void	f11xec (Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], Nag_SparseSym_CheckData check, const double x[], double y[], NagError *fail)

The function may be called by the names: f11xec, nag_sparse_real_symm_matvec or nag_sparse_sym_matvec.

3 Description

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

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: $a [nnz]$ – const double 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 function f11zbc may be used to order the elements in this way.

4: $irow [nnz]$ – const Integer Input

5: $icol [nnz]$ – const Integer Input

On entry: the row and column indices of the nonzero elements supplied in array a.

Constraints:

irow and icol must satisfy these constraints (which may be imposed by a call to 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_CheckData Input

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

A

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

$check = Nag_SparseSym_Check$: 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

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 f11zbc 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: $nnz \leq n \times (n + 1) / 2$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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_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},

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8 Parallelism and Performance

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

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 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 f11xec is proportional to nnz.

9.2 Use of check

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

y = A x

f11xe: FL CL CPP AD

NAG CL Interfacef11xec (real_​symm_​matvec)

▸▿ Contents

1 Purpose

2 Specification