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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dggrqf (f08zf)

Purpose

nag_lapack_dggrqf (f08zf) computes a generalized $RQ$ factorization of a real matrix pair $\left(A,B\right)$, where $A$ is an $m$ by $n$ matrix and $B$ is a $p$ by $n$ matrix.

Syntax

[a, taua, b, taub, info] = f08zf(a, b, 'm', m, 'p', p, 'n', n)
[a, taua, b, taub, info] = nag_lapack_dggrqf(a, b, 'm', m, 'p', p, 'n', n)

Description

nag_lapack_dggrqf (f08zf) forms the generalized $RQ$ factorization of an $m$ by $n$ matrix $A$ and a $p$ by $n$ matrix $B$
 $A = RQ , B= ZTQ ,$
where $Q$ is an $n$ by $n$ orthogonal matrix, $Z$ is a $p$ by $p$ orthogonal matrix and $R$ and $T$ are of the form
 $R = n-mmm0R12() ; if ​ m≤n , nm-nR11nR21() ; if ​ m>n ,$
with ${R}_{12}$ or ${R}_{21}$ upper triangular,
 $T = nnT11p-n0() ; if ​ p≥n , pn-ppT11T12() ; if ​ p
with ${T}_{11}$ upper triangular.
In particular, if $B$ is square and nonsingular, the generalized $RQ$ factorization of $A$ and $B$ implicitly gives the $RQ$ factorization of $A{B}^{-1}$ as
 $AB-1= R T-1 ZT .$

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Anderson E, Bai Z and Dongarra J (1992) Generalized QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271
Hammarling S (1987) The numerical solution of the general Gauss-Markov linear model Mathematics in Signal Processing (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore) 441–456 Oxford University Press
Paige C C (1990) Some aspects of generalized $QR$ factorizations . In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
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 $A$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $p$ by $n$ matrix $B$.

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:     $\mathrm{p}$int64int32nag_int scalar
Default: the first dimension of the array b.
$p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
3:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the arrays a, b.
$n$, the number of columns of the matrices $A$ and $B$.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If $m\le n$, the upper triangle of the subarray ${\mathbf{a}}\left(1:m,n-m+1:n\right)$ contains the $m$ by $m$ upper triangular matrix ${R}_{12}$.
If $m\ge n$, the elements on and above the $\left(m-n\right)$th subdiagonal contain the $m$ by $n$ upper trapezoidal matrix $R$; the remaining elements, with the array taua, represent the orthogonal matrix $Q$ as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2:     $\mathrm{taua}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ – double array
The scalar factors of the elementary reflectors which represent the orthogonal matrix $Q$.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The elements on and above the diagonal of the array contain the $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(p,n\right)$ by $n$ upper trapezoidal matrix $T$ ($T$ is upper triangular if $p\ge n$); the elements below the diagonal, with the array taub, represent the orthogonal matrix $Z$ as a product of elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
4:     $\mathrm{taub}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{p}},{\mathbf{n}}\right)\right)$ – double array
The scalar factors of the elementary reflectors which represent the orthogonal matrix $Z$.
5:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: p, 3: n, 4: a, 5: lda, 6: taua, 7: b, 8: ldb, 9: taub, 10: work, 11: lwork, 12: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed generalized $RQ$ factorization is the exact factorization for nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $E2 = O⁡ε A2 and F2= O⁡ε B2 ,$
and $\epsilon$ is the machine precision.

The orthogonal matrices $Q$ and $Z$ may be formed explicitly by calls to nag_lapack_dorgrq (f08cj) and nag_lapack_dorgqr (f08af) respectively. nag_lapack_dormrq (f08ck) may be used to multiply $Q$ by another matrix and nag_lapack_dormqr (f08ag) may be used to multiply $Z$ by another matrix.
The complex analogue of this function is nag_lapack_zggrqf (f08zt).

Example

This example solves the least squares problem
 $minimize x c-Ax2 subject to Bx=d$
where
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 , B= 1 0 -1 0 0 1 0 -1 ,$
 $c = -1.50 -2.14 1.23 -0.54 -1.68 0.82 and d= 0 0 .$
The constraints $Bx=d$ correspond to ${x}_{1}={x}_{3}$ and ${x}_{2}={x}_{4}$.
The solution is obtained by first computing a generalized $RQ$ factorization of the matrix pair $\left(B,A\right)$. The example illustrates the general solution process.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
```function f08zf_example

fprintf('f08zf example results\n\n');

% Find x that minimizes norm(c-Ax) subject to Bx = d .

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 = [1, 0, -1, 0;
0, 1, 0, -1];
c = [-1.50; -2.14;  1.23; -0.54; -1.68; 0.82];
d = [0; 0];

% Compute the generalized RQ factorization of (B,A) as
% A = ZRQ, B = TQ
[TQ, taub, ZR, taua, info] = f08zf( ...
b, a);

% Set Qx = y. The problem reduces to:
% minimize (Ry - Z^Tc) subject to Ty = d

% Update c = Z^T*c -> minimize (Ry-c)
[cup, info] = f08ag( ...
'Left','Transpose',ZR,taua,c);

% Solve Ty = d for last p elements
T12 = triu(TQ(1:p,n-p+1:n));

[y2, info] = f07te( ...
'Upper', 'No transpose', 'Non-unit', T12, d);

% (from Ry-c) R11*y1 + R12*y2 = c1 --> R11*y1 = c1 - R12*y2
% Update c1
c1 = cup(1:n-p) - ZR(1:n-p,n-p+1:n)*y2;

% Solve R11*y1 = c1 for y1
R11 = triu(ZR(1:n-p,1:n-p));
[y1, info] = f07te( ...
'Upper', 'No transpose', 'Non-unit', R11, c1);

% Contruct y and backtransform for x = Q^Ty
y = [y1;y2];
[~, x, info] = f08ck( ...
'Left', 'Transpose', TQ, taub, y);

fprintf('Constrained least squares solution\n');
disp(x);
res = a*x - c;
fprintf('Residual norm\n%12.2e\n', norm(res));

```
```f08zf example results

Constrained least squares solution
0.4890
0.9975
0.4890
0.9975

Residual norm
2.51e-02
```

Chapter Contents
Chapter Introduction
NAG Toolbox

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