[0];}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{d = maxm(d,0);}} \L{\LB{}\Tab{16}{r.up = U.f[d].center.b;}} \L{\LB{}\Tab{16}{\K{return}(r);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{r.up = RC.up;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(;;)\{}} \L{\LB{}\Tab{16}{rtest = .5*(r.dn+r.up);}} \L{\LB{}\Tab{16}{i1 = grseval(U,cvtdi(rtest));}} \L{\LB{}\Tab{16}{\K{if}(i1.dn \>= w2.up ) r.up = rtest;}} \L{\LB{}\Tab{16}{\K{else} \K{if}(i1.up \<= w2.dn ) r.dn = rtest;}} \L{\LB{}\Tab{16}{\K{else} }} \L{\LB{}\Tab{16}{\{}} \L{\LB{}\Tab{24}{rtest1 = rtest;}} \L{\LB{}\Tab{24}{d = 1;}} \L{\LB{}\Tab{24}{\K{while}(d )\{}} \L{\LB{}\Tab{32}{rtest2 = .5*(rtest+r.up);}} \L{\LB{}\Tab{32}{\K{if}(rtest2 == rtest \|\| rtest2 == r.up)d = 0;}} \L{\LB{}\Tab{32}{i1 = grseval(U,cvtdi(rtest2));}} \L{\LB{}\Tab{32}{\K{if}(i1.dn \>= w2.up ) r.up = rtest2;}} \L{\LB{}\Tab{32}{\K{else} rtest = rtest2; }} \L{\LB{}\Tab{24}{\}}} \L{\LB{}} \L{\LB{}\Tab{24}{rtest = rtest1;}} \L{\LB{}\Tab{24}{d = 1;}} \L{\LB{}\Tab{24}{\K{while}(d )\{}} \L{\LB{}\Tab{32}{rtest2 = .5*(rtest+r.dn);}} \L{\LB{}\Tab{32}{\K{if}(rtest2 == rtest \|\| rtest2 == r.dn)d = 0;}} \L{\LB{}\Tab{32}{i1 = grseval(U,cvtdi(rtest2));}} \L{\LB{}\Tab{32}{\K{if}(i1.up \<= w2.dn ) r.dn = rtest2;}} \L{\LB{}\Tab{32}{\K{else} rtest = rtest2; }} \L{\LB{}\Tab{24}{\}}} \L{\LB{}} \L{\LB{}\Tab{24}{\K{if}(r.dn \<= r.up)\K{return}(r);}} \L{\LB{}\Tab{24}{\K{else} abort();}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{INTERVL}} \L{\LB{\Proc{r1}r1(w)}} \L{\LB{INTERVL w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL r, i1, cvtdi(), grseval(), w2, square();}} \L{\LB{}\Tab{8}{\K{double} rtest, rtest1, rtest2;}} \L{\LB{}\Tab{8}{\K{int} d;}} \L{\LB{}} \L{\LB{}\Tab{8}{w2 = square(w);}} \L{\LB{}\Tab{8}{r = cvtbi(lplusb(U.f[U.n].center,U.f[U.n].r));}} \L{\LB{}\Tab{8}{i1 = rseval(U.f[U.n],r);}} \L{\LB{}\Tab{8}{\K{if}(i1.up \>= w2.dn)\{}} \L{\LB{}\Tab{16}{r.up = 1.e15;}} \L{\LB{}\Tab{16}{d = U.n+1;}} \L{\LB{}\Tab{16}{\K{while}(i1.dn \< w2.up)\{}} \L{\LB{}\Tab{24}{\-\-d;}} \L{\LB{}\Tab{24}{i1 = U.f[d].p.p[0];}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{16}{d = minm(d,U.n);}} \L{\LB{}\Tab{16}{r.dn = U.f[d].center.b;}} \L{\LB{}\Tab{16}{\K{return}(r);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{r.dn = RC.dn;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{for}(;;)\{}} \L{\LB{}\Tab{16}{rtest = 0.5*(r.up+r.dn);}} \L{\LB{}\Tab{16}{i1 = grseval(U,cvtdi(rtest));}} \L{\LB{}\Tab{16}{\K{if}(i1.dn \>= w2.up ) r.dn = rtest;}} \L{\LB{}\Tab{16}{\K{else} \K{if}(i1.up \<= w2.dn ) r.up = rtest;}} \L{\LB{}\Tab{16}{\K{else} \{}} \L{\LB{}\Tab{24}{rtest1 = rtest;}} \L{\LB{}\Tab{24}{d = 1;}} \L{\LB{}\Tab{24}{\K{while}( d )\{}} \L{\LB{}\Tab{32}{rtest2 = .5*(rtest+r.up);}} \L{\LB{}\Tab{32}{\K{if}(rtest2 == rtest \|\| rtest2 == r.up) d = 0;}} \L{\LB{}\Tab{32}{i1 = grseval(U,cvtdi(rtest2));}} \L{\LB{}\Tab{32}{\K{if}(i1.up \<= w2.dn ) r.up = rtest2;}} \L{\LB{}\Tab{32}{\K{else} rtest = rtest2; }} \L{\LB{}\Tab{24}{\}}} \L{\LB{}} \L{\LB{}\Tab{24}{rtest = rtest1;}} \L{\LB{}\Tab{24}{d = 1;}} \L{\LB{}\Tab{24}{\K{while}(d )\{}} \L{\LB{}\Tab{32}{rtest2 = .5*(rtest+r.dn);}} \L{\LB{}\Tab{32}{\K{if}(rtest2 == rtest \|\| rtest2 == r.dn)d = 0;}} \L{\LB{}\Tab{32}{i1 = grseval(U,cvtdi(rtest2));}} \L{\LB{}\Tab{32}{\K{if}(i1.dn \>= w2.up ) r.dn = rtest2;}} \L{\LB{}\Tab{32}{\K{else} rtest = rtest2; }} \L{\LB{}\Tab{24}{\}}} \L{\LB{}} \L{\LB{}\Tab{24}{\K{if}(r.dn \<= r.up)\K{return}(r);}} \L{\LB{}\Tab{24}{\K{else} abort();}} \L{\LB{}\Tab{16}{\}}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{\Proc{vtfinf}vtfinf(a0, t, y0, y1)}} \L{\LB{\K{double} t, a0;}} \L{\LB{INTERVL *y0, *y1;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp;}} \L{\LB{}\Tab{8}{INTERVL i1, ithreehalfs;}} \L{\LB{}\Tab{8}{BND eps, b0;}} \L{\LB{}\Tab{8}{\K{double} pnorm(), pw[SIZE], p[SIZE], alpha;}} \L{\LB{}\Tab{8}{\K{int} i, j, olddeg;}} \L{\LB{}\Tab{8}{alpha = (sqrt(73.0)\-7)\/2.0;}} \L{\LB{}} \L{\LB{}\Tab{8}{olddeg = DEGREE;}} \L{\LB{}\Tab{8}{DEGREE = SIZE\-1;}} \L{\LB{}\Tab{8}{pzer(p);}} \L{\LB{}\Tab{8}{p[0] = 1.0;}} \L{\LB{}\Tab{8}{p[1] = \-a0*exp(log(t)*(\-alpha));}} \L{\LB{}\Tab{8}{eps.b = 1.0;}} \L{\LB{}\Tab{8}{\K{while}(eps.b \> 1.e\-15)\{}} \L{\LB{}\Tab{16}{pcopy(p, pw);}} \L{\LB{}\Tab{16}{myprpower(p, 1.5, p);}} \L{\LB{}\Tab{16}{\K{for}(j=DEGREE; j \>= 2; \-\-j)}} \L{\LB{}\Tab{24}{p[j] = 12.0*p[j]\/((3.0+j*alpha)*(4.0+j*alpha));}} \L{\LB{}\Tab{16}{p[0] = 1.0;}} \L{\LB{}\Tab{16}{p[1] = \-a0*exp(log(t)*(\-alpha));}} \L{\LB{}\Tab{16}{psub(p, pw, pw);}} \L{\LB{}\Tab{16}{eps.b = pnorm(pw);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{y = rs(DEGREE,b0ero,bone);}} \L{\LB{}\Tab{8}{\K{for}(i=2; i\<= DEGREE; ++i) y.p.p[i] = cvtdi(p[i]);}} \L{\LB{}\Tab{8}{b0 = bl1nrs(y);}} \L{\LB{}\Tab{8}{y.p.p[0] = ione;}} \L{\LB{}\Tab{8}{y.p.p[1] = divi(cvtdi(\-a0),iexp(mult(ilog(cvtdi(t)),ALPHA)));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(grtb(b0,lcvtib(ration(3,10))))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"VTINF: condition 1 is WRONG\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{if}(grtb(uabs(y.p.p[1]),lcvtib(ration(23,100))))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"VTINF: condition 2 is WRONG\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{ithreehalfs = divi(ithree,itwo);}} \L{\LB{}\Tab{8}{yp = rspower(y,ithreehalfs);}} \L{\LB{}\Tab{8}{ypp = rs(DEGREE, b0ero, bone);}} \L{\LB{}\Tab{8}{\K{for} (i=2; i \<= DEGREE; ++i)\{}} \L{\LB{}\Tab{16}{i1 = plus(ithree,mult(cvtinti(i),ALPHA));}} \L{\LB{}\Tab{16}{i1 = divi(mult(i1,plus(i1,ione)),cvtinti(12));}} \L{\LB{}\Tab{16}{ypp.p.p[i]= divi(yp.p.p[i],i1);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{ypp.p.p[0]= y.p.p[0];}} \L{\LB{}\Tab{8}{ypp.p.p[1]= y.p.p[1];}} \L{\LB{}\Tab{8}{j = DEGREE+1;}} \L{\LB{}\Tab{8}{i1 = plus(ithree,mult(cvtinti(j),ALPHA));}} \L{\LB{}\Tab{8}{i1 = divi(cvtinti(12),mult(i1,plus(i1,ione)));}} \L{\LB{}\Tab{8}{ypp.h = umultb(yp.h,ucvtib(i1));}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = bl1nrsf(rsminus(y,ypp));}} \L{\LB{}\Tab{8}{y.g = umultb(ucvtib(ration(75,10)),eps);}} \L{\LB{}\Tab{8}{*y0 = izero, *y1 = izero;}} \L{\LB{}\Tab{8}{\K{for}(i=0; i\<=y.p.deg; ++i)\{}} \L{\LB{}\Tab{16}{*y0 = plus(*y0,y.p.p[i]);}} \L{\LB{}\Tab{16}{i1 = plus(ifour,mult(cvtinti(i),ALPHA));}} \L{\LB{}\Tab{16}{*y1 = plus(*y1,divi(yp.p.p[i],neg(i1)));}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{i1 = mult(cvtinti(144),iexp(mult(ilog(cvtdi(t)),neg(ithree))));}} \L{\LB{}\Tab{8}{*y0 = ienlarge(*y0,uplusb(y.g,y.h));}} \L{\LB{}\Tab{8}{*y0 = mult(*y0,i1);}} \L{\LB{}\Tab{8}{ypp = rspower(y,ithreehalfs);}} \L{\LB{}} \L{\LB{}\Tab{8}{i1 = plus(ifour,mult(itwo,ALPHA));}} \L{\LB{}\Tab{8}{*y1 = ienlarge(*y1,uplusb(udivb(ypp.g,lcvtib(i1)),}} \L{\LB{}\Tab{8}{ udivb(ypp.h,lplusb(bfour,lmultb(cvtintb(y.p.deg+1),}} \L{\LB{}\Tab{8}{ lcvtib(ALPHA))))));}} \L{\LB{}\Tab{8}{i1 = mult(cvtinti(1728),iexp(mult(ilog(cvtdi(t)),neg(ifour))));}} \L{\LB{}\Tab{8}{*y1 = mult(*y1,i1);}} \L{\LB{}\Tab{8}{DEGREE = olddeg;}} \L{\LB{}\Tab{8}{freep(y.p), freep(yp.p), freep(ypp.p);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\Proc{tfc1}tfc1(c)}} \L{\LB{\K{double} c;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL i1, i2, i3, i4, i6, i5;}} \L{\LB{}\Tab{8}{\K{double} xin, eps;}} \L{\LB{}\Tab{8}{\K{int} i, j;}} \L{\LB{}} \L{\LB{}\Tab{8}{i = Y.n\-1;}} \L{\LB{}\Tab{8}{;}} \L{\LB{}\Tab{8}{vtfinf(c, Y.f[i].center.b, \&i5, \&i6);}} \L{\LB{}\Tab{8}{i1 = i5;}} \L{\LB{}\Tab{8}{i2 = i6;}} \L{\LB{}\Tab{8}{\K{for}(j=i; j\>=2; \-\-j)\{}} \L{\LB{}\Tab{16}{xin = Y.f[j].center.b;}} \L{\LB{}\Tab{16}{eps = supervtffx(xin, cvtdi(Y.f[j\-1].center.b), i1, i2, \&i1, \&i2);}} \L{\LB{}} \L{\LB{}\Tab{16}{i3 = Y.f[j\-1].p.p[0];}} \L{\LB{}\Tab{16}{i4 = Y.f[j\-1].p.p[1];}} \L{\LB{}\Tab{16}{\K{if}(i1.up \<= i3.dn) \K{return}(\-1);}} \L{\LB{}\Tab{16}{\K{if}(i2.dn \>= i4.up) \K{return}(\-1);}} \L{\LB{}\Tab{16}{\K{if}(i1.dn \>= i3.up) \K{return}(1);}} \L{\LB{}\Tab{16}{\K{if}(i2.up \<= i4.dn) \K{return}(1);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{return}(0);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\Proc{getc1}getc1()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{double} ctest;}} \L{\LB{}\Tab{8}{\K{double} t = 0.05;}} \L{\LB{}\Tab{8}{\K{int} k = 1;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{while}(k ==1 \|\| k == \-1)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"BOUNDS for C1\!n\"\SE{});}} \L{\LB{}\Tab{16}{printival(C1);}} \L{\LB{}\Tab{16}{printivalio(C1);}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{16}{ctest = t*C1.up+(1.0\-t)*C1.dn;}} \L{\LB{}\Tab{16}{k = tfc1(ctest);}} \L{\LB{}\Tab{16}{\K{if}(k == 1) C1.dn = ctest;}} \L{\LB{}\Tab{16}{\K{if}(k == \-1) C1.up = ctest;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\Proc{refiney}refiney()}} \L{\LB{\{}} \L{\LB{}} \L{\LB{}\Tab{8}{INTERVL i1, i2, i3, i4, i6, i5;}} \L{\LB{}\Tab{8}{INTERVL cvtdi(), intersect();}} \L{\LB{}\Tab{8}{\K{double} xin, eps;}} \L{\LB{}\Tab{8}{\K{int} i, j;}} \L{\LB{}} \L{\LB{}\Tab{8}{i = Y.n;}} \L{\LB{}\Tab{8}{vtfinf(C1.up, Y.f[i].center.b, \&i5, \&i6);}} \L{\LB{}\Tab{8}{vtfinf(C1.dn, Y.f[i].center.b, \&i3, \&i4);}} \L{\LB{}\Tab{8}{i1.up = i3.up;}} \L{\LB{}\Tab{8}{i1.dn = i5.dn;}} \L{\LB{}\Tab{8}{i2.dn = i4.dn;}} \L{\LB{}\Tab{8}{i2.up = i6.up;}} \L{\LB{}\Tab{8}{i5 = Y.f[i].p.p[0];}} \L{\LB{}\Tab{8}{i6 = Y.f[i].p.p[1];}} \L{\LB{}\Tab{8}{Y.f[i].p.p[0] = intersect(i1,i5);}} \L{\LB{}\Tab{8}{Y.f[i].p.p[1] = intersect(i2,i6);}} \L{\LB{}\Tab{8}{\K{for}(j=i; j\>=2; \-\-j)\{}} \L{\LB{}\Tab{16}{xin = Y.f[j].center.b;}} \L{\LB{}\Tab{16}{eps = supervtffx(xin, cvtdi(Y.f[j\-1].center.b), i1, i2, \&i1, \&i2);}} \L{\LB{}} \L{\LB{}\Tab{16}{i3 = Y.f[j\-1].p.p[0];}} \L{\LB{}\Tab{16}{i4 = Y.f[j\-1].p.p[1];}} \L{\LB{}\Tab{16}{Y.f[j\-1].p.p[0] = intersect(i1,i3);}} \L{\LB{}\Tab{16}{Y.f[j\-1].p.p[1] = intersect(i2,i4);}} \L{\LB{}\Tab{16}{i1 = Y.f[j\-1].p.p[0];}} \L{\LB{}\Tab{16}{i2 = Y.f[j\-1].p.p[1];}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{\Proc{refine\_numbers}refine\_numbers()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{printivalio(W);}} \L{\LB{}\Tab{8}{printivalio(C1);}} \L{\LB{}\Tab{8}{refiney();}} \L{\LB{}\Tab{8}{printgrsio(Y);}} \L{\LB{}\Tab{8}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{8}{YRS = tfrs(Y);}} \L{\LB{}\Tab{8}{printgrsio(YRS);}} \L{\LB{}\Tab{8}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{8}{U = grstimesx(YRS);}} \L{\LB{}\Tab{8}{printgrsio(U);}} \L{\LB{}\Tab{8}{printivalio(RC = Omega(U));}} \L{\LB{}\Tab{8}{fflush(stdout);}} \L{\LB{}\Tab{8}{printivalio(BC = grseval(U,RC));}} \L{\LB{}\Tab{8}{fflush(stdout);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\Proc{tfw2}tfw2(w)}} \L{\LB{\K{double} w;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL i1, i2, i3, i4, i6, i5;}} \L{\LB{}\Tab{8}{INTERVL cvtdi();}} \L{\LB{}\Tab{8}{\K{double} xin, xout, eps;}} \L{\LB{}\Tab{8}{\K{int} j;}} \L{\LB{}} \L{\LB{}\Tab{8}{vtff0(w, Y.f[1].center, \&i5, \&i6);}} \L{\LB{}\Tab{8}{i1 = i5;}} \L{\LB{}\Tab{8}{i2 = i6;}} \L{\LB{}\Tab{8}{\K{for}(j=1; j\<=Y.n\-1; ++j)\{}} \L{\LB{}\Tab{16}{xin = Y.f[j].center.b;}} \L{\LB{}\Tab{16}{xout = Y.f[j+1].center.b;}} \L{\LB{}} \L{\LB{}\Tab{16}{eps = supervtffx(xin, cvtdi(xout), i1, i2, \&i1, \&i2);}} \L{\LB{}} \L{\LB{}\Tab{16}{i3 = Y.f[j+1].p.p[0];}} \L{\LB{}\Tab{16}{i4 = Y.f[j+1].p.p[1];}} \L{\LB{}\Tab{16}{\K{if}(i1.up \<= i3.dn) \K{return}(\-1);}} \L{\LB{}\Tab{16}{\K{if}(i2.dn \>= i4.up) \K{return}(1);}} \L{\LB{}\Tab{16}{\K{if}(i1.dn \>= i3.up) \K{return}(1);}} \L{\LB{}\Tab{16}{\K{if}(i2.up \<= i4.dn) \K{return}(\-1);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{return}(0);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\Proc{mygetw}mygetw()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{double} wtest;}} \L{\LB{}\Tab{8}{\K{double} t = .05;}} \L{\LB{}\Tab{8}{\K{int} k = 1;}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{while}(k ==1 \|\| k == \-1)\{}} \L{\LB{}\Tab{16}{printf(\S{}\"BOUNDS for W\!n\"\SE{});}} \L{\LB{}\Tab{16}{printival(W);}} \L{\LB{}\Tab{16}{printivalio(W);}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}} \L{\LB{}\Tab{16}{wtest = t*W.up + (1\-t)*W.dn;}} \L{\LB{}\Tab{16}{k = tfw2(wtest);}} \L{\LB{}\Tab{16}{\K{if}(k == 1) W.dn = wtest;}} \L{\LB{}\Tab{16}{\K{if}(k == \-1) W.up = wtest;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{\}}} \L{\LB{}} \L{\LB{\Proc{refineY}refineY()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{INTERVL i1, i2, i3, i4, i5, i6;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}} \L{\LB{}\Tab{8}{vtff0(W.up, Y.f[1].center, \&i1, \&i2);}} \L{\LB{}\Tab{8}{vtff0(W.dn, Y.f[1].center, \&i5, \&i6);}} \L{\LB{}\Tab{8}{Y.f[1].p.p[0] = iunion(i1,i5);}} \L{\LB{}\Tab{8}{Y.f[1].p.p[1] = iunion(i2,i6);}} \L{\LB{}\Tab{8}{\K{for}(i=1; i \<= Y.n\-1; ++i)\{}} \L{\LB{}\Tab{16}{i1 = Y.f[i].p.p[0];}} \L{\LB{}\Tab{16}{i2 = Y.f[i].p.p[1];}} \L{\LB{}\Tab{16}{supervtffx(Y.f[i].center.b, cvtbi(Y.f[i+1].center),}} \L{\LB{}\Tab{16}{ i1,i2,\&i3,\&i4);}} \L{\LB{}\Tab{16}{i1 = Y.f[i+1].p.p[0];}} \L{\LB{}\Tab{16}{i2 = Y.f[i+1].p.p[1];}} \L{\LB{}\Tab{16}{Y.f[i+1].p.p[0] = intersect(i1,i3);}} \L{\LB{}\Tab{16}{Y.f[i+1].p.p[1] = intersect(i2,i4);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{RSERIES}} \L{\LB{\Proc{rstfu2}rstfu2(x,r,y)}} \L{\LB{INTERVL x;}} \L{\LB{\K{double} r;}} \L{\LB{RSERIES *y;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{\K{int} j;}} \L{\LB{}\Tab{8}{INTERVL u0, u1, a;}} \L{\LB{}\Tab{8}{RSERIES u;}} \L{\LB{}} \L{\LB{}} \L{\LB{}\Tab{8}{a = plus(x,cvtdi(r));}} \L{\LB{}\Tab{8}{u0 = grseval(YRS,x);}} \L{\LB{}\Tab{8}{u1 = grsdereval(YRS,x);}} \L{\LB{}\Tab{8}{*y = vtffxi2( x , a, u0, u1);}} \L{\LB{}} \L{\LB{}\Tab{8}{u = rs(y\-\>p.deg, b0ero, y\-\>r);}} \L{\LB{}\Tab{8}{\K{for}(j = y\-\>p.deg; j\>=1; \-\-j)}} \L{\LB{}\Tab{16}{u.p.p[j] = plus(mult(y\-\>p.p[j],x),}} \L{\LB{}\Tab{16}{ mult(y\-\>p.p[j\-1],cvtbi(y\-\>r)));}} \L{\LB{}\Tab{8}{u.p.p[0] = mult(y\-\>p.p[0] ,x);}} \L{\LB{}\Tab{8}{u.h = umultb(y\-\>h,uplusb(uabs(x) ,y\-\>r));}} \L{\LB{}\Tab{8}{u.h = uplusb(u.h,umultb(uabs(y\-\>p.p[y\-\>p.deg]),y\-\>r));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{return}(u);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \L{\LB{RSERIES}} \L{\LB{\Proc{yinf}yinf(t, m)}} \L{\LB{\K{double} t;}} \L{\LB{\K{int} m;}} \L{\LB{\{}} \L{\LB{}\Tab{8}{RSERIES y, yp, ypp;}} \L{\LB{}\Tab{8}{INTERVL i1, ithreehalfs;}} \L{\LB{}\Tab{8}{BND eps, b0;}} \L{\LB{}\Tab{8}{\K{double} pnorm(), pw[SIZE], p[SIZE], alpha;}} \L{\LB{}\Tab{8}{\K{int} i, j, olddeg;}} \L{\LB{}\Tab{8}{alpha = (sqrt(73.0)\-7)\/2.0;}} \L{\LB{}} \L{\LB{}\Tab{8}{olddeg = DEGREE;}} \L{\LB{}\Tab{8}{DEGREE = m;}} \L{\LB{}\Tab{8}{pzer(p);}} \L{\LB{}\Tab{8}{p[0] = 1.0;}} \L{\LB{}\Tab{8}{p[1] = \-(C1.up+C1.dn)*.5*exp(log(t)*(\-alpha));}} \L{\LB{}\Tab{8}{eps.b = 1.0;}} \L{\LB{}\Tab{8}{\K{while}(eps.b \> 1.e\-15)\{}} \L{\LB{}\Tab{16}{pcopy(p, pw);}} \L{\LB{}\Tab{16}{myprpower(p, 1.5, p);}} \L{\LB{}\Tab{16}{\K{for}(j=DEGREE; j \>= 2; \-\-j)}} \L{\LB{}\Tab{24}{p[j] = 12.0*p[j]\/((3.0+j*alpha)*(4.0+j*alpha));}} \L{\LB{}\Tab{16}{p[0] = 1.0;}} \L{\LB{}\Tab{16}{p[1] = \-(C1.up+C1.dn)*.5*exp(log(t)*(\-alpha));}} \L{\LB{}\Tab{16}{psub(p, pw, pw);}} \L{\LB{}\Tab{16}{eps.b = pnorm(pw);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{y = rs(DEGREE,b0ero,bone);}} \L{\LB{}\Tab{8}{\K{for}(i=2; i\<= DEGREE; ++i) y.p.p[i] = cvtdi(p[i]);}} \L{\LB{}\Tab{8}{b0 = bl1nrs(y);}} \L{\LB{}\Tab{8}{y.p.p[0] = ione;}} \L{\LB{}\Tab{8}{y.p.p[1] = divi(neg(C1),iexp(mult(ilog(cvtdi(t)),ALPHA)));}} \L{\LB{}} \L{\LB{}\Tab{8}{\K{if}(grtb(b0,lcvtib(ration(3,10))))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"YINF: condition 1 is WRONG\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{if}(grtb(uabs(y.p.p[1]),lcvtib(ration(23,100))))\{}} \L{\LB{}\Tab{16}{printf(\S{}\"YINF: condition 2 is WRONG\!n\"\SE{});}} \L{\LB{}\Tab{16}{fflush(stdout);}} \L{\LB{}\Tab{16}{abort();}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{ithreehalfs = divi(ithree,itwo);}} \L{\LB{}\Tab{8}{yp = rspower(y,ithreehalfs);}} \L{\LB{}\Tab{8}{ypp = rs(DEGREE, b0ero, bone);}} \L{\LB{}\Tab{8}{\K{for} (i=2; i \<= DEGREE; ++i)\{}} \L{\LB{}\Tab{16}{i1 = plus(ithree,mult(cvtinti(i),ALPHA));}} \L{\LB{}\Tab{16}{i1 = divi(mult(i1,plus(i1,ione)),cvtinti(12));}} \L{\LB{}\Tab{16}{ypp.p.p[i]= divi(yp.p.p[i],i1);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{ypp.p.p[0]= y.p.p[0];}} \L{\LB{}\Tab{8}{ypp.p.p[1]= y.p.p[1];}} \L{\LB{}\Tab{8}{j = DEGREE+1;}} \L{\LB{}\Tab{8}{i1 = plus(ithree,mult(cvtinti(j),ALPHA));}} \L{\LB{}\Tab{8}{i1 = divi(cvtinti(12),mult(i1,plus(i1,ione)));}} \L{\LB{}\Tab{8}{ypp.h = umultb(yp.h,ucvtib(i1));}} \L{\LB{}} \L{\LB{}\Tab{8}{eps = bl1nrsf(rsminus(y,ypp));}} \L{\LB{}\Tab{8}{y.g = umultb(ucvtib(ration(75,10)),eps);}} \L{\LB{}\Tab{8}{y.k = 2;}} \L{\LB{}\Tab{8}{DEGREE = olddeg;}} \L{\LB{}\Tab{8}{freep(yp.p), freep(ypp.p);}} \L{\LB{}\Tab{8}{\K{return}(y);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{GRS}} \L{\LB{\Proc{expandY}expandY()}} \L{\LB{\{}} \L{\LB{}\Tab{8}{GRS newy;}} \L{\LB{}\Tab{8}{\K{double} step;}} \L{\LB{}\Tab{8}{\K{int} i;}} \L{\LB{}\Tab{8}{INTERVL i1;}} \L{\LB{}} \L{\LB{}\Tab{8}{step = Y.f[Y.n].center.b\-Y.f[Y.n\-1].center.b;}} \L{\LB{}\Tab{8}{\K{if}(step \> 0.5) step = 0.5;}} \L{\LB{}\Tab{8}{i = (322.0\-Y.f[Y.n].center.b)\/step;}} \L{\LB{}} \L{\LB{}\Tab{8}{newy = grs(i+Y.n);}} \L{\LB{}\Tab{8}{\K{for}(i=0; i \<= Y.n; ++i)\{}} \L{\LB{}\Tab{16}{newy.f[i] = rscopy(Y.f[i]);}} \L{\LB{}\Tab{16}{freep(Y.f[n].p);}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}\Tab{8}{\K{for}(i=Y.n+1; i \<= newy.n; ++i)\{}} \L{\LB{}\Tab{16}{newy.f[i] = rs(1, cvtdb(newy.f[i\-1].center.b+step),bone);}} \L{\LB{}\Tab{16}{i1.up = zero;}} \L{\LB{}\Tab{16}{i1.dn = (\K{double}) \-2;}} \L{\LB{}\Tab{16}{newy.f[i].p.p[1] = i1;}} \L{\LB{}\Tab{16}{i1.dn = zero;}} \L{\LB{}\Tab{16}{i1.up = one;}} \L{\LB{}\Tab{16}{newy.f[i].p.p[0] = i1;}} \L{\LB{}\Tab{8}{\}}} \L{\LB{}} \L{\LB{}\Tab{8}{free((\K{char} *)Y.f), Y.f = NULL;}} \L{\LB{}\Tab{8}{\K{return}(newy);}} \L{\LB{\}}} \L{\LB{}} \L{\LB{}} \L{\LB{}} \vfill\eject\end