nag_sparse_nherm_sol (f11dsc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_sparse_nherm_sol (f11dsc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_sparse_nherm_sol (f11dsc) 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.

2  Specification

#include <nag.h>
#include <nagf11.h>
void  nag_sparse_nherm_sol (Nag_SparseNsym_Method method, Nag_SparseNsym_PrecType precon, Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], double omega, const Complex b[], Integer m, double tol, Integer maxitn, Complex x[], double *rnorm, Integer *itn, NagError *fail)

3  Description

nag_sparse_nherm_sol (f11dsc) 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.
nag_sparse_nherm_sol (f11dsc) allows the following choices for the preconditioner:
For incomplete LU (ILU) preconditioning see nag_sparse_nherm_fac_sol (f11dqc).
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.
nag_sparse_nherm_sol (f11dsc) is a Black Box function which calls nag_sparse_nherm_basic_setup (f11brc), nag_sparse_nherm_basic_solver (f11bsc) and nag_sparse_nherm_basic_diagnostic (f11btc). If you wish to use an alternative storage scheme, preconditioner, or termination criterion, or require additional diagnostic information, you should call these underlying functions directly.

4  References

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

5  Arguments

1:     method Nag_SparseNsym_MethodInput
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=Nag_SparseNsym_RGMRES, Nag_SparseNsym_CGS, Nag_SparseNsym_BiCGSTAB or Nag_SparseNsym_TFQMR.
2:     precon Nag_SparseNsym_PrecTypeInput
On entry: specifies the type of preconditioning to be used.
No preconditioning.
Symmetric successive-over-relaxation (SSOR).
Constraint: precon=Nag_SparseNsym_NoPrec, Nag_SparseNsym_JacPrec or Nag_SparseNsym_SSORPrec.
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:     a[nnz] const ComplexInput
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 function nag_sparse_nherm_sort (f11znc) may be used to order the elements in this way.
6:     irow[nnz] const IntegerInput
7:     icol[nnz] const IntegerInput
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 nag_sparse_nherm_sort (f11znc)):
  • 1irow[i]n and 1icol[i]n, for i=0,1,,nnz-1;
  • either irow[i-1]<irow[i] or both irow[i-1]=irow[i] and icol[i-1]<icol[i], for i=1,2,,nnz-1.
8:     omega doubleInput
On entry: if precon=Nag_SparseNsym_SSORPrec, 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:     b[n] const ComplexInput
On entry: the right-hand side vector b.
10:   m IntegerInput
On entry: if method=Nag_SparseNsym_RGMRES, m is the dimension of the restart subspace.
If method=Nag_SparseNsym_BiCGSTAB, m is the order  of the polynomial Bi-CGSTAB method.
Otherwise, m is not referenced.
  • if method=Nag_SparseNsym_RGMRES, 0<mminn,50;
  • if method=Nag_SparseNsym_BiCGSTAB, 0<mminn,10.
11:   tol doubleInput
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:   x[n] ComplexInput/Output
On entry: an initial approximation to the solution vector x.
On exit: an improved approximation to the solution vector x.
14:   rnorm double *Output
On exit: the final value of the residual norm rk, where k is the output value of itn.
15:   itn Integer *Output
On exit: the number of iterations carried out.
16:   fail NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.
Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
Dynamic memory allocation failed.
See Section in the Essential Introduction for further information.
On entry, argument value had an illegal value.
The solution has not converged after value iterations.
On entry, m=value and n=value.
Constraint: 0<mminn,value.
On entry, method=value, n=value and m=value.
Constraint: if method=Nag_SparseNsym_BiCGSTAB, 0<mminn,10.
On entry, method=value, n=value and m=value.
Constraint: if method=Nag_SparseNsym_RGMRES, 0<mminn,50.
On entry, maxitn=value.
Constraint: maxitn1 
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
On entry, nnz=value and n=value.
Constraint: 1nnzn2.
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 function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
On entry, i=value, icol[i-1]=value and n=value.
Constraint: icol[i-1]1 and icol[i-1]n.
On entry, i=value, irow[i-1]=value and n=value.
Constraint: irow[i-1]1 and irow[i-1]n.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
On entry, a[i-1] is out of order: i=value.
On entry, the location (irow[I-1],icol[I-1]) is a duplicate: I=value.
On entry, omega=value.
Constraint: 0.0<omega<2.0 
On entry, tol=value.
Constraint: tol<1.0.
The matrix A has a zero diagonal entry in row value.
The matrix A has no diagonal entry in row value.

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

nag_sparse_nherm_sol (f11dsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sparse_nherm_sol (f11dsc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The time taken by nag_sparse_nherm_sol (f11dsc) 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.

10  Example

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

10.1  Program Text

Program Text (f11dsce.c)

10.2  Program Data

Program Data (f11dsce.d)

10.3  Program Results

Program Results (f11dsce.r)

nag_sparse_nherm_sol (f11dsc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015