NAG Toolbox: nag_sparse_complex_herm_precon_ssor_solve (f11jr)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathbf{nz}\right)$ – complex array

The nonzero elements in 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_complex_herm_sort (f11zp) may be used to order the elements in this way.

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

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

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

Constraints:

irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_complex_herm_sort (f11zp)):

• $1\le \mathbf{irow}\left(\mathit{i}\right)\le \mathbf{n}$ 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 $\mathit{i}=2,3,\dots ,\mathbf{nz}$.

4:     $\mathrm{rdiag}\left(\mathbf{n}\right)$ – double array

The elements of the diagonal matrix $D^{-1}$, where $D$ is the diagonal part of $A$. Note that since $A$ is Hermitian the elements of $D^{-1}$ are necessarily real.

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

The relaxation parameter $\omega$.

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

6:     $\mathrm{check}$ – string (length ≥ 1)

Specifies whether or not the input data should be checked.

$\mathbf{check}=\text{'C'}$

Checks are carried out on the values of n, nz, irow, icol and omega.

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

None of these checks are carried out.

Constraint: $\mathbf{check}=\text{'C'}$ or $\text{'N'}$.

7:     $\mathrm{y}\left(\mathbf{n}\right)$ – complex array

The right-hand side vector $y$.

Optional Input Parameters

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

Default: the dimension of the arrays rdiag, y. (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:     $\mathrm{x}\left(\mathbf{n}\right)$ – complex array

The solution vector $x$.

2:     $\mathrm{ifail}$ – 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,

$\mathbf{check}\ne \text{'C'}$ or $\text{'N'}$.

   $\mathbf{ifail}=2$

On entry,	$\mathbf{n}<1$,
or	$\mathbf{nz}<1$,
or	$\mathbf{nz}>\mathbf{n}\times \left(\mathbf{n}+1\right)/2$,
or	omega lies outside the interval $\left(0.0,2.0\right)$.

   $\mathbf{ifail}=3$

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_complex_herm_sort (f11zp) to reorder and sum or remove duplicates.

   $\mathbf{ifail}=4$

On entry,

a row of $A$ has no diagonal entry.

   $\mathbf{ifail}=-99$

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

Timing

Example

Open in the MATLAB editor: f11jr_example

function f11jr_example


fprintf('f11jr example results\n\n');

% Solve SSOR preconditioning problem Mx = y for Hermitian A

% Sparse matrix A and y
nz = int64(23);
n  = int64(9);
a = [ 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];
y = [ 8 + 54i;-10 - 92i; 25 + 27i; 26 - 28i;
      54 + 12i; 26 - 22i; 47 + 65i; 71 - 57i;
      60 + 70i];
irow = 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 = int64([1;1;2;2;3;4;1;5;3;4;6;2;5;6;7;4;6;8;1;5;6;8;9]);


% Fill in the diagonal part for SSOR
rdiag = zeros(n, 1);
for i=1:nz
  if irow(i) == icol(i)
    rdiag(irow(i)) = 1/real(a(i));
  end
end

% Solve Mx = b
omega = 1.1;
check = 'C';
[x, ifail] = f11jr( ...
                    a, irow, icol, rdiag, omega, check, y);

disp('Solution of Mx = y');
disp(x);

f11jr example results

Solution of Mx = y
   1.0977 + 5.9139i
   0.2230 -14.0850i
   2.2315 + 7.0868i
   4.8164 - 6.1807i
   6.7632 + 1.5690i
   3.3531 - 4.7849i
   0.6699 - 1.4646i
   8.8315 - 3.6326i
   4.7685 + 0.1213i