NAG Toolbox: nag_lapack_dormhr (f08ng)

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:     $\mathrm{trans}$ – 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 $\text{'T'}$.

3:     $\mathrm{ilo}$ – int64int32nag_int scalar

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

These must be the same arguments ilo and ihi, respectively, as supplied to nag_lapack_dgehrd (f08ne).

Constraints:

• if $\mathbf{side}=\text{'L'}$ and $\mathbf{m}>0$, $1\le \mathbf{ilo}\le \mathbf{ihi}\le \mathbf{m}$;
• if $\mathbf{side}=\text{'L'}$ and $\mathbf{m}=0$, $\mathbf{ilo}=1$ and $\mathbf{ihi}=0$;
• if $\mathbf{side}=\text{'R'}$ and $\mathbf{n}>0$, $1\le \mathbf{ilo}\le \mathbf{ihi}\le \mathbf{n}$;
• if $\mathbf{side}=\text{'R'}$ and $\mathbf{n}=0$, $\mathbf{ilo}=1$ and $\mathbf{ihi}=0$.

5:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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_dgehrd (f08ne).

6:     $\mathrm{tau}\left(:\right)$ – 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_dgehrd (f08ne).

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

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: trans, 3: m, 4: n, 5: ilo, 6: ihi, 7: a, 8: lda, 9: tau, 10: c, 11: ldc, 12: work, 13: lwork, 14: 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: f08ng_example

function f08ng_example


fprintf('f08ng example results\n\n');

n   = int64(4);
ilo = int64(1);
ihi = n;
a = [ 0.35,  0.45, -0.14, -0.17;
      0.09,  0.07, -0.54,  0.35;
     -0.44, -0.33, -0.03,  0.17;
      0.25, -0.32, -0.13,  0.11];

% Reduce A to upper Hessenberg Form A = QHQ^T
[H, tau, info] = f08ne( ...
                        ilo, ihi, a);

% Form Q
[Q, info] = f08nf( ...
                   ilo, ihi, H, tau);

% Schur factorize H = Y*T*Y^T
job   = 'Schur form';
compz = 'No Vectors';
[~, wr, wi, ~, info] = f08pe( ...
                              job, compz, ilo, ihi, H, Q);

w = wr + i*wi;
disp('Eigenvalues of A');
disp(w);

% Calculate eigenvetors of H corresponding to negative real part eigenvalues
select = (wr < 0);

job = 'Right';
eigsrc = 'QR';
initv = 'No initial vectors';
vl = [];
vr = zeros(n,n);
[~, ~, ~, VR, m, ifaill, ifailr, info] = ...
f08pk( ...
       job, eigsrc, initv, select, H, wr, wi, vl, vr, n);

% Eigenvectors of A = Q*VR
side = 'Left';
trans = 'No transpose';
[V, info] = f08ng( ...
                   side, trans, ilo, ihi, H, tau, VR);

% Combine columns of V into complex eigenvectors Z
j = 0;
for k = 1:n
  if select(k)
    j = j+1;
    if (wi(k)==0)
      % Normalize eigenvector: largest element positive
      [~,l] = max(abs(V(:,j)));
      if V(l,j) < 0;
        V(:,j) = -V(:,j);
      end                      
      Z(:,j) = complex(V(:,j));
    else
      Z(:,j)   = V(:,j) + i*V(:,j+1);
      Z(:,j+1) = V(:,j) - i*V(:,j+1);
      % Normalize: largest element is real
      [~,l] = max(abs(real(Z(:,j)))+abs(imag(Z(:,j))));
      Z(:,j) = Z(:,j)*conj(Z(l,j))/abs(Z(l,j));
      Z(:,j+1) = Z(:,j+1)*conj(Z(l,j+1))/abs(Z(l,j+1));
      j = j+1;
      select(k+1) = false;
    end
  end
end
disp('Eigenvectors corresponding to eigenvalues with negative real part');
disp(Z);

f08ng example results

Eigenvalues of A
   0.7995 + 0.0000i
  -0.0994 + 0.4008i
  -0.0994 - 0.4008i
  -0.1007 + 0.0000i

Eigenvectors corresponding to eigenvalues with negative real part
  -0.2379 + 0.3134i  -0.2379 - 0.3134i   0.1493 + 0.0000i
   0.3100 - 0.6430i   0.3100 + 0.6430i   0.3956 + 0.0000i
   0.1196 - 0.3795i   0.1196 + 0.3795i   0.7075 + 0.0000i
   0.8319 + 0.0000i   0.8319 + 0.0000i   0.8603 + 0.0000i