NAG Toolbox: nag_lapack_dtpqrt (f08bb)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

$l$, the number of rows of the trapezoidal part of $B$ (i.e., $B_2$).

Constraint: $0\le \mathbf{l}\le \min \left(\mathbf{m},\mathbf{n}\right)$.

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

The explicitly chosen block-size to be used in the algorithm for computing the $QR$ factorization. See Further Comments for details.

Constraints:

• $\mathbf{nb}\ge 1$;
• if $\mathbf{n}>0$, $\mathbf{nb}\le \mathbf{n}$.

3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $\max \left(1,\mathbf{n}\right)$.

The $n$ by $n$ upper triangular matrix $A$.

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

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

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

The $m$ by $n$ pentagonal matrix $B$ composed of an $\left(m-l\right)$ by $n$ rectangular matrix $B_1$ above an $l$ by $n$ upper trapezoidal matrix $B_2$.

Optional Input Parameters

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

Default: the first dimension of the array b.

$m$, the number of rows of the matrix $B$.

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

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

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

$n$, the number of columns of the matrix $B$ and the order of the upper triangular matrix $A$.

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

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 $\max \left(1,\mathbf{n}\right)$.

The upper triangle stores the corresponding elements of the $n$ by $n$ upper triangular matrix $R$.

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

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

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

Details of the orthogonal matrix $Q$.

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

The first dimension of the array t will be $\mathbf{nb}$.

The second dimension of the array t will be $\mathbf{n}$.

Further details of the orthogonal matrix $Q$. The number of blocks is $b=⌈\frac{k}{\mathbf{nb}}⌉$, where $k=\min \left(m,n\right)$ and each block is of order nb except for the last block, which is of order $k-\left(b-1\right)\times \mathbf{nb}$. For each of the blocks, an upper triangular block reflector factor is computed: ${\mathit{T}}_1,{\mathit{T}}_2,\dots ,{\mathit{T}}_b$. These are stored in the $\mathbf{nb}$ by $n$ matrix $T$ as $\mathit{T}=\left[{\mathit{T}}_1\left|{\mathit{T}}_2\right|\dots |{\mathit{T}}_b\right]$.

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

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

Accuracy

Further Comments

[t, c, info] = f08bc('Left','Transpose', nb, a(:,1:min(m,n)), t, c);

Example

Open in the MATLAB editor: f08bb_example

function f08bb_example


fprintf('f08bb example results\n\n');

% Minimize ||Ax - b|| using recursive QR for m-by-n A and m-by-p B

m = int64(6);
n = int64(4);
p = int64(2);
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];
b = [-2.67,  0.41;
     -0.55, -3.10;
      3.34, -4.01;
     -0.77,  2.76;
      0.48, -6.17;
      4.10,  0.21];

nb = n;
% Compute the QR Factorisation of first n rows of A
[QRn, Tn, info] = f08ab( ...
			 nb,a(1:n,:));

% Compute C = (C1) = (Q^T)*B
[c1, info] = f08ac( ...
		    'Left', 'Transpose', QRn, Tn, b(1:n,:));

% Compute least-squares solutions by backsubstitution in R*X = C1
[x, info] = f07te( ...
		   'Upper', 'No Transpose', 'Non-Unit', QRn, c1);

% Print first n-row solutions
disp('Solution for n rows');
disp(x(1:n,:));

% Add the remaining rows and perform QR update
nb2 = m-n;
l = int64(0);
[R, Q, T, info] = f08bb( ...
			 l, nb2, QRn, a(n+1:m,:));

% Apply orthogonal transformations to C
[c1,c2,info] = f08bc( ...
		      'Left','Transpose', l, Q, T, c1, b(n+1:m,:));

% Compute least-squares solutions for first n rows: R*X = C1
[x, info] = f07te( ...
		   'Upper', 'No transpose', 'Non-Unit', R, c1);
% Print least-squares solutions for all m rows
disp('Least squares solution');
disp(x(1:n,:));

% Compute and print estimates of the square roots of the residual
% sums of squares
for j=1:p
  rnorm(j) = norm(c2(:,j));
end
fprintf('Square roots of the residual sums of squares\n');
fprintf('%12.2e', rnorm);
fprintf('\n');

f08bb example results

Solution for n rows
    1.5179   -1.5850
    1.8629    0.5531
   -1.4608    1.3485
    0.0398    2.9619

Least squares solution
    1.5339   -1.5753
    1.8707    0.5559
   -1.5241    1.3119
    0.0392    2.9585

Square roots of the residual sums of squares
    2.22e-02    1.38e-02