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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_sparse_complex_herm_precon_ilu_solve (f11jp)

## Purpose

nag_sparse_complex_herm_precon_ilu_solve (f11jp) solves a system of complex linear equations involving the incomplete Cholesky preconditioning matrix generated by nag_sparse_complex_herm_precon_ilu (f11jn).

## Syntax

[x, ifail] = f11jp(a, irow, icol, ipiv, istr, check, y, 'n', n, 'la', la)
[x, ifail] = nag_sparse_complex_herm_precon_ilu_solve(a, irow, icol, ipiv, istr, check, y, 'n', n, 'la', la)

## Description

nag_sparse_complex_herm_precon_ilu_solve (f11jp) solves a system of linear equations
 $Mx=y$
involving the preconditioning matrix $M=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, corresponding to an incomplete Cholesky decomposition of a complex sparse Hermitian matrix stored in symmetric coordinate storage (SCS) format (see Symmetric coordinate storage (SCS) format in the F11 Chapter Introduction), as generated by nag_sparse_complex_herm_precon_ilu (f11jn).
In the above decomposition $L$ is a complex lower triangular sparse matrix with unit diagonal, $D$ is a real diagonal matrix and $P$ is a permutation matrix. $L$ and $D$ are supplied to nag_sparse_complex_herm_precon_ilu_solve (f11jp) through the matrix
 $C=L+D-1-I$
which is a lower triangular $n$ by $n$ complex sparse matrix, stored in SCS format, as returned by nag_sparse_complex_herm_precon_ilu (f11jn). The permutation matrix $P$ is returned from nag_sparse_complex_herm_precon_ilu (f11jn) via the array ipiv.
nag_sparse_complex_herm_precon_ilu_solve (f11jp) may also be used in combination with nag_sparse_complex_herm_precon_ilu (f11jn) to solve a sparse complex Hermitian positive definite system of linear equations directly (see nag_sparse_complex_herm_precon_ilu (f11jn)). This is illustrated in Example.

None.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left({\mathbf{la}}\right)$ – complex array
The values returned in the array a by a previous call to nag_sparse_complex_herm_precon_ilu (f11jn).
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_complex_herm_precon_ilu (f11jn).
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.
${\mathbf{check}}=\text{'N'}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
7:     $\mathrm{y}\left({\mathbf{n}}\right)$ – complex 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_complex_herm_precon_ilu (f11jn).
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 supplied in the preceding call to nag_sparse_complex_herm_precon_ilu (f11jn).

### Output Parameters

1:     $\mathrm{x}\left({\mathbf{n}}\right)$ – complex 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_complex_herm_precon_ilu_solve (f11jp) has been preceded by a valid call to nag_sparse_complex_herm_precon_ilu (f11jn) and that the arrays a, irow, icol, ipiv and istr have not been corrupted between the two calls.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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
 $δM≤cnεPLDLHPT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

### Timing

The time taken for a call to nag_sparse_complex_herm_precon_ilu_solve (f11jp) is proportional to the value of nnzc returned from nag_sparse_complex_herm_precon_ilu (f11jn).

## Example

This example reads in a complex sparse Hermitian positive definite matrix $A$ and a vector $y$. It then calls nag_sparse_complex_herm_precon_ilu (f11jn), with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete Cholesky decomposition of $A$:
 $A=PLDLHPT.$
Finally it calls nag_sparse_complex_herm_precon_ilu_solve (f11jp) to solve the system
 $PLDLHPTx=y.$
```function f11jp_example

fprintf('f11jp 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    = zeros(3*nz,1);
irow = zeros(3*nz, 1, 'int64');
icol = zeros(3*nz, 1, 'int64');
a(1:nz) = [ 6 + 0.i; -1 + 1.i;  6 + 0.i;  0 + 1.i;
5 + 0.i;  5 + 0.i;  2 - 2.i;  4 + 0.i;
1 + 1.i;  2 + 0.i;  6 + 0.i; -4 + 3.i;
0 + 1.i; -1 + 0.i;  6 + 0.i; -1 - 1.i;
0 - 1.i;  9 + 0.i;  1 + 3.i;  1 + 2.i;
-1 + 0.i;  1 + 4.i;  9 + 0.i];
b       = [ 8 + 54i;-10 - 92i; 25 + 27i; 26 - 28i;
54 + 12i; 26 - 22i; 47 + 65i; 71 - 57i;
60 + 70i];
irow(1:nz) = int64([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) = int64([1;1;2;2;3;4;1;5;3;4;6;2;5;6;7;4;6;8;1;5;6;8;9]);

% 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);

```
```f11jp 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

```