NAG Toolbox: nag_lapack_zhpgst (f08ts)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates how the standard form is computed.

$\mathbf{itype}=1$

• if $\mathbf{uplo}=\text{'U'}$, $C=U^{-\mathrm{H}}A{U}^{-1}$;
• if $\mathbf{uplo}=\text{'L'}$, $C=L^{-1}A{L}^{-\mathrm{H}}$.

$\mathbf{itype}=2$ or $3$

• if $\mathbf{uplo}=\text{'U'}$, $C=UA{U}^{\mathrm{H}}$;
• if $\mathbf{uplo}=\text{'L'}$, $C=L^{\mathrm{H}}AL$.

Constraint: $\mathbf{itype}=1$, $2$ or $3$.

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

Indicates whether the upper or lower triangular part of $A$ is stored and how $B$ has been factorized.

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

The upper triangular part of $A$ is stored and $B=U^{\mathrm{H}}U$.

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

The lower triangular part of $A$ is stored and $B=L{L}^{\mathrm{H}}$.

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

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

$n$, the order of the matrices $A$ and $B$.

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

4:     $\mathrm{ap}\left(:\right)$ – 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 upper or lower triangle of 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 $i\ge j$.

5:     $\mathrm{bp}\left(:\right)$ – complex array

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

The Cholesky factor of $B$ as specified by uplo and returned by nag_lapack_zpptrf (f07gr).

Optional Input Parameters

None.

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 upper or lower triangle of ap stores the corresponding upper or lower triangle of $C$ as specified by itype and uplo, using the same packed storage format as described above.

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

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

Error Indicators and Warnings

   $\mathbf{info}=-i$

If $\mathbf{info}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:

1: itype, 2: uplo, 3: n, 4: ap, 5: bp, 6: info.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08ts_example

function f08ts_example


fprintf('f08ts example results\n\n');

% Hermitian matrices A and B stored in packed (Lower) format
uplo = 'L';
n = int64(4);
ap = [-7.36;  0.77 + 0.43i; -0.64 + 0.92i;  3.01 + 6.97i;
              3.49 + 0i;     2.19 - 4.45i;  1.90 - 3.73i;
                             0.12 + 0i;     2.88 + 3.17i;
                                           -2.54 + 0i];
bp = [ 3.23;  1.51 + 1.92i;  1.90 - 0.84i;  0.42 - 2.50i; 
              3.58 + 0i;    -0.23 - 1.11i; -1.18 - 1.37i;
                             4.09 - 0i;     2.33 + 0.14i;
                                            4.29 - 0i];

% Compute Cholesky factorization of B
[lp, info] = f07gr( ...
		    uplo, n, bp);

% Reduce problem to standard form Cy = lambda y,
% where y = L^H x, C = L^-1 A L^-H
itype = int64(1);
[cp, info] = f08ts( ...
		   itype, uplo, n, ap, lp);

% Reduce C to tridiagonal form T = Q^H C Q
[qp, d, e, tau, info] = f08gs( ...
			       uplo, n, cp);

% Calculate eigenvalues of T (same as C)
[w, ~, info] = f08jf(d, e);

disp('Eigenvalues');
disp(w');

f08ts example results

Eigenvalues
   -5.9990   -2.9936    0.5047    3.9990