# NAG Library Routine Document

## 1Purpose

f11drf solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse non-Hermitian matrix, represented in coordinate storage format.

## 2Specification

Fortran Interface
 Subroutine f11drf ( n, nnz, a, irow, icol, y, x,
 Integer, Intent (In) :: n, nnz, irow(nnz), icol(nnz) Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iwork(2*n+1) Real (Kind=nag_wp), Intent (In) :: omega Complex (Kind=nag_wp), Intent (In) :: a(nnz), rdiag(n), y(n) Complex (Kind=nag_wp), Intent (Out) :: x(n) Character (1), Intent (In) :: trans, check
#include nagmk26.h
 void f11drf_ ( const char *trans, const Integer *n, const Integer *nnz, const Complex a[], const Integer irow[], const Integer icol[], const Complex rdiag[], const double *omega, const char *check, const Complex y[], Complex x[], Integer iwork[], Integer *ifail, const Charlen length_trans, const Charlen length_check)

## 3Description

f11drf solves a system of linear equations
 $Mx=y, or MHx=y,$
according to the value of the argument trans, where the matrix
 $M=1ω2-ω D+ω L D-1 D+ω U$
corresponds to symmetric successive-over-relaxation (SSOR) Young (1971) applied to a linear system $Ax=b$, where $A$ is a complex sparse non-Hermitian matrix stored in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction).
In the definition of $M$ given above $D$ is the diagonal part of $A$, $L$ is the strictly lower triangular part of $A$, $U$ is the strictly upper triangular part of $A$, and $\omega$ is a user-defined relaxation parameter.
It is envisaged that a common use of f11drf will be to carry out the preconditioning step required in the application of f11bsf to sparse linear systems. For an illustration of this use of f11drf see the example program given in Section 10. f11drf is also used for this purpose by the Black Box routine f11dsf.

## 4References

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

## 5Arguments

1:     $\mathbf{trans}$ – Character(1)Input
On entry: specifies whether or not the matrix $M$ is transposed.
${\mathbf{trans}}=\text{'N'}$
$Mx=y$ is solved.
${\mathbf{trans}}=\text{'T'}$
${M}^{\mathrm{H}}x=y$ is solved.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
3:     $\mathbf{nnz}$ – IntegerInput
On entry: the number of nonzero elements in the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
4:     $\mathbf{a}\left({\mathbf{nnz}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the nonzero elements in 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 routine f11znf may be used to order the elements in this way.
5:     $\mathbf{irow}\left({\mathbf{nnz}}\right)$ – Integer arrayInput
6:     $\mathbf{icol}\left({\mathbf{nnz}}\right)$ – Integer arrayInput
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 f11znf):
• $1\le {\mathbf{irow}}\left(\mathit{i}\right)\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left(\mathit{i}\right)\le {\mathbf{n}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}$;
• 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 $\mathit{i}=2,3,\dots ,{\mathbf{nnz}}$.
7:     $\mathbf{rdiag}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the elements of the diagonal matrix ${D}^{-1}$, where $D$ is the diagonal part of $A$.
8:     $\mathbf{omega}$ – Real (Kind=nag_wp)Input
On entry: the relaxation parameter $\omega$.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
9:     $\mathbf{check}$ – Character(1)Input
On entry: specifies whether or not the CS representation of the matrix $M$ should be checked.
${\mathbf{check}}=\text{'C'}$
Checks are carried on the values of n, nnz, irow, icol and omega.
${\mathbf{check}}=\text{'N'}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
10:   $\mathbf{y}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the right-hand side vector $y$.
11:   $\mathbf{x}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayOutput
On exit: the solution vector $x$.
12:   $\mathbf{iwork}\left(2×{\mathbf{n}}+1\right)$ – Integer arrayWorkspace
13:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{trans}}\ne \text{'N'}$ or $\text{'T'}$, or ${\mathbf{check}}\ne \text{'C'}$ or $\text{'N'}$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{n}}<1$, or ${\mathbf{nnz}}<1$, or ${\mathbf{nnz}}>{{\mathbf{n}}}^{2}$, or omega lies outside the interval $\left(0.0,2.0\right)$,
${\mathbf{ifail}}=3$
On entry, the arrays irow and icol fail to satisfy the following constraints:
• $1\le {\mathbf{irow}}\left(\mathit{i}\right)\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left(\mathit{i}\right)\le {\mathbf{n}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}$;
• ${\mathbf{irow}}\left(i-1\right)<{\mathbf{irow}}\left(i\right)$ or ${\mathbf{irow}}\left(i-1\right)={\mathbf{irow}}\left(i\right)$ and ${\mathbf{icol}}\left(i-1\right)<{\mathbf{icol}}\left(i\right)$, for $i=2,3,\dots ,{\mathbf{nnz}}$.
Therefore 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. Call f11znf to reorder and sum or remove duplicates.
${\mathbf{ifail}}=4$
On entry, the matrix $A$ has a zero diagonal element. The SSOR preconditioner is not appropriate for this problem.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

If ${\mathbf{trans}}=\text{'N'}$ the computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $δM≤cnεD+ωLD-1D+ωU,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. An equivalent result holds when ${\mathbf{trans}}=\text{'T'}$.

## 8Parallelism and Performance

f11drf is not threaded in any implementation.

### 9.1Timing

The time taken for a call to f11drf is proportional to nnz.

### 9.2Use of check

It is expected that a common use of f11drf will be to carry out the preconditioning step required in the application of f11bsf to sparse linear systems. In this situation f11drf is likely to be called many times with the same matrix $M$. In the interests of both reliability and efficiency, you are recommended to set ${\mathbf{check}}=\text{'C'}$ for the first of such calls, and ${\mathbf{check}}=\text{'N'}$ for all subsequent calls.

## 10Example

This example solves a complex sparse linear system of equations
 $Ax=b,$
using RGMRES with SSOR preconditioning.
The RGMRES algorithm itself is implemented by the reverse communication routine f11bsf, which returns repeatedly to the calling program with various values of the argument irevcm. This argument indicates the action to be taken by the calling program.
• If ${\mathbf{irevcm}}=1$, a matrix-vector product $v=Au$ is required. This is implemented by a call to f11xnf.
• If ${\mathbf{irevcm}}=-1$, a conjugate transposed matrix-vector product $v={A}^{\mathrm{H}}u$ is required in the estimation of the norm of $A$. This is implemented by a call to f11xnf.
• If ${\mathbf{irevcm}}=2$, a solution of the preconditioning equation $Mv=u$ is required. This is achieved by a call to f11drf.
• If ${\mathbf{irevcm}}=4$, f11bsf has completed its tasks. Either the iteration has terminated, or an error condition has arisen.
For further details see the routine document for f11bsf.

### 10.1Program Text

Program Text (f11drfe.f90)

### 10.2Program Data

Program Data (f11drfe.d)

### 10.3Program Results

Program Results (f11drfe.r)

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