NAG Toolbox: nag_lapack_zpstrf (f07kr)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)

Specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.

$\mathbf{uplo}=\text{'U'}$

The upper triangular part of $A$ is stored and $A$ is factorized as $U^{\mathrm{H}}U$, where $U$ is upper triangular.

$\mathbf{uplo}=\text{'L'}$

The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{H}}$, where $L$ is lower triangular.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The $n$ by $n$ Hermitian positive semidefinite matrix $A$.

• If $\mathbf{uplo}=\text{'U'}$, the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
• If $\mathbf{uplo}=\text{'L'}$, the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the first dimension of the array a and the second dimension of the array a.

$n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

2:     $\mathrm{tol}$ – double scalar

Default: $-1$

User defined tolerance. If $\mathbf{tol}<0$, then $n\times {\displaystyle \underset{k=1,n}{\text{max}}}\left|A_{kk}\right|\times \mathit{machine precision}$ will be used. The algorithm terminates at the $r$th step if the $\left(r+1\right)$th step pivot $<\mathbf{tol}$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

If $\mathbf{uplo}=\text{'U'}$, the first rank rows of the upper triangle of $A$ are overwritten with the nonzero elements of the Cholesky factor $U$, and the remaining rows of the triangle are destroyed.

If $\mathbf{uplo}=\text{'L'}$, the first rank columns of the lower triangle of $A$ are overwritten with the nonzero elements of the Cholesky factor $L$, and the remaining columns of the triangle are destroyed.

2:     $\mathrm{piv}\left(\mathbf{n}\right)$ – int64int32nag_int array

piv is such that the nonzero entries of $P$ are $P\left(\mathbf{piv}\left(\mathit{k}\right),\mathit{k}\right)=1$, for $\mathit{k}=1,2,\dots ,n$.

3:     $\mathrm{rank}$ – int64int32nag_int scalar

The computed rank of $A$ given by the number of steps the algorithm completed.

4:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W  $\mathbf{info}=1$

The matrix $A$ is not positive definite. It is either positive semidefinite with computed rank as returned in rank and less than $n$, or it may be indefinite, see Further Comments.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07kr_example

function f07kr_example


fprintf('f07kr example results\n\n');

% Semidefinite matrix A
a = [12.40,       2.39,       5.50+0.05i, 4.47,       11.89;
      2.39,       1.63,       1.04+0.10i, 1.14,        1.81;
      5.50+0.05i, 1.04+0.10i, 2.45,       1.98-0.03i,  5.28-0.02i;
      4.47,       1.14,       1.98-0.03i, 1.71,        4.14;
     11.89,       1.81,       5.28-0.02i, 4.14,       11.63];

% Catch warnings about rank defficient matrix ifail=1
wstat = warning();
warning('OFF');

% Factorize a
uplo = 'l';
[afac, piv, rank, info] = f07kr( ...
                                 uplo, a);

fprintf('\nComputed rank: %d\n\n', rank);
% Zero out columns rank+1 onwards and display
afac(:, rank+1:5) = 0;

ncol   = int64(80);
indent = int64(0);
[ifail] = x04db( ...
                 uplo, 'non-unit', afac, 'bracketed', 'f5.2', 'Factor', ...
                 'Integer', 'Integer', ncol, indent);

fprintf('\n piv:\n   ');
fprintf('%14d', piv);
fprintf('\n');

warning(wstat);

f07kr example results


Computed rank: 3

 Factor
                1             2             3             4             5
 1  ( 3.52, 0.00)
 2  ( 0.68, 0.00) ( 1.08, 0.00)
 3  ( 1.27, 0.00) ( 0.26, 0.00) ( 0.18, 0.00)
 4  ( 1.56, 0.01) (-0.02, 0.08) ( 0.01,-0.05) ( 0.00, 0.00)
 5  ( 3.38, 0.00) (-0.45, 0.00) (-0.17, 0.00) ( 0.00, 0.00) ( 0.00, 0.00)

 piv:
                1             2             4             3             5