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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztpqrt (f08bp)

## Purpose

nag_lapack_ztpqrt (f08bp) computes the $QR$ factorization of a complex $\left(m+n\right)$ by $n$ triangular-pentagonal matrix.

## Syntax

[a, b, t, info] = f08bp(l, nb, a, b, 'm', m, 'n', n)
[a, b, t, info] = nag_lapack_ztpqrt(l, nb, a, b, 'm', m, 'n', n)

## Description

nag_lapack_ztpqrt (f08bp) forms the $QR$ factorization of a complex $\left(m+n\right)$ 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_zgeqrt (f08ap)) is performed on the matrix $C$. The upper triangular matrix $R$, details of the unitary 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_ztpqrt (f08bp) 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_zgeqrt (f08ap)). Given an initial least-squares problem $\stackrel{^}{B}\stackrel{^}{X}=\stackrel{^}{Y}$ where $X$ and $Y$ are $l$ by $\mathit{nrhs}$ matrices, we have $\stackrel{^}{R}\stackrel{^}{X}={\stackrel{^}{Q}}^{\mathrm{H}}\stackrel{^}{Y}$.
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 ${\stackrel{^}{Q}}^{\mathrm{H}}\stackrel{^}{Y}$.
nag_lapack_ztpqrt (f08bp) 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex 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 unitary matrix $Q$.
3:     $\mathrm{t}\left(\mathit{ldt},:\right)$ – complex 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 unitary 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_ztpqrt (f08bp) 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 complex rectangular matrix $C$, nag_lapack_ztpqrt (f08bp) may be followed by a call to nag_lapack_ztpmqrt (f08bq). For example,
```[t, c, info] = f08bq('Left','Transpose', nb, a(:,1:min(m,n)), t, c);
```
forms $C={Q}^{\mathrm{H}}C$, where $C$ is $\left(m+n\right)$ by $p$.
To form the unitary matrix $Q$ explicitly set $p=m+n$, initialize $C$ to the identity matrix and make a call to nag_lapack_ztpmqrt (f08bq) 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 $B$,
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i and$
 $B= -2.09+1.93i 3.26-2.70i 3.34-3.53i -6.22+1.16i -4.94-2.04i 7.94-3.13i 0.17+4.23i 1.04-4.26i -5.19+3.63i -2.31-2.12i 0.98+2.53i -1.39-4.05i .$
A $QR$ factorization is performed on the first $4$ rows of $A$ using nag_lapack_zgeqrt (f08ap) after which the first $4$ rows of $B$ are updated by applying ${Q}^{T}$ using nag_lapack_zgemqrt (f08aq). The remaining row is added by performing a $QR$ update using nag_lapack_ztpqrt (f08bp); $B$ is updated by applying the new ${Q}^{T}$ using nag_lapack_ztpmqrt (f08bq); the solution is finally obtained by triangular solve using $R$ from the updated $QR$.
```function f08bp_example

fprintf('f08bp 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');

```
```f08bp 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
```