NAG Toolbox: nag_sparse_real_gen_solve_jacssor (f11de)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{method}$ – string

The iterative method to be used.

$\mathbf{method}=\text{'RGMRES'}$

Restarted generalized minimum residual method.

$\mathbf{method}=\text{'CGS'}$

Conjugate gradient squared method.

$\mathbf{method}=\text{'BICGSTAB'}$

Bi-conjugate gradient stabilized ($\ell$) method.

$\mathbf{method}=\text{'TFQMR'}$

Transpose-free quasi-minimal residual method.

Constraint: $\mathbf{method}=\text{'RGMRES'}$, $\text{'CGS'}$, $\text{'BICGSTAB'}$ or $\text{'TFQMR'}$.

2:     $\mathrm{precon}$ – string (length ≥ 1)

Specifies the type of preconditioning to be used.

$\mathbf{precon}=\text{'N'}$

No preconditioning.

$\mathbf{precon}=\text{'J'}$

Jacobi.

$\mathbf{precon}=\text{'S'}$

Symmetric successive-over-relaxation.

Constraint: $\mathbf{precon}=\text{'N'}$, $\text{'J'}$ or $\text{'S'}$.

3:     $\mathrm{a}\left(\mathbf{nz}\right)$ – double array

The nonzero elements 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 nag_sparse_real_gen_sort (f11za) may be used to order the elements in this way.

4:     $\mathrm{irow}\left(\mathbf{nz}\right)$ – int64int32nag_int array

5:     $\mathrm{icol}\left(\mathbf{nz}\right)$ – int64int32nag_int array

The row and column indices of the nonzero elements supplied in a.

Constraints:

irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_real_gen_sort (f11za)):

• $1\le \mathbf{irow}\left(\mathit{i}\right)\le \mathbf{n}$ and $1\le \mathbf{icol}\left(\mathit{i}\right)\le \mathbf{n}$, for $\mathit{i}=1,2,\dots ,\mathbf{nz}$;
• $\mathbf{irow}\left(\mathit{i}-1\right)<\mathbf{irow}\left(\mathit{i}\right)$ or $\mathbf{irow}\left(\mathit{i}-1\right)=\mathbf{irow}\left(\mathit{i}\right)$ and $\mathbf{icol}\left(\mathit{i}-1\right)<\mathbf{icol}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,\dots ,\mathbf{nz}$.

6:     $\mathrm{omega}$ – double scalar

If $\mathbf{precon}=\text{'S'}$, omega is the relaxation parameter $\omega$ to be used in the SSOR method. Otherwise omega need not be initialized and is not referenced.

Constraint: $0.0<\mathbf{omega}<2.0$.

7:     $\mathrm{b}\left(\mathbf{n}\right)$ – double array

The right-hand side vector $b$.

8:     $\mathrm{m}$ – int64int32nag_int scalar

If $\mathbf{method}=\text{'RGMRES'}$, m is the dimension of the restart subspace.

If $\mathbf{method}=\text{'BICGSTAB'}$, m is the order $\ell$ of the polynomial Bi-CGSTAB method.

Otherwise, m is not referenced.

Constraints:

• if $\mathbf{method}=\text{'RGMRES'}$, $0<\mathbf{m}\le \min \left(\mathbf{n},50\right)$;
• if $\mathbf{method}=\text{'BICGSTAB'}$, $0<\mathbf{m}\le \min \left(\mathbf{n},10\right)$.

9:     $\mathrm{tol}$ – double scalar

The required tolerance. Let $x_k$ denote the approximate solution at iteration $k$, and $r_k$ the corresponding residual. The algorithm is considered to have converged at iteration $k$ if

$${‖r_k‖}_{\infty }\le \tau \times \left({‖b‖}_{\infty }+{‖A‖}_{\infty }{‖x_k‖}_{\infty }\right)\text{.}$$

If $\mathbf{tol}\le 0.0$, $\tau =\max \sqrt{\epsilon },10\epsilon ,\sqrt{n}\epsilon$ is used, where $\epsilon$ is the machine precision. Otherwise $\tau =\max \left(\mathbf{tol},10\epsilon ,\sqrt{n}\epsilon \right)$ is used.

Constraint: $\mathbf{tol}<1.0$.

10:   $\mathrm{maxitn}$ – int64int32nag_int scalar

The maximum number of iterations allowed.

Constraint: $\mathbf{maxitn}\ge 1$.

11:   $\mathrm{x}\left(\mathbf{n}\right)$ – double array

An initial approximation to the solution vector $x$.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the dimension of the arrays b, x. (An error is raised if these dimensions are not equal.)

$n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 1$.

2:     $\mathrm{nz}$ – 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: $1\le \mathbf{nz}\le {\mathbf{n}}^2$.

Output Parameters

1:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

An improved approximation to the solution vector $x$.

2:     $\mathrm{rnorm}$ – double scalar

The final value of the residual norm ${‖r_k‖}_{\infty }$, where $k$ is the output value of itn.

3:     $\mathrm{itn}$ – int64int32nag_int scalar

The number of iterations carried out.

4:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=1$

On entry,	$\mathbf{method}\ne \text{'RGMRES'},\text{'CGS'},\text{'BICGSTAB'}$, or 'TFQMR',
or	$\mathbf{precon}\ne \text{'N'}$, $\text{'J'}$ or $\text{'S'}$,
or	$\mathbf{n}<1$,
or	$\mathbf{nz}<1$,
or	$\mathbf{nz}>{\mathbf{n}}^2$,
or	$\mathbf{precon}=\text{'S'}$ and omega lies outside the interval $\left(0.0,2.0\right)$,
or	$\mathbf{m}<1$,
or	$\mathbf{m}>\min \left(\mathbf{n},50\right)$, with $\mathbf{method}=\text{'RGMRES'}$,
or	$\mathbf{m}>\min \left(\mathbf{n},10\right)$, with $\mathbf{method}=\text{'BICGSTAB'}$,
or	$\mathbf{tol}\ge 1.0$,
or	$\mathbf{maxitn}<1$,
or	lwork too small.

   $\mathbf{ifail}=2$

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

• $1\le \mathbf{irow}\left(i\right)\le \mathbf{n}$ and $1\le \mathbf{icol}\left(i\right)\le \mathbf{n}$, for $i=1,2,\dots ,\mathbf{nz}$;
• $\mathbf{irow}\left(i-1\right)<\mathbf{irow}\left(i\right)$, or $\mathbf{irow}\left(i-1\right)=\mathbf{irow}\left(i\right)$ and $\mathbf{icol}\left(i-1\right)<\mathbf{icol}\left(i\right)$, for $i=2,3,\dots ,\mathbf{nz}$.

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.

   $\mathbf{ifail}=3$

On entry, the matrix $A$ has a zero diagonal element. Jacobi and SSOR preconditioners are not appropriate for this problem.

W  $\mathbf{ifail}=4$

The required accuracy could not be obtained. However, a reasonable accuracy may have been obtained, and further iterations could not improve the result. You should check the output value of rnorm for acceptability. This error code usually implies that your problem has been fully and satisfactorily solved to within or close to the accuracy available on your system. Further iterations are unlikely to improve on this situation.

   $\mathbf{ifail}=5$

Required accuracy not obtained in maxitn iterations.

   $\mathbf{ifail}=6$

Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.

   $\mathbf{ifail}=7$ (nag_sparse_real_gen_basic_setup (f11bd), nag_sparse_real_gen_basic_solver (f11be) or nag_sparse_real_gen_basic_diag (f11bf))

A serious error has occurred in an internal call to one of the specified functions. Check all function calls and array sizes. Seek expert help.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f11de_example

function f11de_example


fprintf('f11de example results\n\n');

% Sparse matrix A
a    = [        2;  1;-1;-3;-2; 1; 1; 5; 3; 1;-2;-3;-1; 4;-2;-6];
irow = [int64(1); 1; 1; 2; 2; 2; 3; 3; 3; 3; 4; 4; 4; 5; 5; 5];
icol = [int64(1); 2; 4; 2; 3; 5; 1; 3; 4; 5; 1; 4; 5; 2; 3; 5];

% RHS b and initial guess x
b = [0; -7; 33; -19; -28];
x = zeros(5, 1);

% Use RGMRES with SSOR preconditioning
method = 'RGMRES';
precon = 'S';

% Input argument initializations
m      = int64(1);
omega  = 1.05;
tol    = 1e-10;
maxitn = int64(1000);

% Solve Ax = b
[x, rnorm, itn, ifail] = ...
f11de( ...
       method, precon, a, irow, icol, omega, b, m, tol, maxitn, x);

fprintf('Converged in %d iterations\n', itn);
fprintf('Final residual norm = %16.1e\n\n', rnorm);
disp('Solution');
disp(x);

f11de example results

Converged in 13 iterations
Final residual norm =          5.1e-09

Solution
    1.0000
    2.0000
    3.0000
    4.0000
    5.0000