NAG Toolbox: nag_lapack_zhpgvd (f08tq)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Specifies the problem type to be solved.

$\mathbf{itype}=1$

$Az=\lambda Bz$.

$\mathbf{itype}=2$

$ABz=\lambda z$.

$\mathbf{itype}=3$

$BAz=\lambda z$.

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

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

Indicates whether eigenvectors are computed.

$\mathbf{jobz}=\text{'N'}$

Only eigenvalues are computed.

$\mathbf{jobz}=\text{'V'}$

Eigenvalues and eigenvectors are computed.

Constraint: $\mathbf{jobz}=\text{'N'}$ or $\text{'V'}$.

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

If $\mathbf{uplo}=\text{'U'}$, the upper triangles of $A$ and $B$ are stored.

If $\mathbf{uplo}=\text{'L'}$, the lower triangles of $A$ and $B$ are stored.

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

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

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

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

5:     $\mathrm{ap}\left(:\right)$ – 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 $i\ge j$.

6:     $\mathrm{bp}\left(:\right)$ – 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 upper or lower triangle of the $n$ by $n$ Hermitian matrix $B$, packed by columns.

More precisely,

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

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 contents of ap are destroyed.

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

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

The triangular factor $U$ or $L$ from the Cholesky factorization $B=U^{\mathrm{H}}U$ or $B=L{L}^{\mathrm{H}}$, in the same storage format as $B$.

3:     $\mathrm{w}\left(\mathbf{n}\right)$ – double array

The eigenvalues in ascending order.

4:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – complex array

The first dimension, $\mathit{ldz}$, of the array z will be

• if $\mathbf{jobz}=\text{'V'}$, $\mathit{ldz}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldz}=1$.

The second dimension of the array z will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobz}=\text{'V'}$ and $1$ otherwise.

If $\mathbf{jobz}=\text{'V'}$, z contains the matrix $Z$ of eigenvectors. The eigenvectors are normalized as follows:

• if $\mathbf{itype}=1$ or $2$, $Z^{\mathrm{H}}BZ=I$;
• if $\mathbf{itype}=3$, $Z^{\mathrm{H}}{B}^{-1}Z=I$.

If $\mathbf{jobz}=\text{'N'}$, z is not referenced.

5:     $\mathrm{info}$ – 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: jobz, 3: uplo, 4: n, 5: ap, 6: bp, 7: w, 8: z, 9: ldz, 10: work, 11: lwork, 12: rwork, 13: lrwork, 14: iwork, 15: liwork, 16: info.

It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

   $\mathbf{info}>0$

nag_lapack_zpptrf (f07gr) or nag_lapack_zhpevd (f08gq) returned an error code:

$\le \mathbf{n}$	if $\mathbf{info}=i$, nag_lapack_zhpevd (f08gq) failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
$>\mathbf{n}$	if $\mathbf{info}=\mathbf{n}+i$, for $1\le i\le \mathbf{n}$, then the leading minor of order $i$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08tq_example

function f08tq_example


fprintf('f08tq example results\n\n');

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

% Eigenvalues only for AB z = lmbda z
itype = int64(2);
jobz = 'No vectors';
[~, ~, w, ~, info] = f08tq( ...
                            itype, jobz, uplo, n, ap, bp);

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

f08tq example results

Eigenvalues
  -61.7321   -6.6195    0.0725   43.1883