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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgeqrt (f08ap)

Purpose

nag_lapack_zgeqrt (f08ap) recursively computes, with explicit blocking, the $QR$ factorization of a complex $m$ by $n$ matrix.

Syntax

[a, t, info] = f08ap(nb, a, 'm', m, 'n', n)
[a, t, info] = nag_lapack_zgeqrt(nb, a, 'm', m, 'n', n)

Description

nag_lapack_zgeqrt (f08ap) forms the $QR$ factorization of an arbitrary rectangular complex $m$ by $n$ matrix. No pivoting is performed.
It differs from nag_lapack_zgeqrf (f08as) in that it: requires an explicit block size; stores reflector factors that are upper triangular matrices of the chosen block size (rather than scalars); and recursively computes the $QR$ factorization based on the algorithm of Elmroth and Gustavson (2000).
If $m\ge n$, the factorization is given by:
 $A = Q R 0 ,$
where $R$ is an $n$ by $n$ upper triangular matrix (with real diagonal elements) and $Q$ is an $m$ by $m$ unitary matrix. It is sometimes more convenient to write the factorization as
 $A = Q1 Q2 R 0 ,$
which reduces to
 $A = Q1 R ,$
where ${Q}_{1}$ consists of the first $n$ columns of $Q$, and ${Q}_{2}$ the remaining $m-n$ columns.
If $m, $R$ is upper trapezoidal, and the factorization can be written
 $A = Q R1 R2 ,$
where ${R}_{1}$ is upper triangular and ${R}_{2}$ is rectangular.
The matrix $Q$ is not formed explicitly but is represented as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with $Q$ in this representation (see Further Comments).
Note also that for any $k, the information returned represents a $QR$ factorization of the first $k$ columns of the original matrix $A$.

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:     $\mathrm{nb}$int64int32nag_int scalar
The explicitly chosen block size to be used in computing the $QR$ factorization. See Further Comments for details.
Constraints:
• ${\mathbf{nb}}\ge 1$;
• if $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)>0$, ${\mathbf{nb}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $A$.

Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array a.
$n$, the number of columns of the 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If $m\ge n$, the elements below the diagonal store details of the unitary matrix $Q$ and the upper triangle stores the corresponding elements of the $n$ by $n$ upper triangular matrix $R$.
If $m, the strictly lower triangular part stores details of the unitary matrix $Q$ and the remaining elements store the corresponding elements of the $m$ by $n$ upper trapezoidal matrix $R$.
The diagonal elements of $R$ are real.
2:     $\mathrm{t}\left(\mathit{ldt},:\right)$ – complex array
The first dimension of the array t will be ${\mathbf{nb}}$.
The second dimension of the array t will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
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 $\mathbit{T}=\left[{\mathbit{T}}_{1}|{\mathbit{T}}_{2}|\dots |{\mathbit{T}}_{b}\right]$.
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 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 real floating-point operations is approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{8}{3}{m}^{2}\left(3n-m\right)$ if $m.
To apply $Q$ to an arbitrary complex rectangular matrix $C$, nag_lapack_zgeqrt (f08ap) may be followed by a call to nag_lapack_zgemqrt (f08aq). For example,
```[t, c, info] = f08aq('Left', 'Conjugate Transpose', nb, a, t, c);
```
forms $C={Q}^{\mathrm{H}}C$, where $C$ is $m$ by $p$.
To form the unitary matrix $Q$ explicitly, simply initialize the $m$ by $m$ matrix $C$ to the identity matrix and form $C=QC$ using nag_lapack_zgemqrt (f08aq) as above.
The block size, nb, used by nag_lapack_zgeqrt (f08ap) 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 compute a $QR$ factorization with column pivoting, use nag_lapack_ztpqrt (f08bp) or nag_lapack_zgeqpf (f08bs).
The real analogue of this function is nag_lapack_dgeqrt (f08ab).

Example

This example solves 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
 $B = -2.09+1.93i 3.26-2.70i 3.34-3.53i -6.22+1.16i -4.94-2.04i 7.94-3.13i 0.17+4.23i 1.04-4.26i -5.19+3.63i -2.31-2.12i 0.98+2.53i -1.39-4.05i .$
```function f08ap_example

fprintf('f08ap 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];

% Compute the QR Factorisation of A
[QR, T, info] = f08ap(n,a);

% Compute C = (C1) = (Q^H)*B
[c1, info] = f08aq(...
'Left', 'Conjugate Transpose', QR, T, b);

% Compute least-squares solutions by backsubstitution in R*X = C1
[x, info] = f07ts(...
'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');

```
```f08ap example results

Least-squares solutions
-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
```

Chapter Contents
Chapter Introduction
NAG Toolbox

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