NAG Toolbox: nag_lapack_zunmtr (f08fu)

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_zhetrd (f08fs).

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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension, $\mathit{lda}$, of the array a must satisfy

• if $\mathbf{side}=\text{'L'}$, $\mathit{lda}\ge \max \left(1,\mathbf{m}\right)$;
• if $\mathbf{side}=\text{'R'}$, $\mathit{lda}\ge \max \left(1,\mathbf{n}\right)$.

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

Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zhetrd (f08fs).

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_zhetrd (f08fs).

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:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – 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.

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: side, 2: uplo, 3: trans, 4: m, 5: n, 6: a, 7: lda, 8: tau, 9: c, 10: ldc, 11: work, 12: lwork, 13: 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: f08fu_example

function f08fu_example


fprintf('f08fu example results\n\n');

% Lower triangular part of Hermitian matrix A
uplo = 'L';
a = [-2.28 + 0.00i,  0.00 + 0i,     0    + 0i,     0    + 0i;
      1.78 + 2.03i, -1.12 + 0i,     0    + 0i,     0    + 0i;
      2.26 - 0.10i,  0.01 - 0.43i, -0.37 + 0i,     0    + 0i;
     -0.12 - 2.53i, -1.07 - 0.86i,  2.31 + 0.92i, -0.73 + 0i];

% Reduce A to tridiagonal form
[aq, d, e, tau, info] = f08fs( ...
                              uplo, a);

% 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)
trans = 'No transpose';
side = 'Left';
[z, info] = f08fu( ...
                   side, uplo, trans, aq, 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);

f08fu 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