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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dggqrf (f08ze)

Purpose

nag_lapack_dggqrf (f08ze) computes a generalized QR$QR$ factorization of a real matrix pair (A,B)$\left(A,B\right)$, where A$A$ is an n$n$ by m$m$ matrix and B$B$ is an n$n$ by p$p$ matrix.

Syntax

[a, taua, b, taub, info] = f08ze(a, b, 'n', n, 'm', m, 'p', p)
[a, taua, b, taub, info] = nag_lapack_dggqrf(a, b, 'n', n, 'm', m, 'p', p)

Description

nag_lapack_dggqrf (f08ze) forms the generalized QR$QR$ factorization of an n$n$ by m$m$ matrix A$A$ and an n$n$ by p$p$ matrix B$B$
 A = QR ,   B = QTZ , $A =QR , B=QTZ ,$
where Q$Q$ is an n$n$ by n$n$ orthogonal matrix, Z$Z$ is a p$p$ by p$p$ orthogonal matrix and R$R$ and T$T$ are of the form
R =
{
 mm(R11) n − m 0 ,   if ​n ≥ m ;
 nm − nn(R11R12) ,   if ​n < m,
$R = { mm(R11) n-m 0 , if ​n≥m; nm-nn(R11R12) , if ​n
with R11${R}_{11}$ upper triangular,
T =
{
 p − nnn(0T12) ,   if ​n ≤ p,
 pn − p(T11) p T21 ,   if ​n > p,
$T = { p-nnn(0T12) , if ​n≤p, pn-p(T11) p T21 , if ​n>p,$
with T12${T}_{12}$ or T21${T}_{21}$ upper triangular.
In particular, if B$B$ is square and nonsingular, the generalized QR$QR$ factorization of A$A$ and B$B$ implicitly gives the QR$QR$ factorization of B1A${B}^{-1}A$ as
 B − 1A = ZT (T − 1R) . $B-1A= ZT ( T-1 R ) .$

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$QR$ factorizations . In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press

Parameters

Compulsory Input Parameters

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

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
n$n$, the number of rows of the matrices A$A$ and B$B$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     m – int64int32nag_int scalar
Default: The second dimension of the array a.
m$m$, the number of columns of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
3:     p – int64int32nag_int scalar
Default: The second dimension of the array b.
p$p$, the number of columns of the matrix B$B$.
Constraint: p0${\mathbf{p}}\ge 0$.

Input Parameters Omitted from the MATLAB Interface

lda ldb work lwork

Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The elements on and above the diagonal of the array contain the min (n,m)$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,m\right)$ by m$m$ upper trapezoidal matrix R$R$ (R$R$ is upper triangular if nm$n\ge m$); the elements below the diagonal, with the array taua, represent the orthogonal matrix Q$Q$ as a product of min (n,m)$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,m\right)$ elementary reflectors (see Section [Representation of orthogonal or unitary matrices] in the F08 Chapter Introduction).
2:     taua(min (n,m)$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},{\mathbf{m}}\right)$) – double array
The scalar factors of the elementary reflectors which represent the orthogonal matrix Q$Q$.
3:     b(ldb, : $:$) – double array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,p)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If np$n\le p$, the upper triangle of the subarray b( 1 : n , pn + 1 : p ) ${\mathbf{b}}\left(1:n,p-n+1:p\right)$ contains the n$n$ by n$n$ upper triangular matrix T12${T}_{12}$.
If n > p$n>p$, the elements on and above the (np)$\left(n-p\right)$th subdiagonal contain the n$n$ by p$p$ upper trapezoidal matrix T$T$; the remaining elements, with the array taub, represent the orthogonal matrix Z$Z$ as a product of elementary reflectors (see Section [Representation of orthogonal or unitary matrices] in the F08 Chapter Introduction).
4:     taub(min (n,p)$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},{\mathbf{p}}\right)$) – double array
The scalar factors of the elementary reflectors which represent the orthogonal matrix Z$Z$.
5:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: m, 3: p, 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 QR$QR$ factorization is the exact factorization for nearby matrices (A + E)$\left(A+E\right)$ and (B + F)$\left(B+F\right)$, where
 ‖E‖2 = Oε ‖A‖2   and   ‖F‖2 = Oε ‖B‖2 , $‖E‖2 = O⁡ε ‖A‖2 and ‖F‖2= O⁡ε ‖B‖2 ,$
and ε$\epsilon$ is the machine precision.

The orthogonal matrices Q$Q$ and Z$Z$ may be formed explicitly by calls to nag_lapack_dorgqr (f08af) and nag_lapack_dorgrq (f08cj) respectively. nag_lapack_dormqr (f08ag) may be used to multiply Q$Q$ by another matrix and nag_lapack_dormrq (f08ck) may be used to multiply Z$Z$ by another matrix.
The complex analogue of this function is nag_lapack_zggqrf (f08zs).

Example

```function nag_lapack_dggqrf_example
a = [-0.57, -1.28, -0.39;
-1.93,  1.08, -0.31;
2.3,   0.24, -0.4;
-0.02,  1.03, -1.43];
b = [0.5, 0, 0, 0;
0,   1, 0, 0;
0,   0, 2, 0;
0,   0, 0, 5];
d = [1.32; -4.00; 5.52; 3.24];

n = 4;
m = 3;
p = 4;

% Compute the generalized QR factorization of (A,B) as
% A = Q*(R),   B = Q*(T11 T12)*Z
%       (0)          ( 0  T22)
[a, taua, b, taub, info] = nag_lapack_dggqrf(a, b);

% Compute c = (c1) = (Q**T)*d
%             (c2)
[c, info] = nag_lapack_dormqr('Left', 'Transpose', a, taua, d);

% Putting Z*y = w = (w1), set w1 = 0, storing the result in y1
%                   (w2)
y = zeros(p, 1);
if n > m

% Solve T22*w2 = c2 for w2, storing result in y2
[y(m+p-n+1:p), info] = nag_lapack_dtrtrs('Upper', 'No transpose', 'Non-unit', ...
b(m+1:n, m+p-n+1:p), c(m+1:n));

% Compute estimate of the square root of the residual sum of squares
% norm(y) = norm(w2)
rnorm = norm(y(m+p-n+1:p), 1);

% Form c1 - T12*w2 in c
c = c - b(:,m+p-n+1:p)*y(m+p-n+1:p);
end

% Solve R*x = c1 - T12*w2 for x
[c(1:m), info] = nag_lapack_dtrtrs('Upper', 'No transpose', 'Non-unit', a(1:m,:), c(1:m));

% Compute y = (Z^T)*w
[b(max(1, n-p+1):n,:), y, info] = ...
nag_lapack_dormrq('Left', 'Transpose', b(max(1, n-p+1):n,:), taub, y);

fprintf('\nGeneralized least squares solution\n');
disp(transpose(c(1:m)));

fprintf('Residual vector\n');
disp(transpose(y));

fprintf('Square root of the residual sum of squares\n');
disp(rnorm);
```
```

Generalized least squares solution
1.9889   -1.0058   -2.9911

Residual vector
-0.0006   -0.0025   -0.0047    0.0077

Square root of the residual sum of squares
0.0094

```
```function f08ze_example
a = [-0.57, -1.28, -0.39;
-1.93,  1.08, -0.31;
2.3,   0.24, -0.4;
-0.02,  1.03, -1.43];
b = [0.5, 0, 0, 0;
0,   1, 0, 0;
0,   0, 2, 0;
0,   0, 0, 5];
d = [1.32; -4.00; 5.52; 3.24];

n = 4;
m = 3;
p = 4;

% Compute the generalized QR factorization of (A,B) as
% A = Q*(R),   B = Q*(T11 T12)*Z
%       (0)          ( 0  T22)
[a, taua, b, taub, info] = f08ze(a, b);

% Compute c = (c1) = (Q**T)*d
%             (c2)
[c, info] = f08ag('Left', 'Transpose', a, taua, d);

% Putting Z*y = w = (w1), set w1 = 0, storing the result in y1
%                   (w2)
y = zeros(p, 1);
if n > m

% Solve T22*w2 = c2 for w2, storing result in y2
[y(m+p-n+1:p), info] = f07te('Upper', 'No transpose', 'Non-unit', ...
b(m+1:n, m+p-n+1:p), c(m+1:n));

% Compute estimate of the square root of the residual sum of squares
% norm(y) = norm(w2)
rnorm = norm(y(m+p-n+1:p), 1);

% Form c1 - T12*w2 in c
c = c - b(:,m+p-n+1:p)*y(m+p-n+1:p);
end

% Solve R*x = c1 - T12*w2 for x
[c(1:m), info] = f07te('Upper', 'No transpose', 'Non-unit', a(1:m,:), c(1:m));

% Compute y = (Z^T)*w
[b(max(1, n-p+1):n,:), y, info] = ...
f08ck('Left', 'Transpose', b(max(1, n-p+1):n,:), taub, y);

fprintf('\nGeneralized least squares solution\n');
disp(transpose(c(1:m)));

fprintf('Residual vector\n');
disp(transpose(y));

fprintf('Square root of the residual sum of squares\n');
disp(rnorm);
```
```

Generalized least squares solution
1.9889   -1.0058   -2.9911

Residual vector
-0.0006   -0.0025   -0.0047    0.0077

Square root of the residual sum of squares
0.0094

```

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013