NAG Toolbox: nag_sparse_real_symm_solve_jacssor (f11je)

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:     $\mathrm{precon}$ – 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 (SSOR).

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

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

The nonzero elements of the lower triangular part 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_symm_sort (f11zb) 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 array a.

Constraints:

irow and icol must satisfy these constraints (which may be imposed by a call to nag_sparse_real_symm_sort (f11zb)):

• $1\le \mathbf{irow}\left(\mathit{i}\right)\le \mathbf{n}$ and $1\le \mathbf{icol}\left(\mathit{i}\right)\le \mathbf{irow}\left(\mathit{i}\right)$, 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.

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

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

The right-hand side vector $b$.

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

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

The maximum number of iterations allowed.

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

10:   $\mathrm{x}\left(\mathbf{n}\right)$ – 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 lower triangular part of the matrix $A$.

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

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

   $\mathbf{ifail}=1$

On entry,	$\mathbf{method}\ne \text{'CG'}$ or $\text{'SYMMLQ'}$,
or	$\mathbf{precon}\ne \text{'N'}$, $\text{'J'}$ or $\text{'S'}$,
or	$\mathbf{n}<1$,
or	$\mathbf{nz}<1$,
or	$\mathbf{nz}>\mathbf{n}\times \left(\mathbf{n}+1\right)/2$,
or	omega lies outside the interval $\left(0.0,2.0\right)$,
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 $1\le \mathbf{icol}\left(i\right)\le \mathbf{irow}\left(i\right)$, 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 in the lower triangular part of $A$, is out of order, or has duplicate row and column indices. Call nag_sparse_real_symm_sort (f11zb) 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.

   $\mathbf{ifail}=4$

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).

   $\mathbf{ifail}=8$ (nag_sparse_real_symm_basic_setup (f11gd), nag_sparse_real_symm_basic_solver (f11ge) or nag_sparse_real_symm_basic_diag (f11gf))

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: f11je_example

function f11je_example


fprintf('f11je example results\n\n');

% Solve Ax = b using CG and SSOR preconditioning

% Sparse matrix A
n    = int64(7);
nz   = int64(16);
irow = zeros(nz,1,'int64');
icol = irow;

a(1:16)    = [4  1  5  2  2  3 -1  1  4  1 -2  3  2 -1 -2  5];
irow(1:16) = [1  2  2  3  4  4  5  5  5  6  6  6  7  7  7  7];  
icol(1:16) = [1  1  2  3  2  4  1  4  5  2  5  6  1  2  3  7];

% RHS b and initial guess x
b = [15  18  -8   21  11  10  29];
x = zeros(n,1);

% Solve
method = 'CG';
precon = 'S';
omega  = 1.1;
tol    = 1e-06;
maxitn = int64(100);

[x, rnorm, itn, ifail] = ...
  f11je( ...
         method, precon, a, irow, icol, omega, b, tol, maxitn, x);

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

f11je example results

Converged in 6 iterations
Final redidual norm =        5.026e-06

Solution
    1.0000
    2.0000
    3.0000
    4.0000
    5.0000
    6.0000
    7.0000