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_dtpqrt (f08bb)

## Purpose

nag_lapack_dtpqrt (f08bb) computes the $QR$ factorization of a real $\left(m+n\right)$ by $n$ triangular-pentagonal matrix.

## Syntax

[a, b, t, info] = f08bb(l, nb, a, b, 'm', m, 'n', n)
[a, b, t, info] = nag_lapack_dtpqrt(l, nb, a, b, 'm', m, 'n', n)

## Description

nag_lapack_dtpqrt (f08bb) forms the $QR$ factorization of a real $\left(m+n\right)$ by $n$ triangular-pentagonal matrix $C$,
 $C= A B$
where $A$ is an upper triangular $n$ by $n$ matrix and $B$ is an $m$ by $n$ pentagonal matrix consisting of an $\left(m-l\right)$ by $n$ rectangular matrix ${B}_{1}$ on top of an $l$ by $n$ upper trapezoidal matrix ${B}_{2}$:
 $B= B1 B2 .$
The upper trapezoidal matrix ${B}_{2}$ consists of the first $l$ rows of an $n$ by $n$ upper triangular matrix, where $0\le l\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$. If $l=0$, $B$ is $m$ by $n$ rectangular; if $l=n$ and $m=n$, $B$ is upper triangular.
A recursive, explicitly blocked, $QR$ factorization (see nag_lapack_dgeqrt (f08ab)) is performed on the matrix $C$. The upper triangular matrix $R$, details of the orthogonal matrix $Q$, and further details (the block reflector factors) of $Q$ are returned.
Typically the matrix $A$ or ${B}_{2}$ contains the matrix $R$ from the $QR$ factorization of a subproblem and nag_lapack_dtpqrt (f08bb) performs the $QR$ update operation from the inclusion of matrix ${B}_{1}$.
For example, consider the $QR$ factorization of an $l$ by $n$ matrix $\stackrel{^}{B}$ with $l: $\stackrel{^}{B}=\stackrel{^}{Q}\stackrel{^}{R}$, $\stackrel{^}{R}=\left(\begin{array}{cc}\stackrel{^}{{R}_{1}}& \stackrel{^}{{R}_{2}}\end{array}\right)$, where $\stackrel{^}{{R}_{1}}$ is $l$ by $l$ upper triangular and $\stackrel{^}{{R}_{2}}$ is $\left(n-l\right)$ by $n$ rectangular (this can be performed by nag_lapack_dgeqrt (f08ab)). Given an initial least-squares problem $\stackrel{^}{B}\stackrel{^}{X}=\stackrel{^}{Y}$ where $X$ and $Y$ are $l$ by $\mathit{nrhs}$ matrices, we have $\stackrel{^}{R}\stackrel{^}{X}={\stackrel{^}{Q}}^{\mathrm{T}}\stackrel{^}{Y}$.
Now, adding an additional $m-l$ rows to the original system gives the augmented least squares problem
 $BX=Y$
where $B$ is an $m$ by $n$ matrix formed by adding $m-l$ rows on top of $\stackrel{^}{R}$ and $Y$ is an $m$ by $\mathit{nrhs}$ matrix formed by adding $m-l$ rows on top of ${\stackrel{^}{Q}}^{\mathrm{T}}\stackrel{^}{Y}$.
nag_lapack_dtpqrt (f08bb) can then be used to perform the $QR$ factorization of the pentagonal matrix $B$; the $n$ by $n$ matrix $A$ will be zero on input and contain $R$ on output.
In the case where $\stackrel{^}{B}$ is $r$ by $n$, $r\ge n$, $\stackrel{^}{R}$ is $n$ by $n$ upper triangular (forming $A$) on top of $r-n$ rows of zeros (forming first $r-n$ rows of $B$). Augmentation is then performed by adding rows to the bottom of $B$ with $l=0$.

## References

Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel $QR$ Factorization Leads to Better Performance IBM Journal of Research and Development. (Volume 44) 4 605–624
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{l}$int64int32nag_int scalar
$l$, the number of rows of the trapezoidal part of $B$ (i.e., ${B}_{2}$).
Constraint: $0\le {\mathbf{l}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
2:     $\mathrm{nb}$int64int32nag_int scalar
The explicitly chosen block-size to be used in the algorithm for computing the $QR$ factorization. See Further Comments for details.
Constraints:
• ${\mathbf{nb}}\ge 1$;
• if ${\mathbf{n}}>0$, ${\mathbf{nb}}\le {\mathbf{n}}$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ upper triangular matrix $A$.
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ pentagonal matrix $B$ composed of an $\left(m-l\right)$ by $n$ rectangular matrix ${B}_{1}$ above an $l$ by $n$ upper trapezoidal matrix ${B}_{2}$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array b.
$m$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
$n$, the number of columns of the matrix $B$ and the order of the upper triangular matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The upper triangle stores the corresponding elements of the $n$ by $n$ upper triangular matrix $R$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the orthogonal matrix $Q$.
3:     $\mathrm{t}\left(\mathit{ldt},:\right)$ – double array
The first dimension of the array t will be ${\mathbf{nb}}$.
The second dimension of the array t will be ${\mathbf{n}}$.
Further details of the orthogonal matrix $Q$. 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 each of the blocks, an upper triangular block reflector factor is computed: ${\mathbit{T}}_{1},{\mathbit{T}}_{2},\dots ,{\mathbit{T}}_{b}$. These are stored in the ${\mathbf{nb}}$ by $n$ matrix $T$ as $\mathbit{T}=\left[{\mathbit{T}}_{1}|{\mathbit{T}}_{2}|\dots |{\mathbit{T}}_{b}\right]$.
4:     $\mathrm{info}$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 factorization is the exact factorization of a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m.
The block size, nb, used by nag_lapack_dtpqrt (f08bb) is supplied explicitly through the interface. For moderate and large sizes of matrix, the block size can have a marked effect on the efficiency of the algorithm with the optimal value being dependent on problem size and platform. A value of ${\mathbf{nb}}=64\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ is likely to achieve good efficiency and it is unlikely that an optimal value would exceed $340$.
To apply $Q$ to an arbitrary real rectangular matrix $C$, nag_lapack_dtpqrt (f08bb) may be followed by a call to nag_lapack_dtpmqrt (f08bc). For example,
```[t, c, info] = f08bc('Left','Transpose', nb, a(:,1:min(m,n)), t, c);
```
forms $C={Q}^{\mathrm{T}}C$, where $C$ is $\left(m+n\right)$ by $p$.
To form the orthogonal matrix $Q$ explicitly set $p=m+n$, initialize $C$ to the identity matrix and make a call to nag_lapack_dtpmqrt (f08bc) as above.

## Example

This example finds the basic solutions for the linear least squares problems
 $minimize⁡ Axi - bi 2 , i=1,2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 and B= -2.67 0.41 -0.55 -3.10 3.34 -4.01 -0.77 2.76 0.48 -6.17 4.10 0.21 .$
A $QR$ factorization is performed on the first $4$ rows of $A$ using nag_lapack_dgeqrt (f08ab) after which the first $4$ rows of $B$ are updated by applying ${Q}^{T}$ using nag_lapack_dgemqrt (f08ac). The remaining row is added by performing a $QR$ update using nag_lapack_dtpqrt (f08bb); $B$ is updated by applying the new ${Q}^{T}$ using nag_lapack_dtpmqrt (f08bc); the solution is finally obtained by triangular solve using $R$ from the updated $QR$.
```function f08bb_example

fprintf('f08bb 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');

```
```f08bb 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
```

Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015