nag_sparse_nherm_fac_sol (f11dqc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_sparse_nherm_fac_sol (f11dqc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_sparse_nherm_fac_sol (f11dqc) 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, with incomplete LU preconditioning.

2  Specification

#include <nag.h>
#include <nagf11.h>
void  nag_sparse_nherm_fac_sol (Nag_SparseNsym_Method method, Integer n, Integer nnz, const Complex a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipivp[], const Integer ipivq[], const Integer istr[], const Integer idiag[], const Complex b[], Integer m, double tol, Integer maxitn, Complex x[], double *rnorm, Integer *itn, NagError *fail)

3  Description

nag_sparse_nherm_fac_sol (f11dqc) solves a complex sparse non-Hermitian linear system of equations
Ax=b,
using a preconditioned 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_fac_sol (f11dqc) uses the incomplete LU factorization determined by nag_sparse_nherm_fac (f11dnc) as the preconditioning matrix. A call to nag_sparse_nherm_fac_sol (f11dqc) must always be preceded by a call to nag_sparse_nherm_fac (f11dnc). Alternative preconditioners for the same storage scheme are available by calling nag_sparse_nherm_sol (f11dsc).
The matrix A, and the preconditioning matrix M, are represented in coordinate storage (CS) format (see Section 2.1.1 in the f11 Chapter Introduction) in the arrays a, irow and icol, as returned from nag_sparse_nherm_fac (f11dnc). The array a holds the nonzero entries in these matrices, while irow and icol hold the corresponding row and column indices.

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

5  Arguments

1:     methodNag_SparseNsym_MethodInput
On entry: specifies the iterative method to be used.
method=Nag_SparseNsym_RGMRES
Restarted generalized minimum residual method.
method=Nag_SparseNsym_CGS
Conjugate gradient squared method.
method=Nag_SparseNsym_BiCGSTAB
Bi-conjugate gradient stabilized () method.
method=Nag_SparseNsym_TFQMR
Transpose-free quasi-minimal residual method.
Constraint: method=Nag_SparseNsym_RGMRES, Nag_SparseNsym_CGS, Nag_SparseNsym_BiCGSTAB or Nag_SparseNsym_TFQMR.
2:     nIntegerInput
On entry: n, the order of the matrix A. This must be the same value as was supplied in the preceding call to nag_sparse_nherm_fac (f11dnc).
Constraint: n1.
3:     nnzIntegerInput
On entry: the number of nonzero elements in the matrix A. This must be the same value as was supplied in the preceding call to nag_sparse_nherm_fac (f11dnc).
Constraint: 1nnzn2.
4:     a[la]const ComplexInput
On entry: the values returned in the array a by a previous call to nag_sparse_nherm_fac (f11dnc).
5:     laIntegerInput
On entry: the dimension of the arrays a, irow and icol. This must be the same value as was supplied in the preceding call to nag_sparse_nherm_fac (f11dnc).
Constraint: la2×nnz.
6:     irow[la]const IntegerInput
7:     icol[la]const IntegerInput
8:     ipivp[n]const IntegerInput
9:     ipivq[n]const IntegerInput
10:   istr[n+1]const IntegerInput
11:   idiag[n]const IntegerInput
On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_nherm_fac (f11dnc).
ipivp and ipivq are restored on exit.
12:   b[n]const ComplexInput
On entry: the right-hand side vector b.
13:   mIntegerInput
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.
Constraints:
  • if method=Nag_SparseNsym_RGMRES, 0<mminn,50;
  • if method=Nag_SparseNsym_BiCGSTAB, 0<mminn,10.
14:   toldoubleInput
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
rkτ×b+Axk.
If tol0.0, τ=maxε,nε is used, where ε is the machine precision. Otherwise τ=maxtol,10ε,nε is used.
Constraint: tol<1.0.
15:   maxitnIntegerInput
On entry: the maximum number of iterations allowed.
Constraint: maxitn1.
16:   x[n]ComplexInput/Output
On entry: an initial approximation to the solution vector x.
On exit: an improved approximation to the solution vector x.
17:   rnormdouble *Output
On exit: the final value of the residual norm rk, where k is the output value of itn.
18:   itnInteger *Output
On exit: the number of iterations carried out.
19:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ACCURACY
The required accuracy could not be obtained. However a reasonable accuracy may have been achieved.
NE_ALG_FAIL
Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
The solution has not converged after value iterations.
NE_INT
On entry, maxitn=value.
Constraint: maxitn1.
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
NE_INT_2
On entry, la=value and nnz=value.
Constraint: la2×nnz.
On entry, m=value and n=value.
Constraint: m1 and mminn,value.
On entry, nnz=value and n=value.
Constraint: nnzn2.
NE_INTERNAL_ERROR
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.
NE_INVALID_CS
On entry, i=value, icol[i-1]=value and n=value.
Constraint: icol[i-1]1 and icol[i-1]n.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).
On entry, i=value, irow[i-1]=value and n=value.
Constraint: irow[i-1]1 and irow[i-1]n.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).
NE_INVALID_CS_PRECOND
The CS representation of the preconditioner is invalid.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).
NE_NOT_STRICTLY_INCREASING
On entry, a[i-1] is out of order: i=value.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).
On entry, the location (irow[i-1],icol[i-1]) is a duplicate: i=value.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_fac_sol (f11dqc) and nag_sparse_nherm_fac (f11dnc).
NE_REAL
On entry, tol=value.
Constraint: tol<1.0.

7  Accuracy

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

8  Further Comments

The time taken by nag_sparse_nherm_fac_sol (f11dqc) for each iteration is roughly proportional to the value of nnzc returned from the preceding call to nag_sparse_nherm_fac (f11dnc).
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 coefficient matrix A-=M-1A.

9  Example

This example solves a complex sparse non-Hermitian linear system of equations using the CGS method, with incomplete LU preconditioning.

9.1  Program Text

Program Text (f11dqce.c)

9.2  Program Data

Program Data (f11dqce.d)

9.3  Program Results

Program Results (f11dqce.r)


nag_sparse_nherm_fac_sol (f11dqc) (PDF version)
f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012