Using Reduced System Methods

If matrix A has either point or line property A, it can be permuted to a red-black system:

$$\left(\begin{array}{cc} D_R & H \\ K & D_B \end{array} \ri... ...t)= \left(\begin{array}{c} b_R \\ b_B \end{array} \right)$$

where D_R and D_B are scalar (block) diagonal matrices if a point (line) red-black ordering was used. The reduced system corresponding to this matrix is:

(D_R - H D_B^-1 K) u_R = b_R - H D_B^-1 b_B

NSPCG allows two means of using reduced system iterative methods.

1.

The user can apply the RS preconditioners. With these preconditioners, the reduced system is not explicitly computed, and the matrix-vector products using the reduced system matrix are implicitly calculated using the COEF matrix. The user must first fill the integer vector P with a coloring vector for either point or line red-black ordering. For point red-black ordering, the REDBLK subroutine can be used to generate the coloring vector:

       CALL REDBLK (NDIM,N,MAXNZ,COEF,JCOEF,P,IP,NSTORE,IWKSP,IER)

The variable NSTORE takes on the same values as the IPARM parameter, namely

NSTORE = 1	for primary format
= 2	for symmetric diagonal format
= 3	for nonsymmetric diagonal format
= 4	for symmetric coordinate format
= 5	for nonsymmetric coordinate format

P, IP, and IWKSP are integer workspace vectors of length N upon input. Upon output, P contains the coloring vector for red-black ordering if such an ordering exists. IER on output will be either 0, in which case a red- black coloring vector was successfully found, or -8, in which case matrix A does not have point Property A. For line red-black ordering, the user can use the COLOR subroutine described in Section 12 for generating the coloring vector P. COLOR can also be used to generate the coloring vector for point red-black ordering.

2.

If matrix A has point Property A, the user can explicitly compute the reduced system matrix and use any of the preconditioners in NSPCG to solve the reduced system. Storage demands are greater with this method since the reduced system matrix must be stored. To invoke this method, the user simply calls the subroutine RSNSP with the same argument list as NSPCG. The vector P must contain a coloring vector for point red-black ordering and IPERM must be set to 1. The interpretation of the parameters in the argument list is otherwise the same as for NSPCG. Note that since P and IP are used to permute the original system to red-black form, the user cannot apply a further permutation to the reduced system matrix.