NAG Library Function Document

nag_sparse_nsym_matvec (f11xac)

+− 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_nsym_matvec (f11xac) computes a matrix-vector or transposed matrix-vector product involving a real sparse nonsymmetric matrix stored in coordinate storage format.

2 Specification

#include <nag.h>

#include <nagf11.h>

void	nag_sparse_nsym_matvec (Nag_TransType trans, Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], Nag_SparseNsym_CheckData check, const double x[], double y[], NagError *fail)

3 Description

nag_sparse_nsym_matvec (f11xac) computes either the matrix-vector product

y = A x

, or the transposed matrix-vector product

y = A^{T} x

, according to the value of the argument trans, where

A

is an

n

n

sparse nonsymmetric matrix, of arbitrary sparsity pattern. The matrix

A

is stored in coordinate storage (CS) format (see Section 2.1.1 in the f11 Chapter Introduction). The array a stores all nonzero elements 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_nsym_matvec (f11xac) will be to compute the matrix-vector product required in the application of nag_sparse_nsym_basic_solver (f11bec) to sparse linear systems. An illustration of this usage appears in Section 10 in nag_sparse_nsym_precon_ssor_solve (f11ddc).

4 References

None.

5 Arguments

1: trans – Nag_TransTypeInput

On entry: specifies whether or not the matrix

A

is transposed.

$trans = Nag_NoTrans$: $y = A x$ is computed.
$trans = Nag_Trans$: $y = A^{T} x$ is computed.

Constraint:

trans = Nag_NoTrans

Nag_Trans

2: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

3: nnz – IntegerInput

On entry: the number of nonzero elements in the matrix

A

Constraint:

1 \leq nnz \leq n^{2}

4: a[nnz] – const doubleInput

On entry: the nonzero elements in 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_nsym_sort (f11zac) may be used to order the elements in this way.

5: irow[nnz] – const IntegerInput

6: 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 the following constraints (which may be imposed by a call to nag_sparse_nsym_sort (f11zac)):

$1 \leq irow [i] \leq n$ and $1 \leq icol [i] \leq n$ , 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 $i = 1, 2, \dots, nnz - 1$ .

7: check – Nag_SparseNsym_CheckDataInput

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

A

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

$check = Nag_SparseNsym_Check$: Checks are carried on the values of n, nnz, irow and icol.
$check = Nag_SparseNsym_NoCheck$: 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: $nnz \leq n^{2}$ .
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_CS: On entry, $i = ⟨value⟩$ , $icol [i - 1] = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $icol [i - 1] \geq 1$ and $icol [i - 1] \leq n$ .
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_nsym_sort (f11zac) to reorder and sum or remove duplicates.

7 Accuracy

The computed vector

y

satisfies the error bound:

${‖y - A x‖}_{\infty} \leq c (n) ε {‖A‖}_{\infty} {‖x‖}_{\infty}$ , if $trans = Nag_NoTrans$ , or
${‖y - A^{T} x‖}_{\infty} \leq c (n) ε {‖A^{T}‖}_{\infty} {‖x‖}_{\infty}$ , if $trans = Nag_Trans$ ,

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8 Parallelism and Performance

nag_sparse_nsym_matvec (f11xac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sparse_nsym_matvec (f11xac) 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_nsym_matvec (f11xac) is proportional to nnz.

9.2 Use of check

It is expected that a common use of nag_sparse_nsym_matvec (f11xac) will be to compute the matrix-vector product required in the application of nag_sparse_nsym_basic_solver (f11bec) to sparse linear systems. In this situation nag_sparse_nsym_matvec (f11xac) 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_SparseNsym_Check

for the first of such calls, and to set

check = Nag_SparseNsym_NoCheck

for all subsequent calls.

10 Example

This example reads in a sparse matrix

A

and a vector

x

. It then calls nag_sparse_nsym_matvec (f11xac) to compute the matrix-vector product

y = A x

and the transposed matrix-vector product

y = A^{T} x

NAG Library Function Documentnag_sparse_nsym_matvec (f11xac)

+− 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

NAG Library Function Document

nag_sparse_nsym_matvec (f11xac)