# NAG C Library Function Document

## 1Purpose

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.

## 2Specification

 #include #include
 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)

## 3Description

nag_sparse_nherm_sol (f11dsc) solves a complex sparse non-Hermitian system of linear equations:
 $Ax=b,$
using an 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_sol (f11dsc) allows the following choices for the preconditioner:
• – no preconditioning;
• – Jacobi preconditioning (see Young (1971));
• – symmetric successive-over-relaxation (SSOR) preconditioning (see Young (1971)).
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.

## 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
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

## 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.
${\mathbf{method}}=\mathrm{Nag_SparseNsym_CGS}$
${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$
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:    $\mathbf{precon}$Nag_SparseNsym_PrecTypeInput
On entry: specifies the type of preconditioning to be used.
${\mathbf{precon}}=\mathrm{Nag_SparseNsym_NoPrec}$
No preconditioning.
${\mathbf{precon}}=\mathrm{Nag_SparseNsym_JacPrec}$
Jacobi.
${\mathbf{precon}}=\mathrm{Nag_SparseNsym_SSORPrec}$
Symmetric successive-over-relaxation (SSOR).
Constraint: ${\mathbf{precon}}=\mathrm{Nag_SparseNsym_NoPrec}$, $\mathrm{Nag_SparseNsym_JacPrec}$ or $\mathrm{Nag_SparseNsym_SSORPrec}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
4:    $\mathbf{nnz}$IntegerInput
On entry: the number of nonzero elements in the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
5:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$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:    $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const IntegerInput
7:    $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const IntegerInput
On entry: the row and column indices of the nonzero elements supplied in a.
Constraints:
irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_nherm_sort (f11znc)):
• $1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left[\mathit{i}\right]\le {\mathbf{n}}$, for $\mathit{i}=0,1,\dots ,{\mathbf{nnz}}-1$;
• either ${\mathbf{irow}}\left[\mathit{i}-1\right]<{\mathbf{irow}}\left[\mathit{i}\right]$ or both ${\mathbf{irow}}\left[\mathit{i}-1\right]={\mathbf{irow}}\left[\mathit{i}\right]$ and ${\mathbf{icol}}\left[\mathit{i}-1\right]<{\mathbf{icol}}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
8:    $\mathbf{omega}$doubleInput
On entry: if ${\mathbf{precon}}=\mathrm{Nag_SparseNsym_SSORPrec}$, omega is the relaxation parameter $\omega$ to be used in the SSOR method. Otherwise omega need not be initialized and is not referenced.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
9:    $\mathbf{b}\left[{\mathbf{n}}\right]$const ComplexInput
On entry: the right-hand side vector $b$.
10:  $\mathbf{m}$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 ${\mathbf{method}}=\mathrm{Nag_SparseNsym_BiCGSTAB}$, $0<{\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},10\right)$.
11:  $\mathbf{tol}$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: ${\mathbf{tol}}<1.0$.
12:  $\mathbf{maxitn}$IntegerInput
On entry: the maximum number of iterations allowed.
Constraint: ${\mathbf{maxitn}}\ge 1$.
13:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexInput/Output
On entry: an initial approximation to the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
14:  $\mathbf{rnorm}$double *Output
On exit: the final value of the residual norm ${‖{r}_{k}‖}_{\infty }$, where $k$ is the output value of itn.
15:  $\mathbf{itn}$Integer *Output
On exit: the number of iterations carried out.
16:  $\mathbf{fail}$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

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.
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 nag_sparse_nherm_sort (f11znc) to reorder and sum or remove duplicates.
Jacobi and SSOR preconditioners are not appropriate for this problem.
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_ENUM_INT_2
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0<{\mathbf{m}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},〈\mathit{\text{value}}〉\right)$.
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, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
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 ${\mathbf{icol}}\left[i-1\right]\le {\mathbf{n}}$.
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left[i-1\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irow}}\left[i-1\right]\ge 1$ and ${\mathbf{irow}}\left[i-1\right]\le {\mathbf{n}}$.
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: $i=〈\mathit{\text{value}}〉$.
On entry, the location (${\mathbf{irow}}\left[\mathit{I}-1\right],{\mathbf{icol}}\left[\mathit{I}-1\right]$) is a duplicate: $\mathit{I}=〈\mathit{\text{value}}〉$.
NE_REAL
On entry, ${\mathbf{omega}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{omega}}<2.0$
On entry, ${\mathbf{tol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tol}}<1.0$.
NE_ZERO_DIAG_ELEM
The matrix $A$ has a zero diagonal entry in row $〈\mathit{\text{value}}〉$.
The matrix $A$ has no diagonal entry in row $〈\mathit{\text{value}}〉$.

## 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_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.

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 $\stackrel{-}{A}={M}^{-1}A$, for some preconditioning matrix $M$.

## 10Example

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

### 10.1Program Text

Program Text (f11dsce.c)

### 10.2Program Data

Program Data (f11dsce.d)

### 10.3Program Results

Program Results (f11dsce.r)