NAG Toolbox: nag_lapack_dormbr (f08kg)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

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

$\mathbf{vect}=\text{'Q'}$

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

$\mathbf{vect}=\text{'P'}$

$P$ or $P^{\mathrm{T}}$ is applied to $C$.

Constraint: $\mathbf{vect}=\text{'Q'}$ or $\text{'P'}$.

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

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

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

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

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

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

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

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

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

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

$Q$ or $P$ is applied to $C$.

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

$Q^{\mathrm{T}}$ or $P^{\mathrm{T}}$ is applied to $C$.

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

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

If $\mathbf{vect}=\text{'Q'}$, the number of columns in the original matrix $A$.

If $\mathbf{vect}=\text{'P'}$, the number of rows in the original matrix $A$.

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

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

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

• if $\mathbf{vect}=\text{'Q'}$, $\mathit{lda}\ge \max \left(1,\mathit{r}\right)$;
• if $\mathbf{vect}=\text{'P'}$, $\mathit{lda}\ge \max \left(1,\min \left(\mathit{r},\mathbf{k}\right)\right)$.

The second dimension of the array a must be at least $\max \left(1,\min \left(\mathit{r},\mathbf{k}\right)\right)$ if $\mathbf{vect}=\text{'Q'}$ and at least $\max \left(1,\mathit{r}\right)$ if $\mathbf{vect}=\text{'P'}$.

Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dgebrd (f08ke).

6:     $\mathrm{tau}\left(:\right)$ – double array

The dimension of the array tau must be at least $\max \left(1,\min \left(\mathit{r},\mathbf{k}\right)\right)$

Further details of the elementary reflectors, as returned by nag_lapack_dgebrd (f08ke) in its argument tauq if $\mathbf{vect}=\text{'Q'}$, or in its argument taup if $\mathbf{vect}=\text{'P'}$.

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 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 $C$.

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 $C$.

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 $C^{\mathrm{T}}Q$ or $PC$ or $P^{\mathrm{T}}C$ or $CP$ or $C^{\mathrm{T}}P$ as specified by vect, 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: vect, 2: side, 3: trans, 4: m, 5: n, 6: k, 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: f08kg_example

function f08kg_example


fprintf('f08kg example results\n\n');

% Two cases of preceding reduction to bidiagonal form by QR or LQ
% Case 1: m > n, precede by QR
ex1;
% Case 2: m < n, precede by LQ
ex2;



function ex1
  m = 6;
  n = int64(4);
  a = [-0.57  -1.28  -0.39   0.25;
       -1.93   1.08  -0.31  -2.14;
        2.30   0.24   0.40  -0.35;
       -1.93   0.64  -0.66   0.08;
        0.15   0.30   0.15  -2.13;
       -0.02   1.03  -1.43   0.50];
  % Factorize A = QR
  [QR, tau, info] = f08ae(a);

  % Generate Q from QR
  [Q, info] = f08af(QR, tau);

  % Extract R from QR
  R = triu(QR(1:n,1:n));

  % Bidiagonalize R = Q1 B P^T
  [B, d, e, tauq, taup, info] = ...
  f08ke(R);

  % Update Q: Q2 = Q*Q1 (so A = QR = Q2 B P^T)
  vect = 'Q';
  side = 'Right';
  trans = 'No transpose';
  [Q2, info] = f08kg( ...
                      vect, side, trans, n, B, tauq, Q);

  fprintf('Example 1: bidiagonal matrix B\n   Main diagonal  ');
  fprintf(' %7.3f',d);
  fprintf('\n   super-diagonal ');
  fprintf(' %7.3f',e);
  fprintf('\n\n');
  disp('Example 1: Orthogonal matrix Q');
  disp(Q2);

function ex2
  m = int64(4);
  n = int64(6);
  a = [ -5.42   3.28  -3.68   0.27   2.06   0.46;
        -1.65  -3.40  -3.20  -1.03  -4.06  -0.01;
        -0.37   2.35   1.90   4.31  -1.76   1.13;
        -3.15  -0.11   1.99  -2.70   0.26   4.50];

  % Factorize A = LQ
  [LQ, tau, info] = f08ah(a);

  % Generate Q from LQ
  [Q, info] = f08aj(LQ, tau);

  % Extract L from LQ
  L = tril(LQ(1:m,1:m));

  % Bidiagonalize L = Q1 B P^T
  [B, d, e, tauq, taup, info] = ...
  f08ke(L);

  % Update Q: P2 = P^T*Q (so A = LQ = Q1 B P2)
  vect = 'P';
  side = 'Left';
  trans = 'Transpose';
  [P2, info] = f08kg( ...
                      vect, side, trans, n, B, taup, Q);

  fprintf('Example 2: bidiagonal matrix B\n   Main diagonal  ');
  fprintf(' %7.3f',d);
  fprintf('\n   super-diagonal ');
  fprintf(' %7.3f',e);
  fprintf('\n\n');
  disp('Example 2: Orthogonal matrix P^T');
  disp(P2);

f08kg example results

Example 1: bidiagonal matrix B
   Main diagonal     3.618  -2.416   1.921  -1.427
   super-diagonal    1.259  -1.526   1.189

Example 1: Orthogonal matrix Q
   -0.1576   -0.2690    0.2612    0.8513
   -0.5335    0.5311   -0.2922    0.0184
    0.6358    0.3495   -0.0250   -0.0210
   -0.5335    0.0035    0.1537   -0.2592
    0.0415    0.5572   -0.2917    0.4523
   -0.0055    0.4614    0.8585   -0.0532

Example 2: bidiagonal matrix B
   Main diagonal    -7.772   6.157  -6.058   5.793
   super-diagonal    1.193   0.573  -1.914

Example 2: Orthogonal matrix P^T
   -0.7104    0.4299   -0.4824    0.0354    0.2700    0.0603
    0.3583    0.1382   -0.4110    0.4044    0.0951   -0.7148
   -0.0507    0.4244    0.3795    0.7402   -0.2773    0.2203
    0.2442    0.4016    0.4158   -0.1354    0.7666   -0.0137