# NAG Library Routine Document

## 1Purpose

f11jrf solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse Hermitian matrix, represented in symmetric coordinate storage format.

## 2Specification

Fortran Interface
 Subroutine f11jrf ( n, nnz, a, irow, icol, y, x,
 Integer, Intent (In) :: n, nnz, irow(nnz), icol(nnz) Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iwork(n+1) Real (Kind=nag_wp), Intent (In) :: rdiag(n), omega Complex (Kind=nag_wp), Intent (In) :: a(nnz), y(n) Complex (Kind=nag_wp), Intent (Out) :: x(n) Character (1), Intent (In) :: check
#include nagmk26.h
 void f11jrf_ (const Integer *n, const Integer *nnz, const Complex a[], const Integer irow[], const Integer icol[], const double rdiag[], const double *omega, const char *check, const Complex y[], Complex x[], Integer iwork[], Integer *ifail, const Charlen length_check)

## 3Description

f11jrf solves a system of equations
 $Mx=y$
involving the preconditioning matrix
 $M=1ω2-ω D+ω L D-1 D+ω LH$
corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)) on a linear system $Ax=b$, where $A$ is a sparse complex Hermitian matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 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 $A$ and $\omega$ is a user-defined relaxation parameter. Note that since $A$ is Hermitian the matrix $D$ is necessarily real.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathbf{nnz}$ – IntegerInput
On entry: the number of nonzero elements in the lower triangular part of the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
3:     $\mathbf{a}\left({\mathbf{nnz}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the nonzero elements in the lower triangular part 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 routine f11zpf may be used to order the elements in this way.
4:     $\mathbf{irow}\left({\mathbf{nnz}}\right)$ – Integer arrayInput
5:     $\mathbf{icol}\left({\mathbf{nnz}}\right)$ – Integer arrayInput
On entry: the row and column indices of the nonzero elements supplied in array a.
Constraints:
irow and icol must satisfy the following constraints (which may be imposed by a call to f11zpf):
• $1\le {\mathbf{irow}}\left(\mathit{i}\right)\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left(\mathit{i}\right)\le {\mathbf{irow}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}$;
• ${\mathbf{irow}}\left(\mathit{i}-1\right)<{\mathbf{irow}}\left(\mathit{i}\right)$ or ${\mathbf{irow}}\left(\mathit{i}-1\right)={\mathbf{irow}}\left(\mathit{i}\right)$ and ${\mathbf{icol}}\left(\mathit{i}-1\right)<{\mathbf{icol}}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,\dots ,{\mathbf{nnz}}$.
6:     $\mathbf{rdiag}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the elements of the diagonal matrix ${D}^{-1}$, where $D$ is the diagonal part of $A$. Note that since $A$ is Hermitian the elements of ${D}^{-1}$ are necessarily real.
7:     $\mathbf{omega}$ – Real (Kind=nag_wp)Input
On entry: the relaxation parameter $\omega$.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
8:     $\mathbf{check}$ – Character(1)Input
On entry: specifies whether or not the input data should be checked.
${\mathbf{check}}=\text{'C'}$
Checks are carried out 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 $\text{'N'}$.
9:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the right-hand side vector $y$.
10:   $\mathbf{x}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayOutput
On exit: the solution vector $x$.
11:   $\mathbf{iwork}\left({\mathbf{n}}+1\right)$ – Integer arrayWorkspace
12:   $\mathbf{ifail}$ – 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, ${\mathbf{check}}\ne \text{'C'}$ or $\text{'N'}$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{n}}<1$, or ${\mathbf{nnz}}<1$, or ${\mathbf{nnz}}>{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$, 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:
• $1\le {\mathbf{irow}}\left(i\right)\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left(i\right)\le {\mathbf{irow}}\left(i\right)$, for $i=1,2,\dots ,{\mathbf{nnz}}$;
• ${\mathbf{irow}}\left(i-1\right)<{\mathbf{irow}}\left(i\right)$ 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 lower triangular part of $A$, is out of order, or has duplicate row and column indices. Call f11zpf to reorder and sum or remove duplicates.
${\mathbf{ifail}}=4$
 On entry, a row of $A$ has no diagonal entry.
${\mathbf{ifail}}=-99$
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

The computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $δM≤cnεD+ωLD-1D+ωLT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f11jrf is not threaded in any implementation.

### 9.1Timing

The time taken for a call to f11jrf is proportional to nnz.

## 10Example

This example program solves the preconditioning equation $Mx=y$ for a $9$ by $9$ sparse complex Hermitian matrix $A$, given in symmetric coordinate storage (SCS) format.

### 10.1Program Text

Program Text (f11jrfe.f90)

### 10.2Program Data

Program Data (f11jrfe.d)

### 10.3Program Results

Program Results (f11jrfe.r)

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