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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sparse_real_gen_precon_ssor_solve (f11dd)

Purpose

nag_sparse_real_gen_precon_ssor_solve (f11dd) solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse nonsymmetric matrix, represented in coordinate storage format.

Syntax

[x, ifail] = f11dd(trans, a, irow, icol, rdiag, omega, check, y, 'n', n, 'nnz', nnz)
[x, ifail] = nag_sparse_real_gen_precon_ssor_solve(trans, a, irow, icol, rdiag, omega, check, y, 'n', n, 'nnz', nnz)

Description

nag_sparse_real_gen_precon_ssor_solve (f11dd) solves a system of linear equations
Mx = y, or MTx = y,
Mx=y, or MTx=y,
according to the value of the parameter trans, where the matrix
M = 1/(ω(2ω))(D + ωL) D1 (D + ωU)
M=1ω(2-ω) (D+ω L) D-1 (D+ω U)
corresponds to symmetric successive-over-relaxation (SSOR) (see Young (1971)) applied to a linear system Ax = bAx=b, where AA is a real sparse nonsymmetric matrix stored in coordinate storage (CS) format (see Section [Coordinate storage (CS) format] in the F11 Chapter Introduction).
In the definition of MM given above DD is the diagonal part of AA, LL is the strictly lower triangular part of AA, UU is the strictly upper triangular part of AA, and ωω is a user-defined relaxation parameter.
It is envisaged that a common use of nag_sparse_real_gen_precon_ssor_solve (f11dd) will be to carry out the preconditioning step required in the application of nag_sparse_real_gen_basic_solver (f11be) to sparse linear systems. For an illustration of this use of nag_sparse_real_gen_precon_ssor_solve (f11dd) see the example program given in Section [Example]. nag_sparse_real_gen_precon_ssor_solve (f11dd) is also used for this purpose by the Black Box function nag_sparse_real_gen_solve_jacssor (f11de).

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 MM is transposed.
trans = 'N'trans='N'
Mx = yMx=y is solved.
trans = 'T'trans='T'
MTx = yMTx=y is solved.
Constraint: trans = 'N'trans='N' or 'T''T'.
2:     a(nnz) – double array
nnz, the dimension of the array, must satisfy the constraint 1nnzn21nnzn2.
The nonzero elements in the matrix AA, 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_gen_sort (f11za) may be used to order the elements in this way.
3:     irow(nnz) – int64int32nag_int array
4:     icol(nnz) – int64int32nag_int array
nnz, the dimension of the array, must satisfy the constraint 1nnzn21nnzn2.
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_real_gen_sort (f11za)):
  • 1irow(i)n1irowin and 1icol(i)n1icolin, for i = 1,2,,nnzi=1,2,,nnz;
  • either irow(i1) < irow(i)irowi-1<irowi or both irow(i1) = irow(i)irowi-1=irowi and icol(i1) < icol(i)icoli-1<icoli, for i = 2,3,,nnzi=2,3,,nnz.
5:     rdiag(n) – double array
n, the dimension of the array, must satisfy the constraint n1n1.
The elements of the diagonal matrix D1D-1, where DD is the diagonal part of AA.
6:     omega – double scalar
The relaxation parameter ωω.
Constraint: 0.0 < omega < 2.00.0<omega<2.0.
7:     check – string (length ≥ 1)
Specifies whether or not the CS representation of the matrix MM should be checked.
check = 'C'check='C'
Checks are carried on the values of n, nnz, irow, icol and omega.
check = 'N'check='N'
None of these checks are carried out.
Constraint: check = 'C'check='C' or 'N''N'.
8:     y(n) – double array
n, the dimension of the array, must satisfy the constraint n1n1.
The right-hand side vector yy.

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.)
nn, the order of the matrix AA.
Constraint: n1n1.
2:     nnz – 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 AA.
Constraint: 1nnzn21nnzn2.

Input Parameters Omitted from the MATLAB Interface

iwork

Output Parameters

1:     x(n) – double array
The solution vector xx.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,trans'N'trans'N' or 'T''T',
orcheck'C'check'C' or 'N''N'.
  ifail = 2ifail=2
On entry,n < 1n<1,
ornnz < 1nnz<1,
ornnz > n2nnz>n2,
oromega lies outside the interval (0.0,2.0)(0.0,2.0),
  ifail = 3ifail=3
On entry, the arrays irow and icol fail to satisfy the following constraints:
  • 1irow(i)n1irowin and 1icol(i)n1icolin, for i = 1,2,,nnzi=1,2,,nnz;
  • irow(i1) < irow(i)irowi-1<irowi or irow(i1) = irow(i)irowi-1=irowi and icol(i1) < icol(i)icoli-1<icoli, for i = 2,3,,nnzi=2,3,,nnz.
Therefore a nonzero element has been supplied which does not lie in the matrix AA, is out of order, or has duplicate row and column indices. Call nag_sparse_real_gen_sort (f11za) to reorder and sum or remove duplicates.
  ifail = 4ifail=4
On entry, the matrix AA has a zero diagonal element. The SSOR preconditioner is not appropriate for this problem.

Accuracy

If trans = 'N'trans='N' the computed solution xx is the exact solution of a perturbed system of equations (M + δM)x = y(M+δM)x=y, where
|δM|c(n)ε|D + ωL||D1||D + ωU|,
|δM|c(n)ε|D+ωL||D-1||D+ωU|,
c(n)c(n) is a modest linear function of nn, and εε is the machine precision. An equivalent result holds when trans = 'T'trans='T'.

Further Comments

Timing

The time taken for a call to nag_sparse_real_gen_precon_ssor_solve (f11dd) is proportional to nnz.

Use of check

It is expected that a common use of nag_sparse_real_gen_precon_ssor_solve (f11dd) will be to carry out the preconditioning step required in the application of nag_sparse_real_gen_basic_solver (f11be) to sparse linear systems. In this situation nag_sparse_real_gen_precon_ssor_solve (f11dd) is likely to be called many times with the same matrix MM. In the interests of both reliability and efficiency, you are recommended to set check = 'C'check='C' for the first of such calls, and for all subsequent calls set check = 'N'check='N'.

Example

function nag_sparse_real_gen_precon_ssor_solve_example
n = int64(5);
m = int64(2);
nz = int64(16);
method = 'rgmres';
precon = 'P';
iterm = int64(1);
tol = 1e-10;
maxitn = int64(1000);
anorm = 0;
sigmax = 0;
trans = 'N';
omega = 1.1;
check = 'C';
weight = 'N';
monit = int64(0);
lwork = max([n*(m+3)+m*(m+5)+101,7*n+100,(2*n+m)*(m+2)+n+100,10*n+100]);
a = zeros(3*nz, 1);
a(1:nz) = [2; 1; -1; -3; -2; 1; 1; 5; 3; 1; -2; -3; -1; 4; -2; -6];
irow = zeros(3*nz, 1, 'int64');
irow(1:nz) = [1; 1; 1; 2; 2; 2; 3; 3; 3; 3; 4; 4; 4; 5; 5; 5];
icol = zeros(3*nz, 1, 'int64');
icol(1:nz) = [1; 2; 4; 2; 3; 5; 1; 3; 4; 5; 1; 4; 5; 2; 3; 5];
b = [0; -7; 33; -19; -28];
x = zeros(n, 1);
iwork = zeros(2*n + 1, 1, 'int64');
rdiag = zeros(n, 1);
wgt = zeros(n, 1);
irevcm = int64(0);
ckxaf = 'C';
ckddf = 'C';

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

% Calculate reciprocal diagonal matrix elements if necessary
if strcmpi(precon, 'P')

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

  for i = 1:n
    if iwork(i) == 0
      error('Matrix has a missing diagonal element');
    elseif iwork(i) >= 2
      error('Matrix has a multiple diagonal element');
    end
  end

end

% Solve the linear system
while irevcm ~= 4
    [irevcm, x, b, work, ifail] = ...
        nag_sparse_real_gen_basic_solver(irevcm, x, b, wgt, work);

  if (irevcm == -1)
    % Compute transposed matrix-vector product
    [b, ifail] = nag_sparse_real_gen_matvec('T', a(1:nz), irow(1:nz), ...
                                            icol(1:nz), ckxaf, x);
    ckxaf = 'N';
  elseif (irevcm == 1)
    % Compute matrix-vector product
    [b, ifail] = nag_sparse_real_gen_matvec('N', a(1:nz), irow(1:nz), ...
                                            icol(1:nz), ckxaf, x);
    ckxaf = 'N';
  elseif (irevcm == 2)
    % SSOR preconditioning
    [b, ifail] = nag_sparse_real_gen_precon_ssor_solve(trans, a(1:nz), ...
                             irow(1:nz), icol(1:nz), rdiag, omega, ckddf, x);
    ckddf = 'N';
  end
end

% Get information about the computation
[itn, stplhs, stprhs, anorm, sigmax, ifail] = ...
    nag_sparse_real_gen_basic_diag(work);
fprintf('\nConverged in %d iterations\n', itn);
fprintf('Matrix norm = %16.3e\n', anorm);
fprintf('Final residual norm = %16.3e\n', stplhs);
fprintf('      X\n');
disp(x);
 

Converged in 12 iterations
Matrix norm =        1.200e+01
Final residual norm =        3.841e-09
      X
    1.0000
    2.0000
    3.0000
    4.0000
    5.0000


function f11dd_example
n = int64(5);
m = int64(2);
nz = int64(16);
method = 'rgmres';
precon = 'P';
iterm = int64(1);
tol = 1e-10;
maxitn = int64(1000);
anorm = 0;
sigmax = 0;
trans = 'N';
omega = 1.1;
check = 'C';
weight = 'N';
monit = int64(0);
lwork = max([n*(m+3)+m*(m+5)+101,7*n+100,(2*n+m)*(m+2)+n+100,10*n+100]);
a = zeros(3*nz, 1);
a(1:nz) = [2; 1; -1; -3; -2; 1; 1; 5; 3; 1; -2; -3; -1; 4; -2; -6];
irow = zeros(3*nz, 1, 'int64');
irow(1:nz) = [1; 1; 1; 2; 2; 2; 3; 3; 3; 3; 4; 4; 4; 5; 5; 5];
icol = zeros(3*nz, 1, 'int64');
icol(1:nz) = [1; 2; 4; 2; 3; 5; 1; 3; 4; 5; 1; 4; 5; 2; 3; 5];
b = [0; -7; 33; -19; -28];
x = zeros(n, 1);
iwork = zeros(2*n + 1, 1, 'int64');
rdiag = zeros(n, 1);
wgt = zeros(n, 1);
irevcm = int64(0);
ckxaf = 'C';
ckddf = 'C';

% Initialize solver
[lwreq, work, ifail] = ...
     f11bd(method, precon, n, m, tol, maxitn, anorm, sigmax, monit, lwork, ...
           'norm_p', 'I');

% Calculate reciprocal diagonal matrix elements if necessary
if strcmpi(precon, 'P')

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

  for i = 1:n
    if iwork(i) == 0
      error('Matrix has a missing diagonal element');
    elseif iwork(i) >= 2
      error('Matrix has a multiple diagonal element');
    end
  end

end

% Solve the linear system
while irevcm ~= 4
    [irevcm, x, b, work, ifail] = f11be(irevcm, x, b, wgt, work);

  if (irevcm == -1)
    % Compute transposed matrix-vector product
    [b, ifail] = f11xa('T', a(1:nz), irow(1:nz), icol(1:nz), ckxaf, x);
    ckxaf = 'N';
  elseif (irevcm == 1)
    % Compute matrix-vector product
    [b, ifail] = f11xa('N', a(1:nz), irow(1:nz), icol(1:nz), ckxaf, x);
    ckxaf = 'N';
  elseif (irevcm == 2)
    % SSOR preconditioning
    [b, ifail] = f11dd(trans, a(1:nz), irow(1:nz), icol(1:nz), rdiag,...
                       omega, ckddf, x);
    ckddf = 'N';
  end
end

% Get information about the computation
[itn, stplhs, stprhs, anorm, sigmax, ifail] = f11bf(work);
fprintf('\nConverged in %d iterations\n', itn);
fprintf('Matrix norm = %16.3e\n', anorm);
fprintf('Final residual norm = %16.3e\n', stplhs);
fprintf('      X\n');
disp(x);
 

Converged in 12 iterations
Matrix norm =        1.200e+01
Final residual norm =        3.841e-09
      X
    1.0000
    2.0000
    3.0000
    4.0000
    5.0000



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