# NAG Library Routine Document

## 1Purpose

f11xef computes a matrix-vector product involving a real sparse symmetric matrix stored in symmetric coordinate storage format.

## 2Specification

Fortran Interface
 Subroutine f11xef ( n, nnz, a, irow, icol, x, y,
 Integer, Intent (In) :: n, nnz, irow(nnz), icol(nnz) Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a(nnz), x(n) Real (Kind=nag_wp), Intent (Out) :: y(n) Character (1), Intent (In) :: check
#include nagmk26.h
 void f11xef_ (const Integer *n, const Integer *nnz, const double a[], const Integer irow[], const Integer icol[], const char *check, const double x[], double y[], Integer *ifail, const Charlen length_check)

## 3Description

f11xef computes the matrix-vector product
 $y=Ax$
where $A$ is an $n$ by $n$ symmetric sparse matrix, of arbitrary sparsity pattern, stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction). The array a stores all nonzero elements in the lower triangular part of $A$, while arrays irow and icol store the corresponding row and column indices respectively.
It is envisaged that a common use of f11xef will be to compute the matrix-vector product required in the application of f11gef to sparse symmetric linear systems. An illustration of this usage appears in f11jdf.

None.

## 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:     $\mathbf{check}$ – Character(1)Input
On entry: specifies whether or not the SCS representation of the matrix $A$, values of n, nnz, irow and icol should be checked.
${\mathbf{check}}=\text{'C'}$
Checks are carried out on the values of n, nnz, irow and icol.
${\mathbf{check}}=\text{'N'}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
7:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector $x$.
8:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the vector $y$.
9:     $\mathbf{ifail}$ – 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 ${\mathbf{nnz}}>{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
${\mathbf{ifail}}=3$
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 f11zbf to reorder and sum or remove duplicates.
${\mathbf{ifail}}=-99$
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 vector $y$ satisfies the error bound
 $y-Ax∞≤cnεA∞x∞,$
where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f11xef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11xef 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

### 9.1Timing

The time taken for a call to f11xef is proportional to nnz.

### 9.2Use of check

It is expected that a common use of f11xef will be to compute the matrix-vector product required in the application of f11gef to sparse symmetric linear systems. In this situation f11xef is likely to be called many times with the same matrix $A$. 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 reads in a symmetric positive definite sparse matrix $A$ and a vector $x$. It then calls f11xef to compute the matrix-vector product $y=Ax$.

### 10.1Program Text

Program Text (f11xefe.f90)

### 10.2Program Data

Program Data (f11xefe.d)

### 10.3Program Results

Program Results (f11xefe.r)

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