NAG Toolbox: nag_lapack_dggqrf (f08ze)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

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

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

The $n$ by $m$ matrix $A$.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array

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

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

The $n$ by $p$ matrix $B$.

Optional Input Parameters

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

Default: the first dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)

$n$, the number of rows of the matrices $A$ and $B$.

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

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

Default: the second dimension of the array a.

$m$, the number of columns of the matrix $A$.

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

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

Default: the second dimension of the array b.

$p$, the number of columns of the matrix $B$.

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

Output Parameters

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

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

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

The elements on and above the diagonal of the array contain the $\min \left(n,m\right)$ by $m$ upper trapezoidal matrix $R$ ($R$ is upper triangular if $n\ge m$); the elements below the diagonal, with the array taua, represent the orthogonal matrix $Q$ as a product of $\min \left(n,m\right)$ elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).

2:     $\mathrm{taua}\left(\min \left(\mathbf{n},\mathbf{m}\right)\right)$ – double array

The scalar factors of the elementary reflectors which represent the orthogonal matrix $Q$.

3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array

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

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

If $n\le p$, the upper triangle of the subarray $\mathbf{b}\left(1:n,p-n+1:p\right)$ contains the $n$ by $n$ upper triangular matrix $T_{12}$.

If $n>p$, the elements on and above the $\left(n-p\right)$th subdiagonal contain the $n$ by $p$ upper trapezoidal matrix $T$; the remaining elements, with the array taub, represent the orthogonal matrix $Z$ as a product of elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).

4:     $\mathrm{taub}\left(\min \left(\mathbf{n},\mathbf{p}\right)\right)$ – double array

The scalar factors of the elementary reflectors which represent the orthogonal matrix $Z$.

5:     $\mathrm{info}$ – 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: n, 2: m, 3: p, 4: a, 5: lda, 6: taua, 7: b, 8: ldb, 9: taub, 10: work, 11: lwork, 12: 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: f08ze_example

function f08ze_example


fprintf('f08ze example results\n\n');

a = [-0.57, -1.28, -0.39;
     -1.93,  1.08, -0.31;
      2.3,   0.24, -0.4;
     -0.02,  1.03, -1.43];
b = [0.5, 0, 0, 0;
     0,   1, 0, 0;
     0,   0, 2, 0;
     0,   0, 0, 5];
d = [1.32; -4.00; 5.52; 3.24];

n = 4;
m = 3;
p = 4;

% Compute the generalized QR factorization of (A,B) as
% A = Q*(R),   B = Q*(T11 T12)*Z
%       (0)          ( 0  T22)

[a, taua, b, taub, info] = f08ze( ...
                                  a, b);

% Compute c = (c1) = (Q**T)*d
%             (c2)

[c, info] = f08ag( ...
                   'Left', 'Transpose', a, taua, d);

% Putting Z*y = w = (w1), set w1 = 0, storing the result in y1
%                   (w2)
y = zeros(p, 1);
if n > m
  % Solve T22*w2 = c2 for w2, storing result in y2
  y2 = m+p-n+1:p;

  [y(y2), info] = ...
  f07te( ...
         'Upper', 'No transpose', 'Non-unit', b(m+1:n, y2), c(m+1:n));

  % Compute estimate of the square root of the residual sum of squares
  % norm(y) = norm(w2)
  rnorm = norm(y(y2), 1);

  % Form c1 - T12*w2 in c
  c = c - b(:,y2)*y(y2);
end

% Solve R*x = c1 - T12*w2 for x
[c(1:m), info] = f07te( ...
                        'Upper', 'No transpose', 'Non-unit', a(1:m,:), c(1:m));

% Compute y = (Z^T)*w
b1 = max(1, n-p+1):n;
[b(b1,:), y, info] = f08ck( ...
                            'Left', 'Transpose', b(b1,:), taub, y);

fprintf('Generalized least squares solution\n');
fprintf('%12.4f',c(1:m));

fprintf('\n\nResidual vector\n  ');
fprintf('%12.2e',y);

fprintf('\n\nSquare root of the residual sum of squares\n  %12.2e\n', ...
        rnorm);

f08ze example results

Generalized least squares solution
      1.9889     -1.0058     -2.9911

Residual vector
     -6.37e-04   -2.45e-03   -4.72e-03    7.70e-03

Square root of the residual sum of squares
      9.38e-03