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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sparse_complex_herm_basic_diag (f11gt)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_sparse_complex_herm_basic_diag (f11gt) is the third in a suite of three functions for the iterative solution of a complex Hermitian system of simultaneous linear equations (see Golub and Van Loan (1996)). nag_sparse_complex_herm_basic_diag (f11gt) returns information about the computations during an iteration and/or after this has been completed. The first function of the suite, nag_sparse_complex_herm_basic_setup (f11gr), is a setup function, the second function, nag_sparse_complex_herm_basic_solver (f11gs) is the proper iterative solver.
These three functions are suitable for the solution of large sparse complex Hermitian systems of equations.

Syntax

[itn, stplhs, stprhs, anorm, sigmax, its, sigerr, ifail] = f11gt(work)
[itn, stplhs, stprhs, anorm, sigmax, its, sigerr, ifail] = nag_sparse_complex_herm_basic_diag(work)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 22: lwork was removed from the interface

Description

nag_sparse_complex_herm_basic_diag (f11gt) returns information about the solution process. It can be called both during a monitoring step of the solver nag_sparse_complex_herm_basic_solver (f11gs) or after this solver has completed its tasks. Calling nag_sparse_complex_herm_basic_diag (f11gt) at any other time will result in an error condition being raised.
For further information you should read the documentation for nag_sparse_complex_herm_basic_setup (f11gr) and nag_sparse_complex_herm_basic_solver (f11gs).

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     worklwork – complex array
lwork, the dimension of the array, must satisfy the constraint lwork≥120.
The array work as returned by nag_sparse_complex_herm_basic_solver (f11gs) (see also Description in nag_sparse_complex_herm_basic_solver (f11gs)).

Optional Input Parameters

None.

Output Parameters

1:     itn – int64int32nag_int scalar
The number of iterations carried out by nag_sparse_complex_herm_basic_solver (f11gs).
2:     stplhs – double scalar
The current value of the left-hand side of the termination criterion used by nag_sparse_complex_herm_basic_solver (f11gs).
3:     stprhs – double scalar
The current value of the right-hand side of the termination criterion used by nag_sparse_complex_herm_basic_solver (f11gs).
4:     anorm – double scalar
The norm A1=A∞ when either it has been supplied to nag_sparse_complex_herm_basic_setup (f11gr) or it has been estimated by nag_sparse_complex_herm_basic_solver (f11gs) (see also Description and Arguments in nag_sparse_complex_herm_basic_setup (f11gr)).
Otherwise, anorm=0.0 is returned.
5:     sigmax – double scalar
The current estimate of the largest singular value σ1A- of the preconditioned iteration matrix A-=E-1AE-H, when either it has been supplied to nag_sparse_complex_herm_basic_setup (f11gr) or it has been estimated by nag_sparse_complex_herm_basic_solver (f11gs) (see also Description and Arguments in nag_sparse_complex_herm_basic_setup (f11gr)). Note that if its<itn then sigmax contains the final estimate. If, on final exit from nag_sparse_complex_herm_basic_solver (f11gs), its=itn, then the estimation of σ1A- may have not converged: in this case you should look at the value returned in sigerr. Otherwise, sigmax=0.0 is returned.
6:     its – int64int32nag_int scalar
The number of iterations employed so far in the computation of the estimate of σ1A-, the largest singular value of the preconditioned matrix A-=E-1AE-H, when σ1A- has been estimated by nag_sparse_complex_herm_basic_solver (f11gs) using the bisection method (see also Description, Arguments and Further Comments in nag_sparse_complex_herm_basic_setup (f11gr)). Otherwise, its=0 is returned.
7:     sigerr – double scalar
If σ1A- has been estimated by nag_sparse_complex_herm_basic_solver (f11gs) using bisection,
sigerr=maxσ1k-σ1k-1σ1k,σ1k-σ1k-2σ1k ,  
where k=its denotes the iteration number. The estimation has converged if sigerr≤sigtol where sigtol is an input argument to nag_sparse_complex_herm_basic_setup (f11gr).
Otherwise, sigerr=0.0 is returned.
8:     ifail – int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=-i
If ifail=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: itn, 2: stplhs, 3: stprhs, 4: anorm, 5: sigmax, 6: its, 7: sigerr, 8: work, 9: lwork, 10: ifail.
It is possible that ifail refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
   ifail=1
nag_sparse_complex_herm_basic_diag (f11gt) has been called out of sequence. For example, the last call to nag_sparse_complex_herm_basic_solver (f11gs) did not return irevcm=3 or 4.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

None.

Example

See Example in nag_sparse_complex_herm_basic_setup (f11gr).

Open in the MATLAB editor: f11gt_example

function f11gt_example


fprintf('f11gt 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    = complex(zeros(3*nz,1));
irow = zeros(3*nz, 1, 'int64');
icol = zeros(3*nz, 1, 'int64');
a(1:nz) = [ 6 + 0i;         -1 + 1i; 6 + 0i;         0 + 1i; 5 + 0i;
            5 + 0i;          2 - 2i; 4 + 0i;         1 + 1i; 2 + 0i; 6 + 0i;
           -4 + 3i; 0 + 1i; -1 + 0i; 6 + 0i;        -1 - 1i; 0 - 1i; 9 + 0i; 
            1 + 3i; 1 + 2i; -1 + 0i; 1 + 4i; 9 + 0i];

irow(1:nz) = [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) = [1; 1;2; 2;3;  4; 1;5; 3;4;6;  2;5;6;7; 4;6;8;  1;5;6;8;9];

b = [ 8 + 54i; -10 - 92i; 25 + 27i;
     26 - 28i;  54 + 12i; 26 - 22i;
     47 + 65i;  71 - 57i; 60 + 70i];

% Setup IC 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);

% Iterative method setup
method = 'CG    ';
precon = 'Preconditioned';
tol    = (x02aj)^(3/8);
maxitn = int64(20);
anorm  = 0;
sigmax = 0;
maxits = int64(9);
monit  = int64(2);

[lwreq, work, ifail] = ...
  f11gr( ...
         method, precon, int64(n), tol, maxitn, anorm, sigmax, ...
         maxits, monit, 'sigcmp', 's', 'norm_p', '1');

% Reverse communication loop calling f11ge
irevcm = int64(0);
u      = complex(zeros(n,1));
v      = b;
wgt    = zeros(n,1);

while (irevcm ~= 4)
  [irevcm, u, v, work, ifail] = ...
    f11gs( ...
           irevcm, u, v, wgt, work);

  if (irevcm == 1)
    % v = Au
    [v, ifail] = f11xs( ...
                        a(1:nz), irow(1:nz), icol(1:nz), 'N', u);
  elseif (irevcm == 2)
    % Solve (IC)v = u
    [v, ifail] = f11jp( ...
                        a, irow, icol, ipiv, istr, 'N', u);
  elseif (irevcm == 3)
    % Monitoring
    [itn, stplhs, stprhs, anorm, sigmax, its, sigerr, ifail] = ...
    f11gt(work);
    fprintf('\nMonitoring at iteration number %2d\n',itn);
    fprintf('residual norm:              %14.4e\n', stplhs);
    fprintf('\n   Solution Vector\n');
    disp(u);
    fprintf('\n   Residual Vector\n');
    disp(v);
  end
end

% Get information about the computation
[itn, stplhs, stprhs, anorm, sigmax, its, sigerr, ifail] = ...
f11gt(work);

fprintf('\nNumber of iterations for convergence:     %4d\n', itn);
fprintf('Residual norm:                           %14.4e\n', stplhs);
fprintf('Right-hand side of termination criteria: %14.4e\n', stprhs);
fprintf('i-norm of matrix a:                      %14.4e\n', anorm);
fprintf('\n   Solution Vector\n');
disp(u);
fprintf('\n   Residual Vector\n');
disp(v);



f11gt example results


Monitoring at iteration number  2
residual norm:                  1.4937e+01

   Solution Vector
   0.2142 + 4.5333i
  -1.6589 -12.6722i
   2.4101 + 7.4551i
   4.4400 - 6.4174i
   9.1135 + 3.7812i
   4.4419 - 4.0382i
   1.4757 + 1.2662i
   8.4872 - 3.5347i
   5.9948 + 0.9685i


   Residual Vector
  -1.8370 + 3.6956i
  -0.6501 + 0.2546i
  -0.1262 - 0.1362i
  -0.1312 + 0.1413i
  -1.1471 + 0.7339i
  -0.5505 - 1.0535i
   1.7165 - 1.4614i
  -0.3583 + 0.2876i
  -0.3028 - 0.3532i


Monitoring at iteration number  4
residual norm:                  1.4602e+00

   Solution Vector
   1.0061 + 8.9847i
   1.9637 - 7.9768i
   3.0067 + 7.0285i
   3.9830 - 5.9636i
   5.0390 + 5.0432i
   6.0488 - 4.0771i
   6.9710 + 3.0168i
   8.0118 - 1.9806i
   9.0074 + 0.9646i


   Residual Vector
   0.0115 - 0.0282i
   0.0135 - 0.1734i
   0.0182 + 0.0196i
   0.0189 - 0.0204i
  -0.0909 - 0.1090i
  -0.2389 + 0.3244i
   0.1903 - 0.0155i
   0.0516 - 0.0414i
   0.0436 + 0.0509i


Number of iterations for convergence:        5
Residual norm:                               9.0594e-14
Right-hand side of termination criteria:     2.8433e-03
i-norm of matrix a:                          2.2000e+01

   Solution Vector
   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


   Residual Vector
   1.0e-13 *

  -0.0178 + 0.0000i
   0.0355 - 0.2842i
  -0.0355 + 0.0355i
   0.0355 - 0.0711i
  -0.0711 + 0.0355i
  -0.0711 + 0.0000i
   0.0000 + 0.0000i
   0.0000 - 0.0711i
   0.0000 - 0.1421i
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