NAG Toolbox: nag_lapack_dtpmqrt (f08bc)

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:     $\mathrm{l}$ – int64int32nag_int scalar

$l$, the number of rows of the upper trapezoidal part of the pentagonal composite matrix $V$, passed (as b) in a previous call to nag_lapack_dtpqrt (f08bb). This must be the same value used in the previous call to nag_lapack_dtpqrt (f08bb) (see l in nag_lapack_dtpqrt (f08bb)).

Constraint: $0\le \mathbf{l}\le \mathbf{k}$.

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

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

The $m_v$ by $n_v$ matrix $V$; this should remain unchanged from the array b returned by a previous call to nag_lapack_dtpqrt (f08bb).

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

The first dimension of the array t must be at least $\mathbf{nb}$.

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

This must remain unchanged from a previous call to nag_lapack_dtpqrt (f08bb) (see t in nag_lapack_dtpqrt (f08bb)).

6:     $\mathrm{c1}\left(\mathit{ldc1},:\right)$ – double array

The first dimension, $\mathit{ldc1}$, of the array c1 must satisfy

• if $\mathbf{side}=\text{'L'}$, $\mathit{ldc1}\ge \max \left(1,\mathbf{k}\right)$;
• if $\mathbf{side}=\text{'R'}$, $\mathit{ldc1}\ge \max \left(1,\mathbf{m}\right)$.

The second dimension of the array c1 must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{side}=\text{'L'}$ and at least $\max \left(1,\mathbf{k}\right)$ if $\mathbf{side}=\text{'R'}$.

$C_1$, the first part of the composite matrix $C$:

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

then c1 contains the first $k$ rows of $C$;

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

then c1 contains the first $k$ columns of $C$.

7:     $\mathrm{c2}\left(\mathit{ldc2},:\right)$ – double array

The first dimension of the array c2 must be at least $\max \left(1,\mathbf{m}\right)$.

The second dimension of the array c2 must be at least $\max \left(1,\mathbf{n}\right)$.

$C_2$, the second part of the composite matrix $C$.

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

then c2 contains the remaining $m_v$ rows of $C$;

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

then c2 contains the remaining $m_v$ columns of $C$;

Optional Input Parameters

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

Default: the first dimension of the array v.

The number of rows of the matrix $C_2$, that is,

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

then $m_v$, the number of rows of the matrix $V$;

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

then $m_c$, the number of rows of the matrix $C$.

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

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

Default: the first dimension of the array v.

The number of columns of the matrix $C_2$, that is,

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

then $n_c$, the number of columns of the matrix $C$;

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

then $n_v$, the number of columns of the matrix $V$.

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

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

Default: the second dimension of the array t.

$k$, the number of elementary reflectors whose product defines the matrix $Q$.

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

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

Default: the first dimension of the array t.

$\mathit{nb}$, the blocking factor used in a previous call to nag_lapack_dtpqrt (f08bb) to compute the $QR$ factorization of a triangular-pentagonal matrix containing composite matrices $A$ and $B$.

Constraints:

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

Output Parameters

1:     $\mathrm{c1}\left(\mathit{ldc1},:\right)$ – double array

The first dimension, $\mathit{ldc1}$, of the array c1 will be

• if $\mathbf{side}=\text{'L'}$, $\mathit{ldc1}=\max \left(1,\mathbf{k}\right)$;
• if $\mathbf{side}=\text{'R'}$, $\mathit{ldc1}=\max \left(1,\mathbf{m}\right)$.

The second dimension of the array c1 will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{side}=\text{'L'}$ and at least $\max \left(1,\mathbf{k}\right)$ if $\mathbf{side}=\text{'R'}$.

c1 stores the corresponding block of $QC$ or $Q^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$.

2:     $\mathrm{c2}\left(\mathit{ldc2},:\right)$ – double array

The first dimension of the array c2 will be $\max \left(1,\mathbf{m}\right)$.

The second dimension of the array c2 will be $\max \left(1,\mathbf{n}\right)$.

c2 stores the corresponding block of $QC$ or $Q^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$.

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

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

Error Indicators and Warnings

   $\mathbf{info}<0$

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

Example

Open in the MATLAB editor: f08bc_example

function f08bc_example


fprintf('f08bc 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');

f08bc 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