NAG Toolbox: nag_lapack_ztpmqrt (f08bq)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

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

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

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

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

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

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

$Q$ is applied to $C$.

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

$Q^{\mathrm{H}}$ is applied to $C$.

Constraint: $\mathbf{trans}=\text{'N'}$ or $\text{'C'}$.

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_ztpqrt (f08bp). This must be the same value used in the previous call to nag_lapack_ztpqrt (f08bp) (see l in nag_lapack_ztpqrt (f08bp)).

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

4:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – complex 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_ztpqrt (f08bp).

5:     $\mathrm{t}\left(\mathit{ldt},:\right)$ – complex 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_ztpqrt (f08bp) (see t in nag_lapack_ztpqrt (f08bp)).

6:     $\mathrm{c1}\left(\mathit{ldc1},:\right)$ – complex 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:     $\mathrm{c2}\left(\mathit{ldc2},:\right)$ – complex 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_ztpqrt (f08bp) 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:     $\mathrm{c1}\left(\mathit{ldc1},:\right)$ – complex 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 $Q^{\mathrm{H}}C$ or $CQ$ or $C{Q}^{\mathrm{H}}$.

2:     $\mathrm{c2}\left(\mathit{ldc2},:\right)$ – complex 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 $Q^{\mathrm{H}}C$ or $CQ$ or $C{Q}^{\mathrm{H}}$.

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: f08bq_example

function f08bq_example


fprintf('f08bq 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 = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i,  -0.05 + 0.41i;
     -0.98 + 1.98i,  -1.20 + 0.19i,  -0.66 + 0.42i,  -0.81 + 0.56i;
      0.62 - 0.46i,   1.01 + 0.02i,   0.63 - 0.17i,  -1.11 + 0.60i;
     -0.37 + 0.38i,   0.19 - 0.54i,  -0.98 - 0.36i,   0.22 - 0.20i;
      0.83 + 0.51i,   0.20 + 0.01i,  -0.17 - 0.46i,   1.47 + 1.59i;
      1.08 - 0.28i,   0.20 - 0.12i,  -0.07 + 1.23i,   0.26 + 0.26i];
b = [-2.09 + 1.93i,   3.26-2.70i;
      3.34 - 3.53i,  -6.22+1.16i;
     -4.94 - 2.04i,   7.94-3.13i;
      0.17 + 4.23i,   1.04-4.26i;
     -5.19 + 3.63i,  -2.31-2.12i;
      0.98 + 2.53i,  -1.39-4.05i];

nb = n;
% Compute the QR Factorisation of first n rows of A
[QRn, Tn, info] = f08ap( ...
			 nb,a(1:n,:));

% Compute C = (C1) = (Q^H)*B
[c1, info] = f08aq( ...
		    'Left', 'Conjugate Transpose', QRn, Tn, b(1:n,:));

% Compute least-squares solutions by backsubstitution in R*X = C1
[x, info] = f07ts( ...
		   '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] = f08bp( ...
			 l, nb2, QRn, a(n+1:m,:));

% Apply orthogonal transformations to C
[c1,c2,info] = f08bq( ...
		      'Left','Conjugate Transpose', l, Q, T, c1, b(n+1:m,:));

% Compute least-squares solutions for first n rows: R*X = C1
[x, info] = f07ts( ...
		   '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');

f08bq example results

Solution for n rows
  -0.5091 - 1.2428i   0.7569 + 1.4384i
  -2.3789 + 2.8651i   5.1727 - 3.6193i
   1.4634 - 2.2064i  -2.6613 + 2.1339i
   0.4701 + 2.6964i  -2.6933 + 0.2724i

Least squares solution
  -0.5044 - 1.2179i   0.7629 + 1.4529i
  -2.4281 + 2.8574i   5.1570 - 3.6089i
   1.4872 - 2.1955i  -2.6518 + 2.1203i
   0.4537 + 2.6904i  -2.7606 + 0.3318i

Square roots of the residual sums of squares
    6.88e-02    1.87e-01