hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sparse_complex_gen_precon_ssor_solve (f11dr)

▸▿ 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_gen_precon_ssor_solve (f11dr) solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse non-Hermitian matrix, represented in coordinate storage format.

Syntax

[x, ifail] = f11dr(trans, a, irow, icol, rdiag, omega, check, y, 'n', n, 'nz', nz)
[x, ifail] = nag_sparse_complex_gen_precon_ssor_solve(trans, a, irow, icol, rdiag, omega, check, y, 'n', n, 'nz', nz)

Description

nag_sparse_complex_gen_precon_ssor_solve (f11dr) solves a system of linear equations
Mx=y,   or  MHx=y,  
according to the value of the argument trans, where the matrix
M=1ω2-ω D+ω L D-1 D+ω U  
corresponds to symmetric successive-over-relaxation (SSOR) Young (1971) applied to a linear system Ax=b, where A is a complex sparse non-Hermitian matrix stored in coordinate storage (CS) format (see Coordinate storage (CS) format in the F11 Chapter Introduction).
In the definition of M given above D is the diagonal part of A, L is the strictly lower triangular part of A, U is the strictly upper triangular part of A, and ω is a user-defined relaxation parameter.
It is envisaged that a common use of nag_sparse_complex_gen_precon_ssor_solve (f11dr) will be to carry out the preconditioning step required in the application of nag_sparse_complex_gen_basic_solver (f11bs) to sparse linear systems. For an illustration of this use of nag_sparse_complex_gen_precon_ssor_solve (f11dr) see the example program given in Example. nag_sparse_complex_gen_precon_ssor_solve (f11dr) is also used for this purpose by the Black Box function nag_sparse_complex_gen_solve_jacssor (f11ds).

References

Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

Parameters

Compulsory Input Parameters

1:     trans – string (length ≥ 1)
Specifies whether or not the matrix M is transposed.
trans='N'
Mx=y is solved.
trans='T'
MHx=y is solved.
Constraint: trans='N' or 'T'.
2:     anz – complex array
The nonzero elements in 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_gen_sort (f11zn) may be used to order the elements in this way.
3:     irownz – int64int32nag_int array
4:     icolnz – int64int32nag_int array
The row and column indices of the nonzero elements supplied in a.
Constraints:
irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_complex_gen_sort (f11zn)):
  • 1≤irowi≤n and 1≤icoli≤n, for i=1,2,…,nz;
  • either irowi-1<irowi or both irowi-1=irowi and icoli-1<icoli, for i=2,3,…,nz.
5:     rdiagn – complex array
The elements of the diagonal matrix D-1, where D is the diagonal part of A.
6:     omega – double scalar
The relaxation parameter ω.
Constraint: 0.0<omega<2.0.
7:     check – string (length ≥ 1)
Specifies whether or not the CS representation of the matrix M should be checked.
check='C'
Checks are carried on the values of n, nz, irow, icol and omega.
check='N'
None of these checks are carried out.
See also Use of check.
Constraint: check='C' or 'N'.
8:     yn – complex array
The right-hand side vector y.

Optional Input Parameters

1:     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: n≥1.
2:     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 matrix A.
Constraint: 1≤nz≤n2.

Output Parameters

1:     xn – complex array
The solution vector x.
2:     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=1
On entry,trans≠'N' or 'T',
orcheck≠'C' or 'N'.
   ifail=2
On entry,n<1,
ornz<1,
ornz>n2,
oromega lies outside the interval 0.0,2.0,
   ifail=3
On entry, the arrays irow and icol fail to satisfy the following constraints:
  • 1≤irowi≤n and 1≤icoli≤n, for i=1,2,…,nz;
  • irowi-1<irowi or irowi-1=irowi and icoli-1<icoli, for i=2,3,…,nz.
Therefore a nonzero element has been supplied which does not lie in the matrix A, is out of order, or has duplicate row and column indices. Call nag_sparse_complex_gen_sort (f11zn) to reorder and sum or remove duplicates.
   ifail=4
On entry, the matrix A has a zero diagonal element. The SSOR preconditioner is not appropriate for this problem.
   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

If trans='N' the computed solution x is the exact solution of a perturbed system of equations M+δMx=y, where
δM≤cnεD+ωLD-1D+ωU,  
cn is a modest linear function of n, and ε is the machine precision. An equivalent result holds when trans='T'.

Further Comments

Timing

The time taken for a call to nag_sparse_complex_gen_precon_ssor_solve (f11dr) is proportional to nz.

Use of check

It is expected that a common use of nag_sparse_complex_gen_precon_ssor_solve (f11dr) will be to carry out the preconditioning step required in the application of nag_sparse_complex_gen_basic_solver (f11bs) to sparse linear systems. In this situation nag_sparse_complex_gen_precon_ssor_solve (f11dr) is likely to be called many times with the same matrix M. In the interests of both reliability and efficiency, you are recommended to set check='C' for the first of such calls, and check='N' for all subsequent calls.

Example

This example solves a complex sparse linear system of equations
Ax=b,  
using RGMRES with SSOR preconditioning.
The RGMRES algorithm itself is implemented by the reverse communication function nag_sparse_complex_gen_basic_solver (f11bs), which returns repeatedly to the calling program with various values of the argument irevcm. This argument indicates the action to be taken by the calling program.
For further details see the function document for nag_sparse_complex_gen_basic_solver (f11bs).

Open in the MATLAB editor: f11dr_example

function f11dr_example


fprintf('f11dr example results\n\n');

% Solve sparse system Ax = b using preconditioned CGS

% Sparse A and b
n    = int64(5);
m    = int64(2);
nz   = int64(16);
sparseA = [ 2  3   1    1;
            1 -1   1    2;
           -1  0   1    4;
            0  2   2    2;
           -2  1   2    3;
            1  0   2    5;
            0 -1   3    1;
            5  4   3    3;
            3 -1   3    4;
            1  0   3    5;
           -2  2   4    1;
           -3  1   4    4;
            0  3   4    5;
            4 -2   5    2;
           -2  0   5    3;
           -6  1   5    5];
a(1:nz)    = sparseA(:,1) + i*sparseA(:,2);
irow(1:nz) = int64(sparseA(:,3));
icol(1:nz) = int64(sparseA(:,4));
b = [ -3 +  3i;
     -11 +  5i;
      23 + 48i;
     -41 +  2i;
     -28 - 31i];

% Solver setup input argument initialization
method = 'CGS';
precon = 'P';
norm_p = 'I';
tol    = 1e-10;
maxitn = int64(1000);
anorm  = 0;
sigmax = 0;
monit  = int64(0);
lwork  = max([121+n*(3+m)+m*(m+5),120+7*n,120+(2*n+m)*(m+2)+2*n,120+10*n]);

% Initialize solver
[lwreq, work, ifail] = f11br( ...
                              method, precon, n, m, tol, maxitn, anorm, ...
                              sigmax, monit, lwork, 'norm_p', norm_p);

rdiag = zeros(n, 1);
% Calculate reciprocal diagonal matrix elements for preconditioner
if precon == 'P' || precon == 'p'
   dcount = zeros(n, 1, 'int64');

   for j = 1:nz
     if irow(j) == icol(j)
       dcount(irow(j)) = dcount(irow(j)) + 1;
       if a(j) ~= 0
         rdiag(irow(j)) = 1/a(j);
       else
         error('Matrix has a zero diagonal element');
       end
     end
   end

   for j = 1:n
     if dcount(j)==0
       error('Matrix has a missing diagonal element');
     elseif dcount(j) > 1
       error('Matrix has a multiple diagonal element');
     end
   end
end
% Other preconditioner inputs
omega = 1.4;
ckdrf = 'C';

% Solver inputs
irevcm = int64(0);
wgt    = zeros(n, 1);
x      = complex(zeros(n, 1));

% matrix-vector input
ckxnf = 'C';

% solve the linear system by revere communication
while (irevcm ~= 4)
  [irevcm, x, b, work, ifail] = f11bs( ...
                                       irevcm, x, b, wgt, work);

  if (irevcm == -1)
    % b = A^T x
    [b, ifail] = f11xn( ...
                        'T', a, irow, icol, ckxnf, x);
    ckxnf = 'N';
  elseif (irevcm == 1)
    % b = Ax
    [b, ifail] = f11xn( ...
                        'N', a, irow, icol, ckxnf, x);
    ckxnf = 'N';
  elseif (irevcm == 2)
    % SSOR preconditioning
    [b, ifail] = f11dr( ...
                        'N', a, irow, icol, rdiag, omega, ckdrf, x);
    ckdrf = 'N';
  end
end

if ifail == 0
   % Successful termination, get solution details from communication array
  [itn, stplhs, stprhs, anorm, sigmax, work, ifail] = ...
    f11bt(work);

  fprintf('Converged in %d iterations\n', itn);
  fprintf('Matrix norm         = %16.3e\n', anorm);
  fprintf('Final residual norm = %16.4e\n\n', stplhs);
  disp('Solution');
  disp(x);

end



f11dr example results

Converged in 5 iterations
Matrix norm         =        1.500e+01
Final residual norm =       4.6185e-14

Solution
   1.0000 + 2.0000i
   2.0000 + 3.0000i
   3.0000 + 4.0000i
   4.0000 + 5.0000i
   5.0000 + 6.0000i
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015