NAG Toolbox: nag_sparse_complex_gen_solve_ilu (f11dq)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{method}$ – string

Specifies 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:     $\mathrm{nz}$ – int64int32nag_int scalar

The number of nonzero elements in the matrix $A$. This must be the same value as was supplied in the preceding call to nag_sparse_complex_gen_precon_ilu (f11dn).

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

3:     $\mathrm{a}\left(\mathbf{la}\right)$ – complex array

The values returned in the array a by a previous call to nag_sparse_complex_gen_precon_ilu (f11dn).

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

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

6:     $\mathrm{ipivp}\left(\mathbf{n}\right)$ – int64int32nag_int array

7:     $\mathrm{ipivq}\left(\mathbf{n}\right)$ – int64int32nag_int array

8:     $\mathrm{istr}\left(\mathbf{n}+1\right)$ – int64int32nag_int array

9:     $\mathrm{idiag}\left(\mathbf{n}\right)$ – int64int32nag_int array

The values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_complex_gen_precon_ilu (f11dn).

ipivp and ipivq are restored on exit.

10:   $\mathrm{b}\left(\mathbf{n}\right)$ – complex array

The right-hand side vector $b$.

11:   $\mathrm{m}$ – 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 $\mathbf{method}=\text{'BICGSTAB'}$, $0<\mathbf{m}\le \min \left(\mathbf{n},10\right)$.

12:   $\mathrm{tol}$ – 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: $\mathbf{tol}<1.0$.

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

The maximum number of iterations allowed.

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

14:   $\mathrm{x}\left(\mathbf{n}\right)$ – complex array

An initial approximation to the solution vector $x$.

Optional Input Parameters

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

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

$n$, the order of the matrix $A$. This must be the same value as was supplied in the preceding call to nag_sparse_complex_gen_precon_ilu (f11dn).

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

2:     $\mathrm{la}$ – int64int32nag_int scalar

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

The dimension of the arrays a, irow and icol. this must be the same value as was supplied in the preceding call to nag_sparse_complex_gen_precon_ilu (f11dn).

Constraint: $\mathbf{la}\ge 2\times \mathbf{nz}$.

Output Parameters

1:     $\mathrm{x}\left(\mathbf{n}\right)$ – complex 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	$\mathbf{n}<1$,
or	$\mathbf{nz}<1$,
or	$\mathbf{nz}>{\mathbf{n}}^2$,
or	$\mathbf{la}<2\times \mathbf{nz}$,
or	$\mathbf{m}<1$ and $\mathbf{method}=\text{'RGMRES'}$ or $\mathbf{method}=\text{'BICGSTAB'}$,
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 CS representation of $A$ is invalid. Further details are given in the error message. Check that the call to nag_sparse_complex_gen_solve_ilu (f11dq) has been preceded by a valid call to nag_sparse_complex_gen_precon_ilu (f11dn), and that the arrays a, irow, and icol have not been corrupted between the two calls.

   $\mathbf{ifail}=3$

On entry, the CS representation of the preconditioning matrix $M$ is invalid. Further details are given in the error message. Check that the call to nag_sparse_complex_gen_solve_ilu (f11dq) has been preceded by a valid call to nag_sparse_complex_gen_precon_ilu (f11dn) and that the arrays a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between the two calls.

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_complex_gen_basic_setup (f11br), nag_sparse_complex_gen_basic_solver (f11bs) or nag_sparse_complex_gen_basic_diag (f11bt))

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: f11dq_example

function f11dq_example


fprintf('f11dq example results\n\n');

% Define sparse matrix A and RHS B
nz   = int64(24);
n    = int64(8);
a    = complex(zeros(3*nz,1));
irow = zeros(3*nz,1,'int64');
icol = zeros(3*nz,1,'int64');
a(1:nz) = [ 2 + 1i, -1 + 1i,  1 - 3i,  4 + 7i, -3 + 0i,  2 + 4i, ...
           -7 - 5i,  2 + 1i,  3 + 2i, -4 + 2i,  0 + 1i,  5 - 3i, ...
           -1 + 2i,  8 + 6i, -3 - 4i, -6 - 2i,  5 - 2i,  2 + 0i, ...
            0 - 5i, -1 + 5i,  6 + 2i, -1 + 4i,  2 + 0i,  3 + 3i];
b       = [ 7 + 11i;
            1 + 24i;
          -13 - 18i;
          -10 +  3i;
           23 + 14i;
           17 -  7i;
           15 -  3i;
           -3 + 20i];
irow(1:nz) = int64(...
             [1; 1; 1; 2; 2; 2; 3; 3; 4; 4; 4; 4;
              5; 5; 5; 6; 6; 6; 7; 7; 7; 8; 8; 8]);
icol(1:nz) = int64(...
             [1; 4; 8; 1; 2; 5; 3; 6; 1; 3; 4; 7;
              2; 5; 7; 1; 3; 6; 3; 5; 7; 2; 6; 8]);

% ILU preconditioner
%     Input parameters
lfill = int64(0);
dtol  = 0;
milu  = 'No modification';
ipivp = zeros(n, 1, 'int64');
ipivq = zeros(n, 1, 'int64');
% Compute preconditioner
[a, irow, icol, ipivp, ipivq, istr, idiag, nnzc, npivm, ifail] = ...
    f11dn(...
          nz, a, irow, icol, lfill, dtol, milu, ipivp, ipivq);

% Iterative Solution Setup
%      Input parameters
method = 'CGS';
m      = int64(0);
tol    = 1e-10;
maxitn = int64(100);
x      = complex(zeros(n,1));

% Solve Ax = b
[x, rnorm, itn, ifail] = ...
  f11dq( ...
         method, nz, a, irow, icol, ipivp, ipivq, ...
         istr, idiag, b, m, tol, maxitn, x);

fprintf('Converged in %d iterations\n', itn);
fprintf('Final residual norm = %16.3e\n\n', rnorm);
disp('Solution');
disp(x);

f11dq example results

Converged in 4 iterations
Final residual norm =        1.348e-11

Solution
   1.0000 + 1.0000i
   2.0000 - 1.0000i
   3.0000 + 1.0000i
   4.0000 - 1.0000i
   3.0000 - 1.0000i
   2.0000 + 1.0000i
   1.0000 - 1.0000i
  -0.0000 + 3.0000i