NAG FL Interface
f11jef (real_​symm_​solve_​jacssor)

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1 Purpose

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.

2 Specification

Fortran Interface
Subroutine f11jef ( method, precon, n, nnz, a, irow, icol, omega, b, tol, maxitn, x, rnorm, itn, work, lwork, iwork, ifail)
Integer, Intent (In) :: n, nnz, irow(nnz), icol(nnz), maxitn, lwork
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: itn, iwork(n+1)
Real (Kind=nag_wp), Intent (In) :: a(nnz), omega, b(n), tol
Real (Kind=nag_wp), Intent (Inout) :: x(n)
Real (Kind=nag_wp), Intent (Out) :: rnorm, work(lwork)
Character (*), Intent (In) :: method
Character (1), Intent (In) :: precon
C Header Interface
#include <nag.h>
void  f11jef_ (const char *method, const char *precon, const Integer *n, const Integer *nnz, const double a[], const Integer irow[], const Integer icol[], const double *omega, const double b[], const double *tol, const Integer *maxitn, double x[], double *rnorm, Integer *itn, double work[], const Integer *lwork, Integer iwork[], Integer *ifail, const Charlen length_method, const Charlen length_precon)
The routine may be called by the names f11jef or nagf_sparse_real_symm_solve_jacssor.

3 Description

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 A 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:
For incomplete Cholesky (IC) preconditioning see f11jcf.
The matrix A is represented in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction) in the arrays a, irow and icol. The array a holds the nonzero entries in the lower triangular part of the matrix, while irow and icol hold the corresponding row and column indices.

4 References

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

5 Arguments

1: method Character(*) Input
On entry: specifies the iterative method to be used.
Conjugate gradient method.
Lanczos method (SYMMLQ).
Constraint: method='CG' or 'SYMMLQ'.
2: precon Character(1) Input
On entry: specifies the type of preconditioning to be used.
No preconditioning.
Symmetric successive-over-relaxation (SSOR).
Constraint: precon='N', 'J' or 'S'.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n1.
4: nnz Integer Input
On entry: the number of nonzero elements in the lower triangular part of the matrix A.
Constraint: 1nnzn×(n+1)/2.
5: a(nnz) Real (Kind=nag_wp) array Input
On entry: the nonzero elements of the lower triangular part of the matrix A, 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 array Input
7: icol(nnz) Integer array Input
On entry: the row and column indices of the nonzero elements supplied in array a.
irow and icol must satisfy these constraints (which may be imposed by a call to f11zbf):
  • 1irow(i)n and 1icol(i)irow(i), for i=1,2,,nnz;
  • irow(i-1)<irow(i) or irow(i-1)=irow(i) and icol(i-1)<icol(i), for i=2,3,,nnz.
8: omega Real (Kind=nag_wp) Input
On entry: if precon='S', omega is the relaxation parameter ω to be used in the SSOR method. Otherwise omega need not be initialized.
Constraint: 0.0<omega<2.0.
9: b(n) Real (Kind=nag_wp) array Input
On entry: the right-hand side vector b.
10: tol Real (Kind=nag_wp) Input
On entry: the required tolerance. Let xk denote the approximate solution at iteration k, and rk the corresponding residual. The algorithm is considered to have converged at iteration k if
If tol0.0, τ=maxε,10ε,nε is used, where ε is the machine precision. Otherwise τ=max(tol,10ε,nε) is used.
Constraint: tol<1.0.
11: maxitn Integer Input
On entry: the maximum number of iterations allowed.
Constraint: maxitn1.
12: x(n) Real (Kind=nag_wp) array Input/Output
On entry: an initial approximation to the solution vector x.
On exit: an improved approximation to the solution vector x.
13: rnorm Real (Kind=nag_wp) Output
On exit: the final value of the residual norm rk, where k is the output value of itn.
14: itn Integer Output
On exit: the number of iterations carried out.
15: work(lwork) Real (Kind=nag_wp) array Workspace
16: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f11jef is called.
  • if method='CG', lwork6×n+ν+120;
  • if method='SYMMLQ', lwork7×n+ν+120.
where ν=n for precon='J' or 'S', and 0 otherwise.
17: iwork(n+1) Integer array Workspace
18: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, lwork is too small: lwork=value. Minimum required value of lwork=value.
On entry, maxitn=value.
Constraint: maxitn1.
On entry, method=value.
Constraint: method='CG' or 'SYMMLQ'.
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
On entry, nnz=value and n=value.
Constraint: nnzn×(n+1)/2.
On entry, omega=value.
Constraint: omega>0.0 and omega<2.0.
On entry, precon'N', 'J' or 'S': precon=value.
On entry, tol=value.
Constraint: tol<1.0.
On entry, a(i) is out of order: i=value.
On entry, I=value, icol(I)=value and irow(I)=value.
Constraint: icol(I)1 and icol(I)irow(I).
On entry, i=value, irow(i)=value and n=value.
Constraint: irow(i)1 and irow(i)n.
On entry, the location (irow(I),icol(I)) is a duplicate: I=value.
A nonzero element has been supplied which does not lie in the lower triangular part of A, is out of order, or has duplicate row and column indices. Consider calling f11zbf to reorder and sum or remove duplicates.
The matrix A has a zero diagonal entry in row value.
The matrix A has no diagonal entry in row value.
Jacobi and SSOR preconditioners are not appropriate for this problem.
The required accuracy could not be obtained. However a reasonable accuracy has been achieved.
The solution has not converged after value iterations.
The preconditioner appears not to be positive definite. The computation cannot continue.
The matrix of the coefficients a appears not to be positive definite. The computation cannot continue.
A serious error has occurred in an internal call: ifail=value. Check all subroutine calls and array sizes. Seek expert help.
A serious error has occurred in an internal call: IREVCM=value. Check all subroutine calls and array sizes. Seek expert help.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

On successful termination, the final residual rk=b-Axk, where k=itn, satisfies the termination criterion
The value of the final residual norm is returned in rnorm.

8 Parallelism and Performance

f11jef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11jef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

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 A¯=M-1A.

10 Example

This example solves a symmetric positive definite system of equations using the conjugate gradient method, with SSOR preconditioning.

10.1 Program Text

Program Text (f11jefe.f90)

10.2 Program Data

Program Data (f11jefe.d)

10.3 Program Results

Program Results (f11jefe.r)