NAG Library Routine Document
F11JDF solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse symmetric matrix, represented in symmetric coordinate storage format.
|SUBROUTINE F11JDF (
||N, NNZ, A, IROW, ICOL, RDIAG, OMEGA, CHECK, Y, X, IWORK, IFAIL)
||N, NNZ, IROW(NNZ), ICOL(NNZ), IWORK(N+1), IFAIL
||A(NNZ), RDIAG(N), OMEGA, Y(N), X(N)
F11JDF solves a system of equations
involving the preconditioning matrix
corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)
) on a linear system
is a sparse symmetric matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2
in the F11 Chapter Introduction).
In the definition of given above is the diagonal part of , is the strictly lower triangular part of , and is a user-defined relaxation parameter.
It is envisaged that a common use of F11JDF will be to carry out the preconditioning step required in the application of F11GEF
to sparse linear systems. For an illustration of this use of F11JDF see the example program given in Section 9.1
. F11JDF is also used for this purpose by the Black Box routine F11JEF
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York
- 1: N – INTEGERInput
On entry: , the order of the matrix .
- 2: NNZ – INTEGERInput
On entry: the number of nonzero elements in the lower triangular part of .
- 3: A(NNZ) – REAL (KIND=nag_wp) arrayInput
: the nonzero elements in the lower triangular part of 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 F11ZBF
may be used to order the elements in this way.
- 4: IROW(NNZ) – INTEGER arrayInput
- 5: ICOL(NNZ) – INTEGER arrayInput
: the row and column indices of the nonzero elements supplied in array A
must satisfy these constraints (which may be imposed by a call to F11ZBF
- and , for ;
- or and , for .
- 6: RDIAG(N) – REAL (KIND=nag_wp) arrayInput
On entry: the elements of the diagonal matrix , where is the diagonal part of .
- 7: OMEGA – REAL (KIND=nag_wp)Input
On entry: the relaxation parameter .
- 8: CHECK – CHARACTER(1)Input
: specifies whether or not the input data should be checked.
- Checks are carried out on the values of N, NNZ, IROW, ICOL and OMEGA.
- None of these checks are carried out.
- 9: Y(N) – REAL (KIND=nag_wp) arrayInput
On entry: the right-hand side vector .
- 10: X(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the solution vector .
- 11: IWORK() – INTEGER arrayWorkspace
- 12: 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||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 lower triangular part of
, is out of order, or has duplicate row and column indices. Call F11ZBF
to reorder and sum or remove duplicates.
The computed solution
is the exact solution of a perturbed system of equations
is a modest linear function of
is the machine precision
The time taken for a call to F11JDF is proportional to NNZ
It is expected that a common use of F11JDF will be to carry out the preconditioning step required in the application of F11GEF
to sparse symmetric linear systems. In this situation F11JDF 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 to set
for all subsequent calls.
This example solves a sparse symmetric linear system of equations
using the conjugate-gradient (CG) method with SSOR preconditioning.
The CG algorithm itself is implemented by the reverse communication routine F11GEF
, 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 F11XEF.
- If , a solution of the preconditioning equation is required. This is achieved by a call to F11JDF.
- If , F11GEF has completed its tasks. Either the iteration has terminated, or an error condition has arisen.
For further details see the routine document for F11GEF
9.1 Program Text
Program Text (f11jdfe.f90)
9.2 Program Data
Program Data (f11jdfe.d)
9.3 Program Results
Program Results (f11jdfe.r)