NAG Library Routine Document

f11dsf (complex_gen_solve_jacssor)


f11dsf solves a complex sparse non-Hermitian system of linear equations, represented in coordinate storage format, using a restarted generalized minimal residual (RGMRES), conjugate gradient squared (CGS), stabilized bi-conjugate gradient (Bi-CGSTAB), or transpose-free quasi-minimal residual (TFQMR) method, without preconditioning, with Jacobi, or with SSOR preconditioning.


Fortran Interface
Subroutine f11dsf ( method, precon, n, nnz, a, irow, icol, omega, b, m, tol, maxitn, x, rnorm, itn, work, lwork, iwork, ifail)
Integer, Intent (In):: n, nnz, irow(nnz), icol(nnz), m, maxitn, lwork
Integer, Intent (Inout):: ifail
Integer, Intent (Out):: itn, iwork(2*n+1)
Real (Kind=nag_wp), Intent (In):: omega, tol
Real (Kind=nag_wp), Intent (Out):: rnorm
Complex (Kind=nag_wp), Intent (In):: a(nnz), b(n)
Complex (Kind=nag_wp), Intent (Inout):: x(n)
Complex (Kind=nag_wp), Intent (Out):: work(lwork)
Character (*), Intent (In):: method
Character (1), Intent (In):: precon
C Header Interface
#include <nagmk26.h>
void  f11dsf_ (const char *method, const char *precon, const Integer *n, const Integer *nnz, const Complex a[], const Integer irow[], const Integer icol[], const double *omega, const Complex b[], const Integer *m, const double *tol, const Integer *maxitn, Complex x[], double *rnorm, Integer *itn, Complex work[], const Integer *lwork, Integer iwork[], Integer *ifail, const Charlen length_method, const Charlen length_precon)


f11dsf solves a complex sparse non-Hermitian system of linear equations:
using an RGMRES (see Saad and Schultz (1986)), CGS (see Sonneveld (1989)), Bi-CGSTAB() (see Van der Vorst (1989) and Sleijpen and Fokkema (1993)), or TFQMR (see Freund and Nachtigal (1991) and Freund (1993)) method.
f11dsf allows the following choices for the preconditioner:
For incomplete LU (ILU) preconditioning see f11dqf.
The matrix A is represented in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction) in the arrays a, irow and icol. The array a holds the nonzero entries in the matrix, while irow and icol hold the corresponding row and column indices.
f11dsf is a Black Box routine which calls f11brf, f11bsf and f11btf. If you wish to use an alternative storage scheme, preconditioner, or termination criterion, or require additional diagnostic information, you should call these underlying routines directly.


Freund R W (1993) A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems SIAM J. Sci. Comput. 14 470–482
Freund R W and Nachtigal N (1991) QMR: a Quasi-Minimal Residual Method for Non-Hermitian Linear Systems Numer. Math. 60 315–339
Saad Y and Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 7 856–869
Sleijpen G L G and Fokkema D R (1993) BiCGSTAB for linear equations involving matrices with complex spectrum ETNA 1 11–32
Sonneveld P (1989) CGS, a fast Lanczos-type solver for nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 10 36–52
Van der Vorst H (1989) Bi-CGSTAB, a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 13 631–644
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York


1:     method – Character(*)Input
On entry: specifies the iterative method to be used.
Restarted generalized minimum residual method.
Conjugate gradient squared method.
Bi-conjugate gradient stabilized () method.
Transpose-free quasi-minimal residual method.
Constraint: method='RGMRES', 'CGS', 'BICGSTAB' or 'TFQMR'.
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 – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n1.
4:     nnz – IntegerInput
On entry: the number of nonzero elements in the matrix A.
Constraint: 1nnzn2.
5:     annz – Complex (Kind=nag_wp) arrayInput
On entry: the nonzero elements 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 f11znf may be used to order the elements in this way.
6:     irownnz – Integer arrayInput
7:     icolnnz – Integer arrayInput
On entry: the row and column indices of the nonzero elements supplied in a.
irow and icol must satisfy the following constraints (which may be imposed by a call to f11znf):
  • 1irowin and 1icolin, for i=1,2,,nnz;
  • either irowi-1<irowi or both irowi-1=irowi and icoli-1<icoli, 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 and is not referenced.
Constraint: 0.0<omega<2.0.
9:     bn – Complex (Kind=nag_wp) arrayInput
On entry: the right-hand side vector b.
10:   m – IntegerInput
On entry: if method='RGMRES', m is the dimension of the restart subspace.
If method='BICGSTAB', m is the order  of the polynomial Bi-CGSTAB method.
Otherwise, m is not referenced.
  • if method='RGMRES', 0<mminn,50;
  • if method='BICGSTAB', 0<mminn,10.
11:   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=maxtol,10ε,nε is used.
Constraint: tol<1.0.
12:   maxitn – IntegerInput
On entry: the maximum number of iterations allowed.
Constraint: maxitn1.
13:   xn – Complex (Kind=nag_wp) arrayInput/Output
On entry: an initial approximation to the solution vector x.
On exit: an improved approximation to the solution vector x.
14:   rnorm – Real (Kind=nag_wp)Output
On exit: the final value of the residual norm rk, where k is the output value of itn.
15:   itn – IntegerOutput
On exit: the number of iterations carried out.
16:   worklwork – Complex (Kind=nag_wp) arrayWorkspace
17:   lwork – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f11dsf is called.
  • if method='RGMRES', lwork4×n+m×m+n+5+nu+121;
  • if method='CGS', lwork8×n+nu+120;
  • if method='BICGSTAB', lwork2×n×m+3+m×m+2+nu+120;
  • if method='TFQMR', lwork11×n+nu+120.
Where nu=n for precon='J' or 'S' and nu=0 otherwise.
18:   iwork2×n+1 – Integer arrayWorkspace
19:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1 or 1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. 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).

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, m=value and n=value.
Constraint: 0<mminn,value.
On entry, maxitn=value.
Constraint: maxitn1 
On entry, method=value.
Constraint: method='RGMRES', 'CGS' or 'BICGSTAB'.
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
On entry, nnz=value and n=value.
Constraint: 1nnzn2.
On entry, omega=value.
Constraint: 0.0<omega<2.0 
On entry, precon=value.
Constraint: precon='N', 'J' or 'S'.
On entry, tol=value.
Constraint: tol<1.0.
On entry, ai is out of order: i=value.
On entry, i=value, icoli=value and n=value.
Constraint: icoli1 and icolin.
On entry, i=value, irowi=value and n=value.
Constraint: irowi1 and irowin.
On entry, the location (irowI,icolI) is a duplicate: I=value.
A nonzero element has been supplied which does not lie in the matrix A, is out of order, or has duplicate row and column indices. Consider calling f11znf 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 may have been achieved.
You should check the output value of rnorm for acceptability. This error code usually implies that your problem has been fully and satisfactorily solved to within or close to the accuracy available on your system. Further iterations are unlikely to improve on this situation.
The solution has not converged after value iterations.
Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


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.

Parallelism and Performance

f11dsf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11dsf 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.

Further Comments

The time taken by f11dsf for each iteration is roughly proportional to nnz.
The number of iterations required to achieve a prescribed accuracy cannot easily be determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned coefficient matrix A-=M-1A, for some preconditioning matrix M.


This example solves a complex sparse non-Hermitian system of equations using the CGS method, with no preconditioning.

Program Text

Program Text (f11dsfe.f90)

Program Data

Program Data (f11dsfe.d)

Program Results

Program Results (f11dsfe.r)