NAG CL Interface
f11xac (real_gen_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>

void	f11xac (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)

The function may be called by the names: f11xac, nag_sparse_real_gen_matvec or nag_sparse_nsym_matvec.

3 Description

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 \times 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 f11xac will be to compute the matrix-vector product required in the application of f11bec to sparse linear systems. An illustration of this usage appears in Section 10 in f11ddc.

4 References

None.

5 Arguments

1: $trans$ – Nag_TransType Input

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: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 1

3: $nnz$ – Integer Input

On entry: the number of nonzero elements in the matrix

A

Constraint:

1 \leq nnz \leq n^{2}

4: $a [nnz]$ – const double Input

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 f11zac may be used to order the elements in this way.

5: $irow [nnz]$ – const Integer Input

6: $icol [nnz]$ – const Integer Input

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 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_CheckData Input

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

A nonzero element has been supplied which does not lie within the matrix $A$ , is out of order, or has duplicate row and column indices. Consider calling f11zac to reorder and sum or remove duplicates.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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: $I = ⟨ value ⟩$ .

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

Background information to multithreading can be found in the Multithreading documentation.

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

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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. 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 f11xac is proportional to nnz.

9.2 Use of check

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

y = A x

and the transposed matrix-vector product

y = A^{T} x

NAG CL Interfacef11xac (real_​gen_​matvec)

▸▿ 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 CL Interface
f11xac (real_gen_matvec)