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NAG Toolbox: nag_sparse_real_gen_matvec (f11xa)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

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=Ax, or the transposed matrix-vector product y=ATx, according to the value of the argument trans, where A is an n by 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=Ax is computed.
trans='T'
y=ATx is computed.
Constraint: trans='N' or 'T'.
2:     anz – 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:     irownz int64int32nag_int array
4:     icolnz 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)):
  • 1irowin and 1icolin, for i=1,2,,nz;
  • irowi-1<irowi or irowi-1=irowi and icoli-1<icoli, for i=2,3,,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.
See also Use of check.
Constraint: check='C' or 'N'.
6:     xn – double array
The vector x.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the order of the matrix A.
Constraint: n1.
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: 1nzn2.

Output Parameters

1:     yn – 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'N' or 'T',
orcheck'C' or 'N'.
   ifail=2
On entry,n<1,
ornz<1,
ornz>n2.
   ifail=3
On entry, the arrays irow and icol fail to satisfy the following constraints:
  • 1irowin and 1icolin, for i=1,2,,nz;
  • irowi-1<irowi, or irowi-1=irowi and icoli-1<icoli, for i=2,3,,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: where cn 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=Ax and the transposed matrix-vector product y=ATx.
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


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