## Numerical Mathematics and Computing Fifth Edition Ward Cheney & David Kincaid Errata

• Front Righthandside Inside Cover, left column, line -4: Insert "a" in exponent of last term "e^x" to read $\frac{d}{dx} e^{ax} = a e^{ax}$
• Page vi, line 10: Should be: Minghorng Lai
• Page 5, lines -9, -7: should be -0.001 and -547.
• Page 8, Problem 1.1.1, Line 3: Change $m/n$ to $n/m$
• Page 12, Computer Problem 1.1.4: $8^{20}$ change to $8^{10}$
• Page 21, in boxed equations: just before term $f'''$ remove the asterisk
• Page 31, Additional References, lines 2,3,4,6,7: Burden and Faires , ..., Gerald and Wheatley , ... Golub and van Loan , ...,Phillips and Taylor , ... Ralston 
• Page 40, CP 1.2.19b: Should read: $J_{20}(1)\approx 3.87350\, 3009 \times 10^{-25}$ and $J_{19}(1) \approx 1.54847 \, 8441 \times 10^{-23}$
• Page 59, line -12: Should read: $$0< c < (1\,111\,111\,111)_2=2047$$
• Page 59, line -6: Should read: The largest double precision machine number in the {\tt Marc-32} is $(2-2^{-52})2^{1023}\approx • Page 62, line -14: Replace "$(k=23$" with "$(k=24$" • Page 66, Summary (2), Line 5: In exponent of 2,$b_{11}$should be$b_12}$and in last term$b_{12}b_{13}$should be$b_{13}b_{14}$to read: $$(-1)^b_1 \times 2^{(b_2b_3\cdots b_{12})_2} \times 2^{-1023} \times (1.b_{13}b_{14}\cdots b_{64})_2$$ • Page 66, Problem 2.2.1, Line 2: Remove sentence: "Also, determine the machine precision in single and double precision." • Page 67, Problem 2.2.5(c): Change last 23 1s to 0s to read:$0\,11111111\.00000000000000000000000$• Page 67, Problem 2.2.5(d): Change last 23 1s to 0s to read:$1\,11111111\.00000000000000000000000$• Page 88, line 3: Insert "/" between ")(" in middle to read$u = (r - \sqrt{2}/2)/(r + \sqrt{2}/2)$• Page 103, Problem 3.1.3: Replace [1,2] with [4,5] and insert at end: What happens on the interval [1,2]? • Page 116, line 10,11,12: Kincaid and Cheney ,... Crilly, Earnshaw, and Jones ,..., Hastings and Sugihara  • Page 124, Computer Problem 3.2.27, Line 2,3-4: "Generate a picture showing three basins of attraction" .... [Omit "the" and move beginning "1" from line 4 to end of line 3 to fix bad break in$-1 \le Real(z) \le 1$.] • Page 128, line 1: Move minus sign from "x_1$ to "x_0" to read $x_0 = -1$ and $x_1 = 1$
• Page 128, lines 8,12,13,14: Righthand column of output should read: $-7.97, 1.011\times 10^{-4}, 2.990\times 10^{-7}, 2.990\times 10^{-7}$
• Page 133, Additional References, Line 3: Nerinck and Haegemans 
• Page 148, line 11: two fractions in the numericator should switched to read: $\frac{\frac{f(x_2)-f(x_1)}{x_2-x_1} -\frac{f(x_1)-f(x_0)}{x_1-x_0}}{x_2-x_0}$
• Page 178, Problem 4.2.9, Add: (Use 21 uniform nodes, including the end-points of the interval. Compare results using Theorems 1 and 2.)
• Page 183, line +5: Last term is displayed equation should be: $$+\frac34 b_6 h^6+\cdots$$
• Page 183, line +7: Last term is displayed equation should be: $$-\frac{1}{20}b_6 h^6+\cdots$$
• Page 192, Problem 4.3.9b: Last term should have minus sign in it to read $f(x-2h)]$
• Page 212, Example 3, Solution, last line: Reword to read "Hence, 59 or more points will certainly produce the desired accuracy."
• Page 213, Example 4, Solution, last line: Reword to read "So this analysis induces us to take 92 subintervals."
• Page 229, line -4 (line above last displayed equation): Change to read: second column, ${\cal O}(h^6}$ for the third column, and so on. Check the ratio
• Page 230, Additional References, Line 2,4,5: Move Stroud  from line 2 to end of line 5. ...Ghizetti and Ossiccini , ..., Stroud and Secrest  and Stroud .
• Page 236, Next to last displayed equation after "and, thereby," add equation number (2)
• Page237-240, Change equations numbers: From (2)-(6) to (3)-(7)
• Page 251, line 8: Change to read: It will integrate correctly all polynomials up to and including quintic ones..
• Page 256, Additional References, Line 3: Dixon 
• Page 258, Computer Problem 6.2.1: omit superscript "a" before number.
• Page 261, Computer Problem 6.2.13, line 3, 6: omit T.N.L, ... Pessens, de Doncker, Uberhuber, and Kahaner 
• Page 268, line -2 in pseudocode: Should read: for j=n to k step -1 do
• Page 274, line 1: Should read: $\mathbf{e} = \widetilde{\mathbf{x}} - \mathbf{x}$
• Page 274, line -3: Should read: $\mathbf{e} = \widetilde{\mathbf{x}} - \mathbf{x}$
• Page 280, line -7: Remove letter "T" in last displayed equation.
• Page 301, line -5: Insert and align line $d_i \leftarrow d_i - (xmult)c_{i-1}$ after line $xmult \leftarrow a_{i-1}/d_{i-1}$
• Page 303, Line 9: Move minus sign from before $c_{i-1}$ to after $d_i$ to read $\widehat{d}_i = d_i - \left(\frac{a_{i-1}{\widehat{d}_{i-1}\right)c_{i-1}$
• Page 303, Line 14: Move minus sign from before $c_{i-1}$ to after $d_i$ to read $|\widehat{d}_i| = \left | d_i - \left(\frac{a_{i-1}{\widehat{d}_{i-1}}\right) c_{i-1}\right |$
• Page 304, Line 11-12: Should read: Also, one should not use this routing if $n\le 4$.(Why?)
• Page 305, Line 4-5: Should read: Here we assume that all linear arrays are padded with zeros to length $n$ in order not to exceed the array dimensions in the pseudocode.
• Page 306, References, Line 0, 3, 4: Change to read "Additional References",... Dongarra, Duff, Sorenson, and van de Vorst ,... Golub and van Loan 
• Page 310--311: Interchange figures for Problems 19 and 20.
• Page 312, Computer Problem 7.3.23: Entry (n-2,n) is displayed matrix -1 should be 1.
• Page 313: 8.1 Matrix Factorizations [insert "Matrix"]
• Pages 315, 317, 319,321,323, 325, 327, 329, 331, 333, 335, 337: Running Heading: 8.1 Matrix Factorizations
• Page 345, Line 15: font for $Q$ should be $\mathbf{Q}$
• Page 371, Computer Problem 8.3.1e, last line in displayed matrix should omit "2"
• Page 371, Computer Problem 8.4, Lines 2-3: Parlett 
• Page 374, Line 8: Title to pseudocode should read: Modified Power Method Algorithm (with Normalization)
• Page 374, Line -13: Omit word "real" to read: If we want to compute an eigenvalue of $\bA$ that is close to a given number $\mu$
• Page 379, Example: Shifted Inverse Power Method [insert "Inverse"]
• Page 380, Additional References, line 2: Gautschi ,....
• Page 388, Example 2, last line in displayed equation, right-most: Should read $1 \le x \le 20$ [Change limits on x from -1,1 to 1,20
• Page 394, Problem 16, last line in displayed equation, right-most: Change subscript "m" to "n" to read: $0 \le x \le t_n$
• Page 405, Line 7 in pseudocode: Change to read: for i=n-1 to 1 step -1 do
• Page 406, Lines 13, 14, 16 in pseudocode: Change to read: for j=0 to 4n do; x <- a + jh/4; e <- ...; output j,x,e
• Page 406, Lines 1 below pseudocode: Change to read: A typical result from the computer output is $j=19$ ...y
• Page 429, Line -1 above Pseudocode: Omit sentence: Property 5 is exhibited in Computer Problem 28.
• Page 446, Figure 10.1: On both x-axis and t-axis, add minus signs to bottom-most and left-most 4,3,2,1
• Page 447, Figure 10.2: On x-axis, add minus signs to bottom-most 4,3,2,1. On t-axis, add minus signs to left-most 2.
• Page 472, Line 1: ...So six function evalutions give a fifth-order approximation, together...
• Page 472, Line 11: Change to read: ...c_{53}\leftarrow 3680./513.
• Page 517, Line 21: Should read:$\partial \mathbf{F}/\partial \mathbf{X}$
• Page 519, Computer Problem 7a: Remove negative signs in x_13 and x_23 and add then in x'_13 and x'_23.
• Page 544, Problem 12.2.18, Line 3: Should read: $\{u_1, u_2, u_3\}$ Change second $u_1$ to $u_2$ and third $u_1$ to $u_3$.
• Page 562, replace ALGORITHM 1 & 2 and the paragraph after them. Samples of two other random-number generator algorithms follow: ALGORITHM 1. Let $s = 2111111111 x_{n-4} + 1492 x_{n-3} + 1776 x_{n-2} + 5115x_{n-1} +c$, $x_n = s \bmod( 2^{32})$, $c = \lfloor s/2^{32} \rfloor$, which is the so-called {\sl mother-of-all} pseudorandom number generators, invented by George Marsaglia. ALGORITHM 2. Let $x_{n+1}=(1103515245 x_n + 12345) \mod(2^{31})$, which is {\tt rand()} in Unix. These algorithms are suitable for some applications but they may not produce high-quality randomness and may not be suitable for applications requiring accurate statistics or in cryptographics. On the Internet, one can find new and improved pseudorandom number generators which are designed for the fast generations of high quality random numbers with colossal periods and with special distributions. See for example, {\tt www.gnu.org/software/gsl/}. Here are three specific points of caution to remember.
• Page 570, Computer Problem 13.1.4: Reword to read "Test some random number generators found on the Internet or in mathematical software packages.
• Page 622, Line 5: Add to right-most $\equiv \frac{1}{sigma} to read: $$s = \frac{h^2}{k} \equiv \frac{1}{\sigma}$$ • Page 628, Problem 13, Displayed equation: Should read: $$\mathbf{A}\mathbf{V}_{j-1} = \mathbf{V}_j$$ • Page 663, Golden Section Search Algorithm subsection, before Line 5, after displayed golden section ratio, insert: "The mathematical history of this number can be found in Roger  and it ...." Line • Page 683, Line 1: Change to read${\rm inf}_{x \in \IR^n} F(x)$• Page 689, Problem 24, Line 2: Should read:$f(mathbf{x}+t\mathbf{u})$Here insert "t" (not bold) after plus sign. • Page 749, Table C.2, last column, entry -3: should be: 2.225\times 10^{-308}\approx 2^{-1022}$
• Page 759, Line +2 under Other Concepts: the rows and columns of $\mathbf{A}=(a_{ij})_{n\times m}$
• Page 767, Line -2: (Refer to Section 8.2 for definitions.)
• Page 771, Line +7: Change to read: being of little importance ...
• Page 771, Line -6: Sir Isaac Newton's epitaph (1642-1727):
• Page 773, footnote: Move footnote left to align with text above and move some text from second line to the first line in footnote.
• Page 775, Answer Problem 2.2.1c: Should read only: 1c. $[B5000000]_{16}$
• Page 775, Answer Problem 2.2.2d: Should read only: 2d. $[3FA000000000000]_{16},[BFA0000000000000]_{16}$
• Page 775, Answer Problem 2.2.4d: Should read: 4d. $[3E700000]_{16}, [3FCE000000000000]_{16}$
• Page 777, Answer Problem 4.1.3: Should read: $\ell_2(x)=-(x-4)(x^2-1)/8$
• Page 778, Answer Problem 4.2.9: Should read: $4.105E-14$ (Thm 1), $1.1905E-23$ (Thm 2)
• Page 778, Answer Problem 4.3.1: Should read: $-hf''(\xi)$
• Page 778, Answer Problem 4.3.16: Change \phi to \varphi twice
• Page 779, Answer Problem 5.1.1, insert: 1. $\frac{7}{18}$
• Page 779, Answer 5.2.2, insert zero before decimal point three times to read: 0.775, 0.7854, 0.01042
• Page 779, Answer 5.2.12, insert zero before decimal point to read: 0.000025
• Page 779, Answer 5.2.15, insert zero before decimal point to read: 0.3104
• Page 780, Computer Problems 6.2.1: Change from CP 1 to CP 2a to read: 2a. 1.4183.
• Page 780, Problems 7.1: Insert answer to problem 1 to read: 1. Homogeneous: $\alpha = 0$, zero solution; $\alpha = \pm 1$ infinite number of solutions
• Page 783, Answer Problems 7, 9, 11:
7. Eigenvalues/Eigenvectors: 1,(-1,1,0,0); 2,(0,0,-1,1); 5,(-1,-1,2,2); 10,(2,2,1,1). 9. c. 11. d.
• Page 784, Answer Problem 9.2.5: 5. a=-5, b=-26, c=-27, d=27/2 (Here d had the wrong sign.)
• Page 784, Answer Problem 9.2.8: 8a. (m+1)n; 8b. 2n; 8c. (m-1)(n-1); 8d. m -1
• Page 784, Answer Problem 9.3.2: Change "Chapter 10" to "Section 12.2, Equation (2)"
• Page 785, Answer Problem 9.3.20: Omit "of Section 7.3"
• Page 785, Answer Problem 9.3.24: Change Equation (3) to Equation (14)
• Page 785, Answer Problem 10.1.3d: Make integral sign larger
• Page 785, Answer Problem 10.1.6: Change "let" to "Let" and add period to end of sentence.
• Page 786, Answer Computer Problem 10.2.3b: Change (T) to (TS)
• Page 786, Answer Computer Problem 10.2.3c: Change (T) to (TS)
• Page 788, Answer Problem 12.1.7: Add large left [ and right ] in numerator and denominator of fraction
• Page 788, Answer Problem 12.1.11: Change \phi to \varphi
• Page 789, Answer Computer Problem 12.2.7: Change to read: a_{ij}= 0, i\neq j; (m+1), i=j=1; (m+1)/2, i=i>1
• Page 790, Answer Problem 14.1.6: Change \phi to \varphi
• Page 790, Answer Problem 14.1.15: Change \phi to \varphi (three times)
• Page 791, Answer Problem 16.1.7: Change $c_1$ to $A$ and $c_2$ to $B$.
• Page 793, Answer Problem 17.1.13h: Change $(0, \frac{18},{5})$ to $(\frac{18},{5}, 0)$ [Move this entire answer to same line.]
• Page 796, Cash: Change "to appear" to 2003.
• Page 799, Line 4: Change Dover 2003 to 2004

## Solution Manuals for Numerical Mathematics and Computing Fifth Edition Ward Cheney & David Kincaid Errata

• Problem 1.2.8, last three lines: Should read: Therefore, we have $15+log(2e)< log(n!)$. So $n=17$: $15.74\not< 14.55$ and $n=18$: $15.74< 15.81$.
• Problem 2.2.4d, line 1: Should read: ...=(1.11)_2\times 2^{-3}$• Problem 3.2.7, end of first line and beginning of second line: Should read:$x_0=\frac{\pi}{4}$. So$y=cos(\frac{\pi}{4})(x-\frac{\pi}{4})+sin(\frac{\pi}{4})\$
• Problem 4.2.2, line -1: remove extra )
• Problem 6.2.11, line -2: remove extra 0
• Insert answer to 8.3.7, renumber 8.3.7-10 to 8.3.8-11, change answer to 8.3.12, and omit old 8.3.12.
• Problem 8.3.7: 1,(-1,1,0,0); 2,(0,0,-1,1); 5,(-1,-1,2,2); 10,(2,2,1,1)
• Problem 8.3.8: d
• Problem 8.3.9: c
• Problem 8.3.10: a
• Problem 8.3.11: d
• Problem 8.3.12: d
• Computer Problem 12.2.7: Change to read: a_{ij}= 0, i\neq j; (m+1), i=j=1; (m+1)/2, i=i>1

Acknowledgements: We welcome comments and suggestions concerning either the textbook or solution manuals. Send email to Ward Cheney or David Kincaid .

We are grateful to the following individuals and others who have send us email concerning typos and errors in the textbook or solution manuals: Nabeel S.Abo-Ghander, Krishan Agrawal, Keven Anderson, Vladimir Andrijevik, Jon Ashland, Bernand Bialecki, Hassan Basir, Ellen Chen, Roger Crawfis, Jonathan Dautrich, Jason Durheim, Christopher M. Hoss, Victoria Interrante, Sadegh Jokar, Jason Karns, Gary Krenz, Jihoon Kwak, F. Milianazzo, Milan Miklavcic, Sue Minkoff, Valia Guerra Ones, Tony Praseuth, Rolfe G. Petschek, Helia Niroomand Rad, Frank Roberts, Keh-Ming Shyue, Nathan Smith, Mandayam Srinivas, Thiab Taha, David Wallace, Layne T. Watson

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 02/07/2007