Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgemqrt (f08ac)

## Purpose

nag_lapack_dgemqrt (f08ac) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ from a $QR$ factorization computed by nag_lapack_dgeqrt (f08ab).

## Syntax

[c, info] = f08ac(side, trans, v, t, c, 'm', m, 'n', n, 'k', k, 'nb', nb)
[c, info] = nag_lapack_dgemqrt(side, trans, v, t, c, 'm', m, 'n', n, 'k', k, 'nb', nb)

## Description

nag_lapack_dgemqrt (f08ac) is intended to be used after a call to nag_lapack_dgeqrt (f08ab) which performs a $QR$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to form one of the matrix products
 $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on $C$ (which may be any real rectangular matrix).
A common application of this function is in solving linear least squares problems, as described in the F08 Chapter Introduction and illustrated in Example in nag_lapack_dgeqrt (f08ab).

## 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:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – double array
The first dimension, $\mathit{ldv}$, of the array v must satisfy
• if ${\mathbf{side}}=\text{'L'}$, $\mathit{ldv}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{side}}=\text{'R'}$, $\mathit{ldv}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array v must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dgeqrt (f08ab) in the first $k$ columns of its array argument a.
4:     $\mathrm{t}\left(\mathit{ldt},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
Further details of the orthogonal matrix $Q$ as returned by nag_lapack_dgeqrt (f08ab). The number of blocks is $b=⌈\frac{k}{{\mathbf{nb}}}⌉$, where $k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ and each block is of order nb except for the last block, which is of order $k-\left(b-1\right)×{\mathbf{nb}}$. For the $b$ blocks the upper triangular block reflector factors ${\mathbit{T}}_{1},{\mathbit{T}}_{2},\dots ,{\mathbit{T}}_{b}$ are stored in the ${\mathbf{nb}}$ by $n$ matrix $T$ as $\mathbit{T}=\left[{\mathbit{T}}_{1}|{\mathbit{T}}_{2}|\dots |{\mathbit{T}}_{b}\right]$.
5:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array
The first dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $C$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array c.
$m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array c.
$n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{k}$int64int32nag_int scalar
Default: the second dimension of the arrays v, t.
$k$, the number of elementary reflectors whose product defines the matrix $Q$. Usually ${\mathbf{k}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({m}_{A},{n}_{A}\right)$ where ${m}_{A}$, ${n}_{A}$ are the dimensions of the matrix $A$ supplied in a previous call to nag_lapack_dgeqrt (f08ab).
Constraints:
• if ${\mathbf{side}}=\text{'L'}$, ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$;
• if ${\mathbf{side}}=\text{'R'}$, ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
4:     $\mathrm{nb}$int64int32nag_int scalar
Default: the first dimension of the array t.
The block size used in the $QR$ factorization performed in a previous call to nag_lapack_dgeqrt (f08ab); this value must remain unchanged from that call.
Constraints:
• ${\mathbf{nb}}\ge 1$;
• if ${\mathbf{k}}>0$, ${\mathbf{nb}}\le {\mathbf{k}}$.

### Output Parameters

1:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array
The first dimension of the array c will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
c stores $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.
2:     $\mathrm{info}$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_zgemqrt (f08aq).

## Example

See Example in nag_lapack_dgeqrt (f08ab).
```function f08ac_example

fprintf('f08ac 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];

% Compute the QR Factorisation of A
[QR, T, info] = f08ab(n,a);

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

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

% Print least-squares solutions
disp('Least-squares solutions');
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(x(n+1:m,j));
end
fprintf('\nSquare roots of the residual sums of squares\n');
fprintf('%12.2e', rnorm);
fprintf('\n');

```
```f08ac example results

Least-squares solutions
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
```