NAG Toolbox: nag_lapack_dopmtr (f08gg)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

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

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

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

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

$Q$ or $Q^{\mathrm{T}}$ 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_dsptrd (f08ge).

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

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

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

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

$Q$ is applied to $C$.

$\mathbf{trans}=\text{'T'}$

$Q^{\mathrm{T}}$ is applied to $C$.

Constraint: $\mathbf{trans}=\text{'N'}$ or $\text{'T'}$.

4:     $\mathrm{ap}\left(:\right)$ – double 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_dsptrd (f08ge).

5:     $\mathrm{tau}\left(:\right)$ – double 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_dsptrd (f08ge).

6:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double 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:     $\mathrm{ap}\left(:\right)$ – double 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:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double 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 $Q^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ 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: f08gg_example

function f08gg_example


fprintf('f08gg example results\n\n');

% Symmetric matrix A stored in symmetric packed format (Lower)
uplo = 'L';
n = int64(4);
ap = [2.07;     3.87;     4.20;    -1.15;
               -0.21;     1.87;     0.63;
                          1.15;     2.06;
                                   -1.81];

% Reduce A to tridiagonal form
[apf, d, e, tau, info] = f08ge( ...
                                uplo, n, ap);

% Two smallest eigenvalues
range = 'I';
order = 'B';
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);

% Get corresponding eigenvectors of T
[v, ifailv, info] = f08jk( ...
                           d, e, m, w, iblock, isplit);

% Transform Q*V to get eigenvectors of A 
side = 'Left';
trans = 'No transpose';
[~, z, info] = f08gg( ...
                      side, uplo, trans, apf, tau, v);

% Normalize eigenvectors: largest element positive
for j = 1:m
  [~,k] = max(abs(z(:,j)));
  if z(k,j) < 0;
    z(:,j) = -z(:,j);
  end
end                            

% Display results
fprintf(' Eigenvalues numbered 1 to 2 are:\n   ');
fprintf(' %7.4f',w(1:m));
fprintf('\n\n');

[ifail] = x04ca( ...
                 'General', ' ', z, 'Corresponding eigenvectors of A');

f08gg example results

 Eigenvalues numbered 1 to 2 are:
    -5.0034 -1.9987

 Corresponding eigenvectors of A
          1       2
 1   0.5658 -0.2328
 2  -0.3478  0.7994
 3  -0.4740 -0.4087
 4   0.5781  0.3737