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

C Header Interface

#include <nag.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)

The routine may be called by the names f11xef or nagf_sparse_real_symm_matvec.

3 Description

f11xef computes the matrix-vector product

y = A x

where

A

is an

n \times 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.

4 References

None.

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

2: $nnz$ – Integer Input

On entry: the number of nonzero elements in the lower triangular part of

A

Constraint:

1 \leq nnz \leq n \times (n + 1) / 2

3: $a (nnz)$ – Real (Kind=nag_wp) array Input

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: $irow (nnz)$ – Integer array Input

5: $icol (nnz)$ – Integer array Input

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):

$1 \leq irow (i) \leq n$ and $1 \leq icol (i) \leq irow (i)$ , for $i = 1, 2, \dots, nnz$ ;
$irow (i - 1) < irow (i)$ or $irow (i - 1) = irow (i)$ and $icol (i - 1) < icol (i)$ , for $i = 2, 3, \dots, nnz$ .

6: $check$ – 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.

$check ='C'$: Checks are carried out on the values of n, nnz, irow and icol.
$check ='N'$: None of these checks are carried out.

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $check = ⟨ value ⟩$ .
Constraint: $check ='C'$ or $'N'$ .

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $nnz = ⟨ value ⟩$ .
Constraint: $nnz \geq 1$ .

On entry, $nnz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $nnz \leq n \times (n + 1) / 2$ .

$ifail = 3$: On entry, $a (i)$ is out of order: $i = ⟨ value ⟩$ .

On entry, $I = ⟨ value ⟩$ , $icol (I) = ⟨ value ⟩$ and $irow (I) = ⟨ value ⟩$ .
Constraint: $icol (I) \geq 1$ and $icol (I) \leq irow (I)$ .

On entry, $i = ⟨ value ⟩$ , $irow (i) = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $irow (i) \geq 1$ and $irow (i) \leq n$ .

On entry, the location ( $irow (I), icol (I)$ ) is a duplicate: $I = ⟨ value ⟩$ .
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. Consider calling f11zbf to reorder and sum or remove duplicates.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The computed vector

y

satisfies the error bound

{‖ y - A x ‖}_{\infty} \leq c (n) ε {‖ A ‖}_{\infty} {‖ x ‖}_{\infty},

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8 Parallelism 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 Further Comments

9.1 Timing

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

9.2 Use 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

check ='C'

for the first of such calls, and to set

check ='N'

for all subsequent calls.

10 Example

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 = A x

f11xe: FL CL CPP AD

NAG FL Interfacef11xef (real_​symm_​matvec)

▸▿ Contents

1 Purpose

2 Specification