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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dtpmqrt (f08bc)

## Purpose

nag_lapack_dtpmqrt (f08bc) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ from a $QR$ factorization computed by nag_lapack_dtpqrt (f08bb).

## Syntax

[c1, c2, info] = f08bc(side, trans, l, v, t, c1, c2, 'm', m, 'n', n, 'k', k, 'nb', nb)
[c1, c2, info] = nag_lapack_dtpmqrt(side, trans, l, v, t, c1, c2, 'm', m, 'n', n, 'k', k, 'nb', nb)

## Description

nag_lapack_dtpmqrt (f08bc) is intended to be used after a call to nag_lapack_dtpqrt (f08bb) which performs a $QR$ factorization of a triangular-pentagonal matrix containing an upper triangular matrix $A$ over a pentagonal matrix $B$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to form the matrix products
 $QC , QTC , CQ ​ or ​ CQT ,$
where the real 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 ${Q}^{\mathrm{T}}C$) 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_dtpqrt (f08bb)); the number of columns of $V$ is ${n}_{v}$ (i.e., n as passed to nag_lapack_dtpqrt (f08bb)).
If $Q$ is being applied from the right ($CQ$ or $C{Q}^{\mathrm{T}}$) 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_dtpqrt (f08bb).

## 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 ${Q}^{\mathrm{T}}$ 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
The ${m}_{v}$ by ${n}_{v}$ matrix $V$; this should remain unchanged from the array b returned by a previous call to nag_lapack_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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
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 ${\mathbf{side}}=\text{'L'}$, $\mathit{ldc1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• 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:     $\mathrm{c2}\left(\mathit{ldc2},:\right)$ – double 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_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 ${\mathbf{side}}=\text{'L'}$, $\mathit{ldc1}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$;
• 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 ${Q}^{\mathrm{T}}C$ 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
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 ${Q}^{\mathrm{T}}C$ 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

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 complex analogue of this function is nag_lapack_ztpmqrt (f08bq).

## Example

See Example in nag_lapack_dtpqrt (f08bb).
```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
```