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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_ztpmqrt (f08bq)

Purpose

nag_lapack_ztpmqrt (f08bq) multiplies an arbitrary complex matrix $C$ by the complex unitary matrix $Q$ from a $QR$ factorization computed by nag_lapack_ztpqrt (f08bp).

Syntax

[c1, c2, info] = f08bq(side, trans, l, v, t, c1, c2, 'm', m, 'n', n, 'k', k, 'nb', nb)
[c1, c2, info] = nag_lapack_ztpmqrt(side, trans, l, v, t, c1, c2, 'm', m, 'n', n, 'k', k, 'nb', nb)

Description

nag_lapack_ztpmqrt (f08bq) is intended to be used after a call to nag_lapack_ztpqrt (f08bp) which performs a $QR$ factorization of a triangular-pentagonal matrix containing an upper triangular matrix $A$ over a pentagonal matrix $B$. The unitary matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to form the matrix products
 $QC , QHC , CQ ​ or ​ CQH ,$
where the complex rectangular ${m}_{c}$ by ${n}_{c}$ matrix $C$ is split into component matrices ${C}_{1}$ and ${C}_{2}$.
If $Q$ is being applied from the left ($QC$ or ${Q}^{\mathrm{H}}C$) then
 $C = C1 C2$
where ${C}_{1}$ is $k$ by ${n}_{c}$, ${C}_{2}$ is ${m}_{v}$ by ${n}_{c}$, ${m}_{c}=k+{m}_{v}$ is fixed and ${m}_{v}$ is the number of rows of the matrix $V$ containing the elementary reflectors (i.e., m as passed to nag_lapack_ztpqrt (f08bp)); the number of columns of $V$ is ${n}_{v}$ (i.e., n as passed to nag_lapack_ztpqrt (f08bp)).
If $Q$ is being applied from the right ($CQ$ or $C{Q}^{\mathrm{H}}$) then
 $C = C1 C2$
where ${C}_{1}$ is ${m}_{c}$ by $k$, and ${C}_{2}$ is ${m}_{c}$ by ${m}_{v}$ and ${n}_{c}=k+{m}_{v}$ is fixed.
The matrices ${C}_{1}$ and ${C}_{2}$ are overwriten by the result of the matrix product.
A common application of this routine is in updating the solution of a linear least squares problem as illustrated in Example in nag_lapack_ztpqrt (f08bp).

References

Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
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 ${\mathbf{side}}=\text{'L'}$, $\mathit{ldc1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{side}}=\text{'R'}$, $\mathit{ldc1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c1 must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$ 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c2 must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${C}_{2}$, 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 ${\mathbf{side}}=\text{'L'}$, $\mathit{ldc1}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• if ${\mathbf{side}}=\text{'R'}$, $\mathit{ldc1}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c1 will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$ 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c2 will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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

The computed result differs from the exact result by a matrix $E$ such that
 $E2 = Oε C2 ,$
where $\epsilon$ is the machine precision.

The total number of floating-point operations is approximately $2nk\left(2m-k\right)$ if ${\mathbf{side}}=\text{'L'}$ and $2mk\left(2n-k\right)$ if ${\mathbf{side}}=\text{'R'}$.
The real analogue of this function is nag_lapack_dtpmqrt (f08bc).

Example

See Example in nag_lapack_ztpqrt (f08bp).
```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
```