NAG Toolbox: nag_sparse_complex_herm_solve_ilu (f11jq)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{method}$ – string

Specifies the iterative method to be used.

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

Conjugate gradient method.

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

Lanczos method (SYMMLQ).

Constraint: $\mathbf{method}=\text{'CG'}$ or $\text{'SYMMLQ'}$.

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

The number of nonzero elements in the lower triangular part of the matrix $A$. This must be the same value as was supplied in the preceding call to nag_sparse_complex_herm_precon_ilu (f11jn).

Constraint: $1\le \mathbf{nz}\le \mathbf{n}\times \left(\mathbf{n}+1\right)/2$.

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

The values returned in the array a by a previous call to nag_sparse_complex_herm_precon_ilu (f11jn).

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

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

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

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

The values returned in arrays irow, icol, ipiv and istr by a previous call to nag_sparse_complex_herm_precon_ilu (f11jn).

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

The right-hand side vector $b$.

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:   $\mathrm{x}\left(\mathbf{n}\right)$ – 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 ipiv, 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_herm_precon_ilu (f11jn).

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_herm_precon_ilu (f11jn).

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,	$\mathbf{method}\ne \text{'CG'}$ or $\text{'SYMMLQ'}$,
or	$\mathbf{n}<1$,
or	$\mathbf{nz}<1$,
or	$\mathbf{nz}>\mathbf{n}\times \left(\mathbf{n}+1\right)/2$,
or	la too small,
or	$\mathbf{tol}\ge 1.0$,
or	$\mathbf{maxitn}<1$,
or	lwork too small.

   $\mathbf{ifail}=2$

On entry, the SCS representation of $A$ is invalid. Further details are given in the error message. Check that the call to nag_sparse_complex_herm_solve_ilu (f11jq) has been preceded by a valid call to nag_sparse_complex_herm_precon_ilu (f11jn), and that the arrays a, irow, and icol have not been corrupted between the two calls.

   $\mathbf{ifail}=3$

On entry, the SCS representation of $M$ is invalid. Further details are given in the error message. Check that the call to nag_sparse_complex_herm_solve_ilu (f11jq) has been preceded by a valid call to nag_sparse_complex_herm_precon_ilu (f11jn), and that the arrays a, irow, icol, ipiv and istr have not been corrupted between the two calls.

W  $\mathbf{ifail}=4$

The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations could not improve the result.

   $\mathbf{ifail}=5$

Required accuracy not obtained in maxitn iterations.

   $\mathbf{ifail}=6$

The preconditioner appears not to be positive definite.

   $\mathbf{ifail}=7$

The matrix of the coefficients appears not to be positive definite (conjugate gradient method only).

   $\mathbf{ifail}=8$

A serious error has occurred in an internal call to an auxiliary function. 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: f11jq_example

function f11jq_example


fprintf('f11jq example results\n\n');

% Solve sparse Hermitian system Ax = b using CG method with
% Incomplete Cholesky preconditioning (IC)

% Define A and b 
n  = int64(9);
nz = int64(23);
a    = zeros(3*nz,1);
irow = zeros(3*nz, 1, 'int64');
icol = zeros(3*nz, 1, 'int64');
a(1:nz) = [ 6 + 0.i; -1 + 1.i;  6 + 0.i;  0 + 1.i;
            5 + 0.i;  5 + 0.i;  2 - 2.i;  4 + 0.i;
            1 + 1.i;  2 + 0.i;  6 + 0.i; -4 + 3.i;
            0 + 1.i; -1 + 0.i;  6 + 0.i; -1 - 1.i;
            0 - 1.i;  9 + 0.i;  1 + 3.i;  1 + 2.i;
           -1 + 0.i;  1 + 4.i;  9 + 0.i];
b       = [ 8 + 54i;-10 - 92i; 25 + 27i; 26 - 28i;
           54 + 12i; 26 - 22i; 47 + 65i; 71 - 57i;
           60 + 70i];
irow(1:nz) = int64([1;2;2;3;3;4;5;5;6;6;6;7;7;7;7;8;8;8;9;9;9;9;9]);
icol(1:nz) = int64([1;1;2;2;3;4;1;5;3;4;6;2;5;6;7;4;6;8;1;5;6;8;9]);

% Calculate incomplete Cholesky factorization
lfill  = int64(0);
dtol   = 0;
mic    = 'N';
dscale = 0;
ipiv   = zeros(n, 1, 'int64');

[a, irow, icol, ipiv, istr, nnzc, npivm, ifail] = ...
  f11jn( ...
         nz, a, irow, icol, lfill, dtol, mic, dscale, ipiv);

% Solve Ax = b
method = 'CG';
tol    = 1e-06;
maxitn = int64(100);
x      = complex(zeros(n, 1));

[x, rnorm, itn, ifail] = ...
  f11jq( ...
         method, nz, a, irow, icol, ipiv, istr, b, tol, maxitn, x);

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

f11jq example results

Converged in 5 iterations
Final redidual norm =        3.197e-14

Solution
   1.0000 + 9.0000i
   2.0000 - 8.0000i
   3.0000 + 7.0000i
   4.0000 - 6.0000i
   5.0000 + 5.0000i
   6.0000 - 4.0000i
   7.0000 + 3.0000i
   8.0000 - 2.0000i
   9.0000 + 1.0000i