NAG Library Routine Document
F11JEF solves a real sparse symmetric system of linear equations, represented in symmetric coordinate storage format, using a conjugate gradient or Lanczos method, without preconditioning, with Jacobi or with SSOR preconditioning.
|SUBROUTINE F11JEF (
||METHOD, PRECON, N, NNZ, A, IROW, ICOL, OMEGA, B, TOL, MAXITN, X, RNORM, ITN, WORK, LWORK, IWORK, IFAIL)
||N, NNZ, IROW(NNZ), ICOL(NNZ), MAXITN, ITN, LWORK, IWORK(N+1), IFAIL
||A(NNZ), OMEGA, B(N), TOL, X(N), RNORM, WORK(LWORK)
F11JEF solves a real sparse symmetric linear system of equations
using a preconditioned conjugate gradient method (see Barrett et al. (1994)
), or a preconditioned Lanczos method based on the algorithm SYMMLQ (see Paige and Saunders (1975)
). The conjugate gradient method is more efficient if
is positive definite, but may fail to converge for indefinite matrices. In this case the Lanczos method should be used instead. For further details see Barrett et al. (1994)
The routine allows the following choices for the preconditioner:
- no preconditioning;
- Jacobi preconditioning (see Young (1971));
- symmetric successive-over-relaxation (SSOR) preconditioning (see Young (1971)).
For incomplete Cholesky (IC) preconditioning see F11JCF
is represented in symmetric coordinate storage (SCS) format (see Section 2.1.2
in the F11 Chapter Introduction) in the arrays A
. The array A
holds the nonzero entries in the lower triangular part of the matrix, while IROW
hold the corresponding row and column indices.
Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia
Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York
- 1: METHOD – CHARACTER(*)Input
: specifies the iterative method to be used.
- Conjugate gradient method.
- Lanczos method (SYMMLQ).
- 2: PRECON – CHARACTER(1)Input
: specifies the type of preconditioning to be used.
- No preconditioning.
- Symmetric successive-over-relaxation (SSOR).
, or .
- 3: N – INTEGERInput
On entry: , the order of the matrix .
- 4: NNZ – INTEGERInput
On entry: the number of nonzero elements in the lower triangular part of the matrix .
- 5: A(NNZ) – REAL (KIND=nag_wp) arrayInput
: the nonzero elements of 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.
- 6: IROW(NNZ) – INTEGER arrayInput
- 7: 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 .
- 8: OMEGA – REAL (KIND=nag_wp)Input
is the relaxation parameter
to be used in the SSOR method. Otherwise OMEGA
need not be initialized.
- 9: B(N) – REAL (KIND=nag_wp) arrayInput
On entry: the right-hand side vector .
- 10: TOL – REAL (KIND=nag_wp)Input
: the required tolerance. Let
denote the approximate solution at iteration
the corresponding residual. The algorithm is considered to have converged at iteration
is used, where
is the machine precision
- 11: MAXITN – INTEGERInput
On entry: the maximum number of iterations allowed.
- 12: X(N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: an initial approximation to the solution vector .
On exit: an improved approximation to the solution vector .
- 13: RNORM – REAL (KIND=nag_wp)Output
: the final value of the residual norm
is the output value of ITN
- 14: ITN – INTEGEROutput
On exit: the number of iterations carried out.
- 15: WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace
- 16: LWORK – INTEGERInput
: the dimension of the array WORK
as declared in the (sub)program from which F11JEF is called.
- if , ;
- if , .
where for or , and otherwise.
- 17: IWORK() – INTEGER arrayWorkspace
- 18: 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 ,|
|or||LWORK too small.|
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.
On entry, the matrix has a zero diagonal element. Jacobi and SSOR preconditioners are not appropriate for this problem.
The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations could not improve the result.
Required accuracy not obtained in MAXITN
The preconditioner appears not to be positive definite.
The matrix of the coefficients appears not to be positive definite (conjugate gradient method only).
- (F11GDF, F11GEF or F11GFF)
A serious error has occurred in an internal call to one of the specified routines. Check all subroutine calls and array sizes. Seek expert help.
On successful termination, the final residual
, satisfies the termination criterion
The value of the final residual norm is returned in RNORM
The time taken by F11JEF for each iteration is roughly proportional to NNZ
. One iteration with the Lanczos method (SYMMLQ) requires a slightly larger number of operations than one iteration with the conjugate gradient method.
The number of iterations required to achieve a prescribed accuracy cannot be easily determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients .
This example solves a symmetric positive definite system of equations using the conjugate gradient method, with SSOR preconditioning.
9.1 Program Text
Program Text (f11jefe.f90)
9.2 Program Data
Program Data (f11jefe.d)
9.3 Program Results
Program Results (f11jefe.r)