nag_sparse_real_gen_matvec (f11xa) computes a matrix-vector or transposed matrix-vector product involving a real sparse nonsymmetric matrix stored in coordinate storage format.

Syntax

[y, ifail] = f11xa(trans, a, irow, icol, check, x, 'n', n, 'nz', nz)

[y, ifail] = nag_sparse_real_gen_matvec(trans, a, irow, icol, check, x, 'n', n, 'nz', nz)

Description

nag_sparse_real_gen_matvec (f11xa) 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 Coordinate storage (CS) format 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_real_gen_matvec (f11xa) will be to compute the matrix-vector product required in the application of nag_sparse_real_gen_basic_solver (f11be) to sparse linear systems. An illustration of this usage appears in Example in nag_sparse_real_gen_precon_ssor_solve (f11dd).

References

None.

Parameters

Compulsory Input Parameters

1: $trans$ – string (length ≥ 1)

Specifies whether or not the matrix

A

is transposed.

$trans ='N'$: $y = A x$ is computed.
$trans ='T'$: $y = A^{T} x$ is computed.

Constraint:

trans ='N'

'T'

2: $a (nz)$ – double array

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_real_gen_sort (f11za) may be used to order the elements in this way.

3: $irow (nz)$ – int64int32nag_int array

4: $icol (nz)$ – int64int32nag_int array

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_real_gen_sort (f11za)):

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

5: $check$ – string (length ≥ 1)

Specifies whether or not the CS representation of the matrix

A

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

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

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array x.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 1$ .
2: $nz$ – int64int32nag_int scalar: Default: the dimension of the arrays a, irow, icol. (An error is raised if these dimensions are not equal.)
The number of nonzero elements in the matrix $A$ .

Constraint: $1 \leq nz \leq n^{2}$ .

Output Parameters

1: $y (n)$ – double array: The vector $y$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$trans \neq'N'$ or $'T'$ ,
or	$check \neq'C'$ or $'N'$ .

$ifail = 2$

On entry,	$n < 1$ ,
or	$nz < 1$ ,
or	$nz > n^{2}$ .

$ifail = 3$

On entry, the arrays irow and icol fail to satisfy the following constraints:

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

Therefore 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. Call nag_sparse_real_gen_sort (f11za) to reorder and sum or remove duplicates.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The computed vector

y

satisfies the error bound:

${‖y - A x‖}_{\infty} \leq c (n) ε {‖A‖}_{\infty} {‖x‖}_{\infty}$ , if $trans ='N'$ , or
${‖y - A^{T} x‖}_{\infty} \leq c (n) ε {‖A^{T}‖}_{\infty} {‖x‖}_{\infty}$ , if $trans ='T'$ ,

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision

Further Comments

Timing

The time taken for a call to nag_sparse_real_gen_matvec (f11xa) is proportional to nz.

Use of check

It is expected that a common use of nag_sparse_real_gen_matvec (f11xa) will be to compute the matrix-vector product required in the application of nag_sparse_real_gen_basic_solver (f11be) to sparse linear systems. In this situation nag_sparse_real_gen_matvec (f11xa) 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.

Example

This example reads in a sparse matrix

A

and a vector

x

. It then calls nag_sparse_real_gen_matvec (f11xa) to compute the matrix-vector product

y = A x

and the transposed matrix-vector product

y = A^{T} x

Open in the MATLAB editor: f11xa_example

function f11xa_example


fprintf('f11xa example results\n\n');

% Sparse Matrix vector Products y=Ax, y = A^Tx

% 5x5 sparse matrix A and vector x
a    =         [2; 1; 1;-1; 4; 1; 1; 1; 2;-2; 3];
irow = int64([1; 1; 2; 2; 3; 3; 3; 4; 4; 5; 5]);
icol = int64([1; 2; 3; 4; 1; 3; 5; 4; 5; 2; 5]);

x    = [0.7; 0.16; 0.52; 0.77; 0.28];

% Calculate matrix-vector product
trans = 'N';
check = 'C';
[y, ifail] = f11xa( ...
                    trans, a, irow, icol, check, x);
fprintf('Matrix-vector product:\n');
disp(y);

% Calculate transposed matrix-vector product
trans = 'T';
[y, ifail] = f11xa( ...
                    trans, a, irow, icol, check, x);
fprintf('Transposed matrix-vector product:\n');
disp(y);

f11xa example results

Matrix-vector product:
    1.5600
   -0.2500
    3.6000
    1.3300
    0.5200

Transposed matrix-vector product:
    3.4800
    0.1400
    0.6800
    0.6100
    2.9000

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox