NAG Toolbox: nag_lapack_zhpgv (f08tn)

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: rwork, 12: 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_zhpev (f08gn) returned an error code:

$\le \mathbf{n}$	if $\mathbf{info}=i$, nag_lapack_zhpev (f08gn) 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: f08tn_example

function f08tn_example


fprintf('f08tn 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 of Az = lambda*Bz
itype = int64(1);
jobz = 'No vectors';
[~, up, w, z, info] = f08tn( ...
                            itype, jobz, uplo, n, ap, bp);

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

% Unpack to calculate 1 norms
[a, ap, ifail] = f01zb( ...
                        'Unpack', uplo, 'N', complex(zeros(n,n)), ap);
[b, bp, ifail] = f01zb( ...
                        'Unpack', uplo, 'N', complex(zeros(n,n)), bp);
anorm = norm(a + a' - diag(diag(a)),1);
bnorm = norm(b + b' - diag(diag(b)),1);

% Estimate condition number
[rcond, info] = f07uu( ...
                       '1', uplo, 'N', n, up);

rcondb = rcond^2;
fprintf('Estimate of reciprocal condition number for B\n%12.1e\n', rcondb);

% Error bounds
t1 = x02aj/rcondb;
t2 = anorm/bnorm;
errbnd(1:n) = t1*t2 + t1.*abs(w);

fprintf('\n');
disp('Error estimate for the eigenvalues');
fprintf('%12.1e',errbnd);
fprintf('\n');

f08tn example results

Eigenvalues
   -5.9990   -2.9936    0.5047    3.9990

Estimate of reciprocal condition number for B
     2.5e-03

Error estimate for the eigenvalues
     3.4e-13     2.0e-13     9.6e-14     2.5e-13