Examples

Given a linear system Au=b with

$$A = \left[\begin{array}{rrrr} 4 & -1 & -1 & 0 \\ -1 & 4 & ... ...\left[\begin{array}{c} 6 \\ 0 \\ 0 \\ 6 \end{array} \right],$$

a program to solve this problem with an initial guess of u^T = (0, 0, 0, 0) using JCG() with symmetric sparse storage and printing the final approximate solution vector follows.

 
         INTEGER IA(5), JA(8), IPARM(12), IWKSP(12)
         REAL    A(8), RHS(4), u(4), WKSP(24), RPARM(12)
         DATA A(1),A(2),A(3),A(4) / 4.0,-1.0,-1.0,4.0 /
         DATA A(5),A(6),A(7),A(8) / -1.0,4.0,-1.0,4.0 /
         DATA JA(1),JA(2),JA(3),JA(4) / 1,2,3,2 /
         DATA JA(5),JA(6),JA(7),JA(8) / 4,3,4,4 / 
         DATA IA(1),IA(2),IA(3),IA(4),IA(5) / 1,4,6,8,9 /
         DATA RHS(1),RHS(2),RHS(3),RHS(4)  / 6.0,0.0,0.0,6.0 /
         DATA N /4/, NW /24/, ITMAX /4/, LEVEL/1/, IDGTS/2/ 
   C
         CALL DFAULT (IPARM, RPARM)
         IPARM(1) = ITMAX
         IPARM(2) = LEVEL
         IPARM(12) = IDGTS
         CALL VFILL (N, U, 0.E0)
         CALL JCG (N,IA,JA,A,RHS,U,IWKSP,NW,WKSP,IPARM,RPARM,IER)
         STOP
         END

The output for this run would be

 
  BEGINNING OF ITPACK SOLUTION MODULE JCG
 
  JCG HAS CONVERGED IN     2 ITERATIONS.
 
       APPROX. NO. OF DIGITS (EST. REL. ERROR) = 14.6  (DIGIT1)
    APPROX. NO. OF DIGITS (EST. REL. RESIDUAL) = 14.3  (DIGIT2)
 
 
      SOLUTION VECTOR.
 
                 1              2              3              4
       -------------------------------------------------------------
         2.00000E+00    1.00000E+00    1.00000E+00    2.00000E+00

Textbook methods such as the Jacobi (J), Gauss-Seidel (GS), Successive Overrelaxation (SOR--fixed relaxation factor omega), Symmetric Successive Overrelaxation (SSOR--fixed relaxation factor omega), and the RS method can be obtained from this package by resetting appropriate parameters after the subroutine DFAULT() is called but before ITPACK routines are called.

Method	Use	Parameters
J	JSI()	${\bf IPARM(6)=0, IPARM(7)=2}$
GS	SOR()	${\bf IPARM(6)=0}$
SOR--fixed omega	SOR()	${\bf IPARM(6)=0, RPARM(5)=OMEGA}$
SSOR--fixed omega	SSORSI()	${\bf IPARM(6)=0, RPARM(5)=OMEGA}$
RS	RSSI()	${\bf IPARM(6)=0}$

These methods were not included as separate routines because they are usually slower than the accelerated methods included in this package. On the black unknowns, the Cyclic Chebyshev Semi-Iterative (CCSI) method of Golub and Varga [2] gives the same result as the RSSI method. The CCSI and RSSI methods converge at the same rate, and each of them converges twice as fast as the JSI method. This is a theoretical result [6] and does not count the time involved in establishing the red-black indexing and the red-black partitioned system. Similarly, the Cyclic Conjugate Gradient (CCG) method with respect to the black unknowns, considered by Reid [16] (see also Hageman and Young [6]), gives the same results as the RSCG method. Also, the CCG and the RSCG methods converge at the same rate, and each of them converges, theoretically, exactly twice as fast as the JCG method. Hence, the accelerated RS methods are preferable to the accelerated J methods when using a red-black indexing.