NAG CL Interface
f11dqc (complex_​gen_​solve_​ilu)

1 Purpose

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>
void  f11dqc (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)
The function may be called by the names: f11dqc, nag_sparse_complex_gen_solve_ilu or nag_sparse_nherm_fac_sol.

3 Description

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.
f11dqc uses the incomplete LU factorization determined by f11dnc as the preconditioning matrix. A call to f11dqc must always be preceded by a call to f11dnc. Alternative preconditioners for the same storage scheme are available by calling 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 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: method Nag_SparseNsym_Method Input
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: n Integer Input
On entry: n, the order of the matrix A. This must be the same value as was supplied in the preceding call to f11dnc.
Constraint: n1.
3: nnz Integer Input
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 f11dnc.
Constraint: 1nnzn2.
4: a[la] const Complex Input
On entry: the values returned in the array a by a previous call to f11dnc.
5: la Integer Input
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 f11dnc.
Constraint: la2×nnz.
6: irow[la] const Integer Input
7: icol[la] const Integer Input
8: ipivp[n] const Integer Input
9: ipivq[n] const Integer Input
10: istr[n+1] const Integer Input
11: idiag[n] const Integer Input
On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11dnc.
ipivp and ipivq are restored on exit.
12: b[n] const Complex Input
On entry: the right-hand side vector b.
13: m Integer Input
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: tol double 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
rkτ×b+Axk.  
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.
15: maxitn Integer Input
On entry: the maximum number of iterations allowed.
Constraint: maxitn1.
16: x[n] Complex Input/Output
On entry: an initial approximation to the solution vector x.
On exit: an improved approximation to the solution vector x.
17: rnorm double * Output
On exit: the final value of the residual norm rk, where k is the output value of itn.
18: itn Integer * Output
On exit: the number of iterations carried out.
19: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dqc and f11dnc.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dnc and f11dqc.
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.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_CS
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, n=value.
Constraint: irow[i-1]1 and irow[i-1]n.
NE_INVALID_CS_PRECOND
The CS representation of the preconditioner is invalid.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING
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.
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 Parallelism and Performance

f11dqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11dqc 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 f11dqc for each iteration is roughly proportional to the value of nnzc returned from the preceding call to 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.

10 Example

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

10.1 Program Text

Program Text (f11dqce.c)

10.2 Program Data

Program Data (f11dqce.d)

10.3 Program Results

Program Results (f11dqce.r)