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# NAG Toolbox: nag_sparse_real_symm_precon_ichol_solve (f11jb)

## Purpose

nag_sparse_real_symm_precon_ichol_solve (f11jb) solves a system of linear equations involving the incomplete Cholesky preconditioning matrix generated by nag_sparse_real_symm_precon_ichol (f11ja).

## Syntax

[x, ifail] = f11jb(a, irow, icol, ipiv, istr, check, y, 'n', n, 'la', la)
[x, ifail] = nag_sparse_real_symm_precon_ichol_solve(a, irow, icol, ipiv, istr, check, y, 'n', n, 'la', la)

## Description

nag_sparse_real_symm_precon_ichol_solve (f11jb) solves a system of linear equations
 $Mx=y$
involving the preconditioning matrix $M=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, corresponding to an incomplete Cholesky decomposition of a sparse symmetric matrix stored in symmetric coordinate storage (SCS) format (see Symmetric coordinate storage (SCS) format in the F11 Chapter Introduction), as generated by nag_sparse_real_symm_precon_ichol (f11ja).
In the above decomposition $L$ is a lower triangular sparse matrix with unit diagonal, $D$ is a diagonal matrix and $P$ is a permutation matrix. $L$ and $D$ are supplied to nag_sparse_real_symm_precon_ichol_solve (f11jb) through the matrix
 $C=L+D-1-I$
which is a lower triangular n by n sparse matrix, stored in SCS format, as returned by nag_sparse_real_symm_precon_ichol (f11ja). The permutation matrix $P$ is returned from nag_sparse_real_symm_precon_ichol (f11ja) via the array ipiv.
It is envisaged that a common use of nag_sparse_real_symm_precon_ichol_solve (f11jb) will be to carry out the preconditioning step required in the application of nag_sparse_real_symm_basic_solver (f11ge) to sparse symmetric linear systems. nag_sparse_real_symm_precon_ichol_solve (f11jb) is used for this purpose by the Black Box function nag_sparse_real_symm_solve_ichol (f11jc).
nag_sparse_real_symm_precon_ichol_solve (f11jb) may also be used in combination with nag_sparse_real_symm_precon_ichol (f11ja) to solve a sparse symmetric positive definite system of linear equations directly (see Direct Solution of Systems in nag_sparse_real_symm_precon_ichol (f11ja)). This use of nag_sparse_real_symm_precon_ichol_solve (f11jb) is demonstrated in Example.

None.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left({\mathbf{la}}\right)$ – double array
The values returned in the array a by a previous call to nag_sparse_real_symm_precon_ichol (f11ja).
2:     $\mathrm{irow}\left({\mathbf{la}}\right)$int64int32nag_int array
3:     $\mathrm{icol}\left({\mathbf{la}}\right)$int64int32nag_int array
4:     $\mathrm{ipiv}\left({\mathbf{n}}\right)$int64int32nag_int array
5:     $\mathrm{istr}\left({\mathbf{n}}+1\right)$int64int32nag_int array
The values returned in arrays irow, icol, ipiv and istr by a previous call to nag_sparse_real_symm_precon_ichol (f11ja).
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, irow, icol, ipiv and istr.
${\mathbf{check}}=\text{'N'}$
No checks are carried out.
See also Use of check.
Constraint: ${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
7:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
The right-hand side vector $y$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the arrays ipiv, y. (An error is raised if these dimensions are not equal.)
$n$, the order of the matrix $M$. This must be the same value as was supplied in the preceding call to nag_sparse_real_symm_precon_ichol (f11ja).
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathrm{la}$int64int32nag_int scalar
Default: the dimension of the arrays a, irow, icol. (An error is raised if these dimensions are not equal.)
The dimension of the arrays a, irow and icol. this must be the same value returned by the preceding call to nag_sparse_real_symm_precon_ichol (f11ja).

### Output Parameters

1:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double 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

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{check}}\ne \text{'C'}$ or $\text{'N'}$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{n}}<1$.
${\mathbf{ifail}}=3$
On entry, the SCS representation of the preconditioning matrix $M$ is invalid. Further details are given in the error message. Check that the call to nag_sparse_real_symm_precon_ichol_solve (f11jb) has been preceded by a valid call to nag_sparse_real_symm_precon_ichol (f11ja) and that the arrays a, irow, icol, ipiv and istr have not been corrupted between the two calls.
${\mathbf{ifail}}=-99$
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

The computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $δM≤cnεPLDLTPT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## Further Comments

### Timing

The time taken for a call to nag_sparse_real_symm_precon_ichol_solve (f11jb) is proportional to the value of nnzc returned from nag_sparse_real_symm_precon_ichol (f11ja).

### Use of check

It is expected that a common use of nag_sparse_real_symm_precon_ichol_solve (f11jb) will be to carry out the preconditioning step required in the application of nag_sparse_real_symm_basic_solver (f11ge) to sparse symmetric linear systems. In this situation nag_sparse_real_symm_precon_ichol_solve (f11jb) 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 ${\mathbf{check}}=\text{'C'}$ for the first of such calls, and to set ${\mathbf{check}}=\text{'N'}$ for all subsequent calls.

## Example

This example reads in a symmetric positive definite sparse matrix $A$ and a vector $y$. It then calls nag_sparse_real_symm_precon_ichol (f11ja), with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete Cholesky decomposition of $A$:
 $A=PLDLTPT.$
Then it calls nag_sparse_real_symm_precon_ichol_solve (f11jb) to solve the system
 $PLDLTPTx=y.$
It then repeats the exercise for the same matrix permuted with the bandwidth-reducing Reverse Cuthill–McKee permutation, calculated with nag_sparse_sym_rcm (f11ye).
```function f11jb_example

fprintf('f11jb example results\n\n');

% Solve sparse symmetric system Ax = b using CG method with
% Incomplete Cholesky preconditioning (IC)

% Define A and b
n  = int64(9);
nz = int64(23);
a = zeros(3*nz,1);
irow = zeros(3*nz,1,'int64');
icol = irow;
a(1:nz)    = [ 4    -1     6     1     2     3     2     4 ...
1     2     6    -4     1    -1     6    -1 ...
-1     3     1     1    -1     1     4 ];
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          = [4.10 -2.94  1.41 ...
2.53  4.35  1.29 ...
5.01  0.52  4.57];

% Setup IC factorization

lfill  = int64(-1);
dtol   = 0;
mic    = 'N';
dscale = 0;
ipiv   = zeros(n, 1, 'int64');
[a, irow, icol, ipiv, istr, nnzc, npivm, ifail] = ...
f11ja(...
nz, a, irow, icol, lfill, dtol, mic, dscale, ipiv);

% Solve Ax = b, using Cholesky factorization
[x, ifail] = f11jb(...
a, irow, icol, ipiv, istr, 'C', b);
fprintf('\n   Solution of linear System\n');
fprintf('%16.4f\n', x);

% Redo calculation after reverse Cuthill-McKee permutation for
% bandwidth reduction

% Reverse Cuthill-McKee
[irow,icol,a,perm_fwd,perm_inv] =  do_rcm(n,nz,irow,icol,a);

% Repeat factorization and solution
[a, irow, icol, ipiv, istr, nnzc, npivm, ifail] = ...
f11ja(...
nz, a, irow, icol, lfill, dtol, mic, dscale, ipiv);

% Permute rhs vector and solution (PAP^T)(Px) = Pb.
y = b(perm_fwd(:));
[u, ifail] = f11jb(...
a, irow, icol, ipiv, istr, 'C', y);

x = u(perm_inv(:));
fprintf('\n   Solution of linear System with reverse Cuthill-McKee\n');
fprintf('%16.4f\n', x);

function [irow_out,icol_out,a_out,perm_fwd,perm_inv] = ...
do_rcm(n,nz,irow,icol,a)

% Reverse Cuthill-McKee for symmetric matrix

% Add upper triangle, sort and remove duplicate diagonal elements
irow(nz+1:2*nz) = icol(1:nz);
icol(nz+1:2*nz) = irow(1:nz);
a(nz+1:2*nz)    = a(1:nz);
nnz = nz + nz;
[nnz,a,icol,irow,istr,ifail] = f11za(n,nnz,a,icol,irow,'R','F');

% Reverse Cuthill-McKee
lopts(1:5) = [false false true true true];
mask = zeros(n,1,'int64');
[perm_fwd, info, ifail] = f11ye(istr, irow, lopts, mask,'n',n);

% Inverse perm
perm_inv(perm_fwd(1:n)) = [1:n];

% Apply permutation on column/row indices
icol(1:nnz) = perm_inv(icol(1:nnz));
irow(1:nnz) = perm_inv(irow(1:nnz));

% collect only lower triangle
j = 0;
for i=1:nnz
if icol(i)<=irow(i)
j = j + 1;
a(j) = a(i);
irow(j) = irow(i);
icol(j) = icol(i);
end
end

nnz = int64(j);
% Sort collected lower triangle
[nnz, a, irow, icol, istr, ifail] = ...
f11zb(...
n, nnz, a, irow, icol, 'S', 'K');

% copy to output arrays
a_out = zeros(3*nz,1);
irow_out = zeros(3*nz,1,'int64');
icol_out = irow_out;
a_out(1:nnz) = a(1:nnz);
irow_out(1:nnz) = irow(1:nnz);
icol_out(1:nnz) = icol(1:nnz);
```
```f11jb example results

Solution of linear System
0.7000
0.1600
0.5200
0.7700
0.2800
0.2100
0.9300
0.2000
0.9000

Solution of linear System with reverse Cuthill-McKee
0.7000
0.1600
0.5200
0.7700
0.2800
0.2100
0.9300
0.2000
0.9000
```

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