NAG Toolbox: nag_lapack_zhpsv (f07pn)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

If $\mathbf{uplo}=\text{'U'}$, the upper triangle of $A$ is stored.

If $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A$ is stored.

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

2:     $\mathrm{ap}\left(:\right)$ – complex array

The dimension of the array ap must be at least $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$

The $n$ by $n$ Hermitian matrix $A$, packed by columns.

More precisely,

• if $\mathbf{uplo}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{ap}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

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

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

Note: to solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $\mathit{ldb}=\max \left(1,\mathbf{n}\right)$.

The $n$ by $r$ right-hand side matrix $B$.

Optional Input Parameters

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

Default: the first dimension of the array b.

$n$, the number of linear equations, i.e., the order of the matrix $A$.

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

2:     $\mathrm{nrhs\_p}$ – int64int32nag_int scalar

Default: the second dimension of the array b.

$r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.

Constraint: $\mathbf{nrhs\_p}\ge 0$.

Output Parameters

1:     $\mathrm{ap}\left(:\right)$ – complex array

The dimension of the array ap will be $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$

The block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{H}}$ or $A=LD{L}^{\mathrm{H}}$ as computed by nag_lapack_zhptrf (f07pr), stored as a packed triangular matrix in the same storage format as $A$.

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

Details of the interchanges and the block structure of $D$. More precisely,

• if $\mathbf{ipiv}\left(i\right)=k>0$, $d_{ii}$ is a $1$ by $1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
• if $\mathbf{uplo}=\text{'U'}$ and $\mathbf{ipiv}\left(i-1\right)=\mathbf{ipiv}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& d_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i-1\right)$th row and column of $A$ were interchanged with the $l$th row and column;
• if $\mathbf{uplo}=\text{'L'}$ and $\mathbf{ipiv}\left(i\right)=\mathbf{ipiv}\left(i+1\right)=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& d_{i+1,i}\\ d_{i+1,i}& d_{i+1,i+1}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i+1\right)$th row and column of $A$ were interchanged with the $m$th row and column.

3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

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

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

Note: to solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $\mathit{ldb}=\max \left(1,\mathbf{n}\right)$.

If $\mathbf{info}=\mathbf{0}$, the $n$ by $r$ solution matrix $X$.

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  $\mathbf{info}>0$

Element $\_$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be computed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07pn_example

function f07pn_example


fprintf('f07pn example results\n\n');

% Hermitian indefinite matrix A (Upper triangular part stored in packed format)
uplo = 'Upper';
n  = int64(4);
ap = [-1.84;
       0.11 - 0.11i;  -4.63 + 0i;
      -1.78 - 1.18i;  -1.84 + 0.03i;  -8.87 + 0i;
       3.91 - 1.50i;   2.21 + 0.21i;   1.58 - 0.90i;  -1.36 + 0i];

% RHS
b = [ 2.98 - 10.18i;
     -9.58 +  3.88i;
     -0.77 - 16.05i;
      7.79 +  5.48i];

% Solve
[apf, ipiv, x, info] = f07pn( ...
                              uplo, ap, b);

disp('Solution');
disp(x);

[ifail] = x04dc( ...
                 uplo, 'Non-unit', n, apf, 'Details of factorization');

fprintf('\nPivot indices\n   ');
fprintf('%11d', ipiv);
fprintf('\n');

f07pn example results

Solution
   2.0000 + 1.0000i
   3.0000 - 2.0000i
  -1.0000 + 2.0000i
   1.0000 - 1.0000i

 Details of factorization
             1          2          3          4
 1     -7.1028     0.2997     0.3397    -0.1518
        0.0000     0.1578     0.0303     0.3743

 2                -5.4176     0.5637     0.3100
                   0.0000     0.2850     0.0433

 3                           -1.8400     3.9100
                              0.0000    -1.5000

 4                                      -1.3600
                                         0.0000

Pivot indices
             1          2         -1         -1