NAG Library Function Document

1Purpose

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.

2Specification

 #include #include
 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)

3Description

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($\ell$) (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.

4References

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$\left(\ell \right)$ 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

5Arguments

1:    $\mathbf{method}$Nag_SparseNsym_MethodInput
On entry: specifies the iterative method to be used.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$
Restarted generalized minimum residual method.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_CGS}$
Conjugate gradient squared method.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$
Bi-conjugate gradient stabilized ($\ell$) method.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_TFQMR}$
Transpose-free quasi-minimal residual method.
Constraint: ${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$, $\mathrm{Nag_SparseNsym_CGS}$, $\mathrm{Nag_SparseNsym_BiCGSTAB}$ or $\mathrm{Nag_SparseNsym_TFQMR}$.
2:    $\mathbf{n}$IntegerInput
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: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{nnz}$IntegerInput
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: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
4:    $\mathbf{a}\left[{\mathbf{la}}\right]$const ComplexInput
On entry: the values returned in the array a by a previous call to nag_sparse_nherm_fac (f11dnc).
5:    $\mathbf{la}$IntegerInput
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: ${\mathbf{la}}\ge 2×{\mathbf{nnz}}$.
6:    $\mathbf{irow}\left[{\mathbf{la}}\right]$const IntegerInput
7:    $\mathbf{icol}\left[{\mathbf{la}}\right]$const IntegerInput
8:    $\mathbf{ipivp}\left[{\mathbf{n}}\right]$const IntegerInput
9:    $\mathbf{ipivq}\left[{\mathbf{n}}\right]$const IntegerInput
10:  $\mathbf{istr}\left[{\mathbf{n}}+1\right]$const IntegerInput
11:  $\mathbf{idiag}\left[{\mathbf{n}}\right]$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:  $\mathbf{b}\left[{\mathbf{n}}\right]$const ComplexInput
On entry: the right-hand side vector $b$.
13:  $\mathbf{m}$IntegerInput
On entry: if ${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$, m is the dimension of the restart subspace.
If ${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$, m is the order $\ell$ of the polynomial Bi-CGSTAB method.
Otherwise, m is not referenced.
Constraints:
• if ${\mathbf{method}}=\mathrm{Nag_SparseNsym_RGMRES}$, $0<{\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},50\right)$;
• if ${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$, $0<{\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},10\right)$.
14:  $\mathbf{tol}$doubleInput
On entry: the required tolerance. Let ${x}_{k}$ denote the approximate solution at iteration $k$, and ${r}_{k}$ the corresponding residual. The algorithm is considered to have converged at iteration $k$ if
 $rk∞≤τ×b∞+A∞xk∞.$
If ${\mathbf{tol}}\le 0.0$, $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\sqrt{\epsilon },10\epsilon ,\sqrt{n}\epsilon \right)$ is used, where $\epsilon$ is the machine precision. Otherwise $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{tol}},10\epsilon ,\sqrt{n}\epsilon \right)$ is used.
Constraint: ${\mathbf{tol}}<1.0$.
15:  $\mathbf{maxitn}$IntegerInput
On entry: the maximum number of iterations allowed.
Constraint: ${\mathbf{maxitn}}\ge 1$.
16:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexInput/Output
On entry: an initial approximation to the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
17:  $\mathbf{rnorm}$double *Output
On exit: the final value of the residual norm ${‖{r}_{k}‖}_{\infty }$, where $k$ is the output value of itn.
18:  $\mathbf{itn}$Integer *Output
On exit: the number of iterations carried out.
19:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error 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.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
The solution has not converged after $〈\mathit{\text{value}}〉$ iterations.
NE_INT
On entry, ${\mathbf{maxitn}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxitn}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{la}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{la}}\ge 2×{\mathbf{nnz}}$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$ and ${\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},〈\mathit{\text{value}}〉\right)$.
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_CS
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[i-1\right]=〈\mathit{\text{value}}〉$, and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{icol}}\left[i-1\right]\ge 1$ and ${\mathbf{icol}}\left[i-1\right]\le {\mathbf{n}}$.
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, $i=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left[i-1\right]=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irow}}\left[i-1\right]\ge 1$ and ${\mathbf{irow}}\left[i-1\right]\le {\mathbf{n}}$.
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 (f11dnc) and nag_sparse_nherm_fac_sol (f11dqc).
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_STRICTLY_INCREASING
On entry, ${\mathbf{a}}\left[i-1\right]$ is out of order: $i=〈\mathit{\text{value}}〉$.
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 (${\mathbf{irow}}\left[i-1\right],{\mathbf{icol}}\left[i-1\right]$) is a duplicate: $i=〈\mathit{\text{value}}〉$.
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, ${\mathbf{tol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tol}}<1.0$.

7Accuracy

On successful termination, the final residual ${r}_{k}=b-A{x}_{k}$, where $k={\mathbf{itn}}$, satisfies the termination criterion
 $rk∞≤τ×b∞+A∞xk∞.$
The value of the final residual norm is returned in rnorm.

8Parallelism and Performance

nag_sparse_nherm_fac_sol (f11dqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sparse_nherm_fac_sol (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.

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 $\stackrel{-}{A}={M}^{-1}A$.

10Example

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

10.1Program Text

Program Text (f11dqce.c)

10.2Program Data

Program Data (f11dqce.d)

10.3Program Results

Program Results (f11dqce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017