NAG Library Routine Document
F11DRF solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse non-Hermitian matrix, represented in coordinate storage format.
|SUBROUTINE F11DRF (
||TRANS, N, NNZ, A, IROW, ICOL, RDIAG, OMEGA, CHECK, Y, X, IWORK, IFAIL)
||N, NNZ, IROW(NNZ), ICOL(NNZ), IWORK(2*N+1), IFAIL
||A(NNZ), RDIAG(N), Y(N), X(N)
F11DRF solves a system of linear equations
according to the value of the parameter TRANS
, where the matrix
corresponds to symmetric successive-over-relaxation (SSOR) Young (1971)
applied to a linear system
is a complex sparse non-Hermitian matrix stored in coordinate storage (CS) format (see Section 2.1.1
in the F11 Chapter Introduction).
In the definition of given above is the diagonal part of , is the strictly lower triangular part of , is the strictly upper triangular part of , and is a user-defined relaxation parameter.
It is envisaged that a common use of F11DRF will be to carry out the preconditioning step required in the application of F11BSF
to sparse linear systems. For an illustration of this use of F11DRF see the example program given in Section 9
. F11DRF is also used for this purpose by the Black Box routine F11DSF
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York
- 1: TRANS – CHARACTER(1)Input
: specifies whether or not the matrix
- is solved.
- is solved.
- 2: N – INTEGERInput
On entry: , the order of the matrix .
- 3: NNZ – INTEGERInput
On entry: the number of nonzero elements in the matrix .
- 4: A(NNZ) – COMPLEX (KIND=nag_wp) arrayInput
: the nonzero elements in the matrix
, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The routine F11ZNF
may be used to order the elements in this way.
- 5: IROW(NNZ) – INTEGER arrayInput
- 6: ICOL(NNZ) – INTEGER arrayInput
: the row and column indices of the nonzero elements supplied in A
must satisfy the following constraints (which may be imposed by a call to F11ZNF
- and , for ;
- either or both and , for .
- 7: RDIAG(N) – COMPLEX (KIND=nag_wp) arrayInput
On entry: the elements of the diagonal matrix , where is the diagonal part of .
- 8: OMEGA – REAL (KIND=nag_wp)Input
On entry: the relaxation parameter .
- 9: CHECK – CHARACTER(1)Input
: specifies whether or not the CS representation of the matrix
should be checked.
- Checks are carried on the values of N, NNZ, IROW, ICOL and OMEGA.
- None of these checks are carried out.
- 10: Y(N) – COMPLEX (KIND=nag_wp) arrayInput
On entry: the right-hand side vector .
- 11: X(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: the solution vector .
- 12: IWORK() – INTEGER arrayWorkspace
- 13: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|On entry,|| or ,|
|or|| or .|
|or||OMEGA lies outside the interval ,|
On entry, the arrays IROW
fail to satisfy the following constraints:
- and , for ;
- or and , for .
Therefore a nonzero element has been supplied which does not lie in the matrix
, is out of order, or has duplicate row and column indices. Call F11ZNF
to reorder and sum or remove duplicates.
On entry, the matrix has a zero diagonal element. The SSOR preconditioner is not appropriate for this problem.
the computed solution
is the exact solution of a perturbed system of equations
is a modest linear function of
is the machine precision
. An equivalent result holds when
The time taken for a call to F11DRF is proportional to NNZ
It is expected that a common use of F11DRF will be to carry out the preconditioning step required in the application of F11BSF
to sparse linear systems. In this situation F11DRF is likely to be called many times with the same matrix
. In the interests of both reliability and efficiency, you are recommended to set
for the first of such calls, and
for all subsequent calls.
This example solves a complex sparse linear system of equations
using RGMRES with SSOR preconditioning.
The RGMRES algorithm itself is implemented by the reverse communication routine F11BSF
, which returns repeatedly to the calling program with various values of the parameter IREVCM
. This parameter indicates the action to be taken by the calling program.
- If , a matrix-vector product is required. This is implemented by a call to F11XNF.
- If , a conjugate transposed matrix-vector product is required in the estimation of the norm of . This is implemented by a call to F11XNF.
- If , a solution of the preconditioning equation is required. This is achieved by a call to F11DRF.
- If , F11BSF has completed its tasks. Either the iteration has terminated, or an error condition has arisen.
For further details see the routine document for F11BSF
9.1 Program Text
Program Text (f11drfe.f90)
9.2 Program Data
Program Data (f11drfe.d)
9.3 Program Results
Program Results (f11drfe.r)