# NAG Library Routine Document

## 1Purpose

f11jdf solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse symmetric matrix, represented in symmetric coordinate storage format.

## 2Specification

Fortran Interface
 Subroutine f11jdf ( 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) :: a(nnz), rdiag(n), omega, y(n) Real (Kind=nag_wp), Intent (Out) :: x(n) Character (1), Intent (In) :: check
#include <nagmk26.h>
 void f11jdf_ (const Integer *n, const Integer *nnz, const double a[], const Integer irow[], const Integer icol[], const double rdiag[], const double *omega, const char *check, const double y[], double x[], Integer iwork[], Integer *ifail, const Charlen length_check)

## 3Description

f11jdf solves a system of equations
 $Mx=y$
involving the preconditioning matrix
 $M=1ω2-ω D+ω L D-1 D+ω LT$
corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)) on a linear system $Ax=b$, where $A$ is a sparse symmetric 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.
It is envisaged that a common use of f11jdf will be to carry out the preconditioning step required in the application of f11gef to sparse linear systems. For an illustration of this use of f11jdf see the example program given in Section 10.1. f11jdf is also used for this purpose by the Black Box routine f11jef.

## 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 $A$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
3:     $\mathbf{a}\left({\mathbf{nnz}}\right)$ – Real (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 f11zbf 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 these constraints (which may be imposed by a call to f11zbf):
• $1\le {\mathbf{irow}}\left(\mathit{i}\right)\le {\mathbf{n}}$ 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 $A$.
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:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the right-hand side vector $y$.
10:   $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (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 $0$, . 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  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  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 $\text{'N'}$: ${\mathbf{check}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}\ge 1$.
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$
On entry, ${\mathbf{omega}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{omega}}<2.0$
${\mathbf{ifail}}=3$
On entry, ${\mathbf{a}}\left(i\right)$ is out of order: $i=〈\mathit{\text{value}}〉$.
On entry, $\mathit{I}=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left(\mathit{I}\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{irow}}\left(\mathit{I}\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{icol}}\left(\mathit{I}\right)\ge 1$ and ${\mathbf{icol}}\left(\mathit{I}\right)\le {\mathbf{irow}}\left(\mathit{I}\right)$.
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left(i\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irow}}\left(i\right)\ge 1$ and ${\mathbf{irow}}\left(i\right)\le {\mathbf{n}}$.
On entry, the location (${\mathbf{irow}}\left(\mathit{I}\right),{\mathbf{icol}}\left(\mathit{I}\right)$) is a duplicate: $\mathit{I}=〈\mathit{\text{value}}〉$.
Consider calling f11zbf to reorder and sum or remove duplicates.
${\mathbf{ifail}}=4$
The matrix $A$ has no diagonal entry in row $〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
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

f11jdf is not threaded in any implementation.

### 9.1Timing

The time taken for a call to f11jdf is proportional to nnz.

### 9.2Use of check

It is expected that a common use of f11jdf will be to carry out the preconditioning step required in the application of f11gef to sparse symmetric linear systems. In this situation f11jdf 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 to set ${\mathbf{check}}=\text{'N'}$ for all subsequent calls.

## 10Example

This example solves a sparse symmetric linear system of equations
 $Ax=b,$
using the conjugate-gradient (CG) method with SSOR preconditioning.
The CG algorithm itself is implemented by the reverse communication routine f11gef, 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 f11xef.
• If ${\mathbf{irevcm}}=2$, a solution of the preconditioning equation $Mv=u$ is required. This is achieved by a call to f11jdf.
• If ${\mathbf{irevcm}}=4$, f11gef has completed its tasks. Either the iteration has terminated, or an error condition has arisen.
For further details see the routine document for f11gef.

### 10.1Program Text

Program Text (f11jdfe.f90)

### 10.2Program Data

Program Data (f11jdfe.d)

### 10.3Program Results

Program Results (f11jdfe.r)