NAG Library Function Document

nag_sparse_sym_matvec (f11xec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

+− 9 Further Comments

9.1 Timing

9.2 Use of check

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_sparse_sym_matvec (f11xec) computes a matrix-vector product involving a real sparse symmetric matrix stored in symmetric coordinate storage format.

2 Specification

#include <nag.h>

#include <nagf11.h>

void	nag_sparse_sym_matvec (Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], Nag_SparseSym_CheckData check, const double x[], double y[], NagError *fail)

3 Description

nag_sparse_sym_matvec (f11xec) computes the matrix-vector product

y = A x

where

A

is an

n

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 nag_sparse_sym_matvec (f11xec) will be to compute the matrix-vector product required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse symmetric linear systems. An illustration of this usage appears in nag_sparse_sym_precon_ssor_solve (f11jdc).

4 References

None.

5 Arguments

1: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

2: nnz – IntegerInput

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] – const doubleInput

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 function nag_sparse_sym_sort (f11zbc) may be used to order the elements in this way.

4: irow[nnz] – const IntegerInput

5: icol[nnz] – const IntegerInput

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 nag_sparse_sym_sort (f11zbc)):

$1 \leq irow [i] \leq n$ and $1 \leq icol [i] \leq irow [i]$ , for $i = 0, 1, \dots, nnz - 1$ ;
$irow [i - 1] < irow [i]$ or $irow [i - 1] = irow [i]$ and $icol [i - 1] < icol [i]$ , for $i = 1, 2, \dots, nnz - 1$ .

6: check – Nag_SparseSym_CheckDataInput

On entry: specifies whether or not the SCS representation of the matrix

A

, values of n, nnz, irow and icol should be checked.

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

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
On entry, $nnz = ⟨value⟩$ .
Constraint: $nnz \geq 1$ .
NE_INT_2: On entry, $nnz = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $nnz \leq n \times (n + 1) / 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_SCS: On entry, $I = ⟨value⟩$ , $icol [I - 1] = ⟨value⟩$ and $irow [I - 1] = ⟨value⟩$ .
Constraint: $icol [I - 1] \geq 1$ and $icol [I - 1] \leq irow [I - 1]$ .
On entry, $i = ⟨value⟩$ , $irow [i - 1] = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $irow [i - 1] \geq 1$ and $irow [i - 1] \leq n$ .
NE_NOT_STRICTLY_INCREASING: On entry, $a [i - 1]$ is out of order: $i = ⟨value⟩$ .
On entry, the location ( $irow [I - 1], icol [I - 1]$ ) is a duplicate: $I = ⟨value⟩$ . Consider calling nag_sparse_sym_sort (f11zbc) to reorder and sum or remove duplicates.

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

nag_sparse_sym_matvec (f11xec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sparse_sym_matvec (f11xec) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

9.1 Timing

The time taken for a call to nag_sparse_sym_matvec (f11xec) is proportional to nnz.

9.2 Use of check

It is expected that a common use of nag_sparse_sym_matvec (f11xec) will be to compute the matrix-vector product required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse symmetric linear systems. In this situation nag_sparse_sym_matvec (f11xec) 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 = Nag_SparseSym_Check

for the first of such calls, and to set

check = Nag_SparseSym_NoCheck

for all subsequent calls.

10 Example

This example reads in a symmetric positive definite sparse matrix

A

and a vector

x

. It then calls nag_sparse_sym_matvec (f11xec) to compute the matrix-vector product

y = A x

NAG Library Function Documentnag_sparse_sym_matvec (f11xec)