NAG Toolbox: nag_lapack_zupmtr (f08gu)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates how $Q$ or $Q^{\mathrm{H}}$ is to be applied to $C$.

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

$Q$ or $Q^{\mathrm{H}}$ is applied to $C$ from the left.

$\mathbf{side}=\text{'R'}$

$Q$ or $Q^{\mathrm{H}}$ is applied to $C$ from the right.

Constraint: $\mathbf{side}=\text{'L'}$ or $\text{'R'}$.

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

This must be the same argument uplo as supplied to nag_lapack_zhptrd (f08gs).

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

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

Indicates whether $Q$ or $Q^{\mathrm{H}}$ is to be applied to $C$.

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

$Q$ is applied to $C$.

$\mathbf{trans}=\text{'C'}$

$Q^{\mathrm{H}}$ is applied to $C$.

Constraint: $\mathbf{trans}=\text{'N'}$ or $\text{'C'}$.

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

The dimension of the array ap must be at least $\max \left(1,\mathbf{m}\times \left(\mathbf{m}+1\right)/2\right)$ if $\mathbf{side}=\text{'L'}$ and at least $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$ if $\mathbf{side}=\text{'R'}$

Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zhptrd (f08gs).

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

The dimension of the array tau must be at least $\max \left(1,\mathbf{m}-1\right)$ if $\mathbf{side}=\text{'L'}$ and at least $\max \left(1,\mathbf{n}-1\right)$ if $\mathbf{side}=\text{'R'}$

Further details of the elementary reflectors, as returned by nag_lapack_zhptrd (f08gs).

6:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – complex array

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

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

The $m$ by $n$ matrix $C$.

Optional Input Parameters

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

Default: the first dimension of the array c.

$m$, the number of rows of the matrix $C$; $m$ is also the order of $Q$ if $\mathbf{side}=\text{'L'}$.

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

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

Default: the second dimension of the array c.

$n$, the number of columns of the matrix $C$; $n$ is also the order of $Q$ if $\mathbf{side}=\text{'R'}$.

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

Output Parameters

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

The dimension of the array ap will be $\max \left(1,\mathbf{m}\times \left(\mathbf{m}+1\right)/2\right)$ if $\mathbf{side}=\text{'L'}$ and at least $\max \left(1,\mathbf{n}\times \left(\mathbf{n}+1\right)/2\right)$ if $\mathbf{side}=\text{'R'}$

Is used as internal workspace prior to being restored and hence is unchanged.

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

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

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

c stores $QC$ or $Q^{\mathrm{H}}C$ or $CQ$ or $C{Q}^{\mathrm{H}}$ as specified by side and trans.

3:     $\mathrm{info}$ – 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: side, 2: uplo, 3: trans, 4: m, 5: n, 6: ap, 7: tau, 8: c, 9: ldc, 10: work, 11: 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.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08gu_example

function f08gu_example


fprintf('f08gu example results\n\n');

% Hermitian matrix A stored in symmetric packed format (Lower)
uplo = 'L';
n = int64(4);
ap = [-2.28 + 0i;   1.78 + 2.03i;   2.26 - 0.10i;  -0.12 - 2.53i;
                   -1.12 + 0i;      0.01 - 0.43i;  -1.07 - 0.86i;
                                   -0.37 + 0i;      2.31 + 0.92i;
                                                   -0.73 + 0i];

% Reduce to tridiagonal form
[apf, d, e, tau, info] = f08gs( ...
                                uplo, n, ap);

% Calculate two smallest eigenvalues
range = 'Indices';
order = 'Block';
vl = 0;
vu = 0;
il = int64(1);
iu = int64(2);
abstol = 0;
[m, nsplit, w, iblock, isplit, info] = ...
  f08jj( ...
         range, order, vl, vu, il, iu, abstol, d, e);

% Corresponding eigenvectors of T
[tz, ifailv, info] = f08jx( ...
                            d, e, m, w, iblock, isplit);

% Transform to eigenvalues of A (by premultiplying by Q)
side = 'Left';
trans = 'No transpose';
[~, z, info] = f08gu( ...
                      side, uplo, trans, apf, tau, tz);

% Normalize vectors, largest element is real and positive.
for i = 1:m
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

disp(' Selected eigenvalues of A:');
disp(w(1:m));
disp(' Corresponding eigenvectors:');
disp(z);

f08gu example results

 Selected eigenvalues of A:
   -6.0002
   -3.0030

 Corresponding eigenvectors:
   0.7299 + 0.0000i  -0.2120 + 0.1497i
  -0.1663 - 0.2061i   0.7307 + 0.0000i
  -0.4165 - 0.1417i  -0.3291 + 0.0479i
   0.1743 + 0.4162i   0.5200 + 0.1329i