NAG Toolbox: nag_sparse_complex_gen_solve_bdilu (f11du)

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

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

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

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

6:     $\mathrm{nb}$ – int64int32nag_int scalar

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

8:     $\mathrm{indb}\left(\mathbf{lindb}\right)$ – int64int32nag_int array

9:     $\mathrm{ipivp}\left(\mathbf{lindb}\right)$ – int64int32nag_int array

10:   $\mathrm{ipivq}\left(\mathbf{lindb}\right)$ – int64int32nag_int array

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

12:   $\mathrm{idiag}\left(\mathbf{lindb}\right)$ – 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_bdilu (f11dt).

The arrays istb, indb and a together with the scalars n, nz, la, nb and lindb must be the same values that were supplied in the preceding call to nag_sparse_complex_gen_precon_bdilu (f11dt).

The values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_complex_gen_precon_bdilu (f11dt).

The arrays istb, indb and the scalars nb and lindb must be the same values that were supplied in the preceding call to nag_sparse_complex_gen_precon_bdilu (f11dt).

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

The right-hand side vector $b$.

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

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

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

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

The maximum number of iterations allowed.

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

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

An initial approximation to the solution vector $x$.

Optional Input Parameters

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

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

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

Default: For la, the dimension of the arrays a, icol, irow. (An error is raised if these dimensions are not equal.)For lindb, the dimension of the arrays idiag, indb, ipivp, ipivq. (An error is raised if these dimensions are not equal.)For n, the dimension of the arrays b, x. (An error is raised if these dimensions are not equal.)

The values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_complex_gen_precon_bdilu (f11dt).

The arrays istb, indb and a together with the scalars n, nz, la, nb and lindb must be the same values that were supplied in the preceding call to nag_sparse_complex_gen_precon_bdilu (f11dt).

Default: For la, the dimension of the arrays a, icol, irow. (An error is raised if these dimensions are not equal.)For lindb, the dimension of the arrays idiag, indb, ipivp, ipivq. (An error is raised if these dimensions are not equal.)For n, the dimension of the arrays b, x. (An error is raised if these dimensions are not equal.)

The values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_complex_gen_precon_bdilu (f11dt).

The arrays istb, indb and the scalars nb and lindb must be the same values that were supplied in the preceding call to nag_sparse_complex_gen_precon_bdilu (f11dt).

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

   $\mathbf{ifail}=1$

Constraint: $1\le \mathbf{nb}\le \mathbf{n}$.

Constraint: $\mathbf{istb}\left(\mathit{b}+1\right)>\mathbf{istb}\left(\mathit{b}\right)$, for $\mathit{b}=1,2,\dots ,\mathbf{nb}$.

Constraint: if $\mathbf{method}=\text{'RGMRES'}$, $1\le \mathbf{m}\le \min \left(\mathbf{n},\_\right)$.

Constraint: $\mathbf{istb}\left(1\right)\ge 1$.

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

Constraint: $\mathbf{lindb}\ge \mathbf{istb}\left(\mathbf{nb}+1\right)-1$.

Constraint: $1\le \mathbf{indb}\left(\mathit{m}\right)\le \mathbf{n}$, for $\mathit{m}=1,2,\dots ,\mathbf{istb}\left(\mathbf{nb}+1\right)-1$ 

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

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

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

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

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

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

lwork is too small.

   $\mathbf{ifail}=2$

Constraint: $1\le \mathbf{icol}\left(\mathit{i}\right)\le \mathbf{n}$, for $\mathit{i}=1,2,\dots ,\mathbf{nz}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_complex_gen_precon_bdilu (f11dt) and nag_sparse_complex_gen_solve_bdilu (f11du).

Constraint: $1\le \mathbf{irow}\left(\mathit{i}\right)\le \mathbf{n}$, for $\mathit{i}=1,2,\dots ,\mathbf{nz}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_complex_gen_precon_bdilu (f11dt) and nag_sparse_complex_gen_solve_bdilu (f11du).

On entry, element $\_$ of a was out of order.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_complex_gen_precon_bdilu (f11dt) and nag_sparse_complex_gen_solve_bdilu (f11du).

On entry, location $\_$ of $\left(\mathbf{irow},\mathbf{icol}\right)$ was a duplicate.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_complex_gen_precon_bdilu (f11dt) and nag_sparse_complex_gen_solve_bdilu (f11du).

   $\mathbf{ifail}=3$

The CS representation of the preconditioner is invalid.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_complex_gen_precon_bdilu (f11dt) and nag_sparse_complex_gen_solve_bdilu (f11du).

   $\mathbf{ifail}=4$

The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved. 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.

   $\mathbf{ifail}=5$

The solution has not converged after $\_$ iterations.

   $\mathbf{ifail}=6$

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

   $\mathbf{ifail}=-99$

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: f11du_example

function f11du_example


fprintf('f11du example results\n\n');

n   = int64(9);
nz  = int64(33);
a    = zeros(20*nz, 1);
irow = zeros(20*nz, 1, 'int64');
icol = zeros(20*nz, 1, 'int64');
a(1:nz) = [ 96 - 64i; -20 + 22i; -36 + 14i; 
           -12 + 10i;  96 - 64i; -20 + 22i; -36 + 14i;
           -12 + 10i;  96 - 64i; -36 + 14i;
           -28 + 18i;  96 - 64i; -20 + 22i; -36 + 14i;
           -28 + 18i; -12 + 10i;  96 - 64i; -20 + 22i; -36 + 14i;
           -28 + 18i; -12 + 10i;  96 - 64i; -36 + 14i;
           -28 + 18i;  96 - 64i; -20 + 22i;
           -28 + 18i; -12 + 10i;  96 - 64i; -20 + 22i; 
           -28 + 18i; -12 + 10i;  96 - 64i];
irow(1:nz) = [1; 1; 1;  2; 2; 2; 2;  3; 3; 3;  4; 4; 4; 4;  5; 5; 5; 5; 5; 
              6; 6; 6; 6;  7; 7; 7;  8; 8; 8; 8;  9; 9; 9];
icol(1:nz) = [1; 2; 4;  1; 2; 3; 5;  2; 3; 6;  1; 4; 5; 7;  2; 4; 5; 6; 8; 
              3; 5; 6; 9;  4; 7; 8;  5; 7; 8; 9;  6; 8; 9];

% 3 blocks
nb    = int64(3);
nover = 1;
lfill = [int64(0); 0; 0];
dtol  = [0; 0; 0];
pstrat = {'n'; 'n'; 'n'};
milu   = {'n'; 'n'; 'n'};

% Define diagonal block indices.
% In this example use blocks of MB consecutive rows and initialise
% assuming no overlap.
mb    = idivide(n+nb-1, nb);
istb  = zeros(nb+1, 1, 'int64');
indb  = zeros(3*n, 1, 'int64');
ipivp = zeros(3*n, 1, 'int64');
ipivq = zeros(3*n, 1, 'int64');
istb(1:nb) = [1:mb:nb*mb];
istb(nb+1) = n+1;
indb(1:n)  = [1:n];

% Modify indb and istb to account for overlap.
[istb, indb, ifail] = f11du_overlap(n, nz, irow, icol, nb, ...
                                    istb, indb, 3*n, nover);
if (ifail == -999)
  error('indb is too small, size of indb = %d', numel(indb));
end

% Calculate Factorisation
[a, irow, icol, ipivp, ipivq, istr, idiag, nnzc, npivm, ifail] = ...
     f11dt(n, nz, a, irow, icol, istb, indb, ...
           lfill, dtol, milu, ipivp, ipivq, 'pstrat', pstrat);

% RHS b and initial guess x
x = complex(zeros(n,1));
b = x;
b(1:n) = 100+4i;

% Solver algorithmic parameters
method = 'rgmres';
m      = int64(2);
tol    = 1e-6;
maxitn = int64(100);

% Solve Ax = b
[x, rnorm, itn, ifail] = f11du( ...
                                method, nz, a, irow, icol, nb, istb, indb, ...
                                ipivp, ipivq, istr, idiag, b, m, tol, ...
                                maxitn, x);

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



function [istb, indb, ifail] = f11du_overlap(n, nz, irow, icol, nb, ...
                                             istb, indb, lindb, nover)

  ifail = 0;

  % This function takes a set of row indices indb defining the diagonal
  % blocks to be used in f11dt to define a block Jacobi or additive Schwarz
  % preconditioner, and expands them to allow for nover levels of overlap.
  % The pointer array istb is also updated accordingly, so that the returned
  % values of istb and indb can be passed to f11dt to define overlapping
  % diagonal blocks.
  iwork = zeros(3*n+1, 1, 'int64');

  % Find the number of non-zero elements in each row of the matrix A, and
  % the start address of each row. Store the start addresses in
  % iwork(n+1,...,2*n+1).
  for k=1:nz
    iwork(irow(k)) = iwork(irow(k)) + 1;
  end
  iwork(n+1) = 1;
  for i = 1:n
    iwork(n+i+1) = iwork(n+i) + iwork(i);
  end

  % Loop over blocks
  for k=1:nb
    % Initialize marker array.
    iwork(1:n) = 0;

    % Mark the rows already in block k in the workspace array.
    for l = istb(k):istb(k+1)-1
      iwork(indb(l)) = 1;
    end

    % Loop over levels of overlap.
    for iover=1:nover
      % Initialize counter of new row indices to be added.
      ind = 0;

      % Loop over the rows currently in the diagonal block.
      for l = istb(k):istb(k+1)-1
        row = indb(l);

        % Loop over non-zero elements in row
        for i = iwork(n+row):iwork(n+row+1)-1

          % If the column index of the non-zero element is not in the
          % existing set for this block, store it to be added later, and
          % mark it in the marker array.
          if (iwork(icol(i))==0)
            iwork(icol(i)) = 1;
            ind = ind + 1;
            iwork(2*n+1+ind) = icol(i);
          end
        end
      end

      % Shift the indices in indb and add the new entries for block k.
      % Change istb accordingly.
      nadd = ind;
      if (istb(nb+1)+nadd-1>lindb) Then
        ifail = -999;
        return;
      end

      for i = istb(nb+1) - 1:-1:istb(k+1)
        indb(i+nadd) = indb(i);
      end
      n21 = 2*n + 1;
      ik = istb(k+1) - 1;
      indb(ik+1:ik+nadd) = iwork(n21+1:n21+nadd);
      istb(k+1:nb+1) = istb(k+1:nb+1) + nadd;
    end
  end

f11du example results

Converged in 8 iterations
Final redidual norm =        6.492e-04

Solution
   2.2040 + 1.6972i
   2.3511 + 1.9275i
   1.5931 + 1.4368i
   2.8641 + 1.9762i
   3.0687 + 2.2645i
   2.0467 + 1.6948i
   2.2065 + 1.3244i
   2.3724 + 1.5170i
   1.6254 + 1.1783i