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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dggrqf (f08zf)

Purpose

nag_lapack_dggrqf (f08zf) computes a generalized RQRQ factorization of a real matrix pair (A,B)(A,B), where AA is an mm by nn matrix and BB is a pp by nn 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 RQRQ factorization of an mm by nn matrix AA and a pp by nn matrix BB 
A = RQ ,   B = ZTQ ,
A = RQ ,   B= ZTQ ,
where QQ is an nn by nn orthogonal matrix, ZZ is a pp by pp orthogonal matrix and RR and TT are of the form
R =
{
n − mmm(0R12);   if ​m ≤ n,
nm − n(R11) n R21 ;   if ​m > n,
R = { n-mmm(0R12) ;   if ​ mn , nm-n(R11) n R21 ;   if ​ m>n ,
with R12R12 or R21R21 upper triangular,
T =
{
nn(T11) p − n 0 ;   if ​p ≥ n,
pn − pp(T11T12);   if ​p < n,
T = { nn(T11) p-n 0 ;   if ​ pn , pn-pp(T11T12) ;   if ​ p<n ,
with T11T11 upper triangular.
In particular, if BB is square and nonsingular, the generalized RQRQ factorization of AA and BB implicitly gives the RQRQ factorization of AB1AB-1 as
AB1 = (RT1) ZT .
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 QRQR 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,m)max(1,m)
The second dimension of the array must be at least max (1,n)max(1,n)
The mm by nn matrix AA.
2:     b(ldb, : :) – double array
The first dimension of the array b must be at least max (1,p)max(1,p)
The second dimension of the array must be at least max (1,n)max(1,n)
The pp by nn matrix BB.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
mm, the number of rows of the matrix AA.
Constraint: m0m0.
2:     p – int64int32nag_int scalar
Default: The first dimension of the array b.
pp, the number of rows of the matrix BB.
Constraint: p0p0.
3:     n – int64int32nag_int scalar
Default: The second dimension of the arrays a, b.
nn, the number of columns of the matrices AA and BB.
Constraint: n0n0.

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,m)max(1,m)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,m)ldamax(1,m).
If mnmn, the upper triangle of the subarray a(1 : m,nm + 1 : n)a1:mn-m+1:n contains the mm by mm upper triangular matrix R12R12.
If mnmn, the elements on and above the (mn)(m-n)th subdiagonal contain the mm by nn upper trapezoidal matrix RR; the remaining elements, with the array taua, represent the orthogonal matrix QQ as a product of min (m,n)min(m,n) elementary reflectors (see Section [Representation of orthogonal or unitary matrices] in the F08 Chapter Introduction).
2:     taua(min (m,n)min(m,n)) – double array
The scalar factors of the elementary reflectors which represent the orthogonal matrix QQ.
3:     b(ldb, : :) – double array
The first dimension of the array b will be max (1,p)max(1,p)
The second dimension of the array will be max (1,n)max(1,n)
ldbmax (1,p)ldbmax(1,p).
The elements on and above the diagonal of the array contain the min (p,n)min(p,n) by nn upper trapezoidal matrix TT (TT is upper triangular if pnpn); the elements below the diagonal, with the array taub, represent the orthogonal matrix ZZ as a product of elementary reflectors (see Section [Representation of orthogonal or unitary matrices] in the F08 Chapter Introduction).
4:     taub(min (p,n)min(p,n)) – double array
The scalar factors of the elementary reflectors which represent the orthogonal matrix ZZ.
5:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii 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 RQRQ factorization is the exact factorization for nearby matrices (A + E)(A+E) and (B + F)(B+F), where
E2 = Oε A2   and   F2 = Oε B2 ,
E2 = Oε A2   and   F2= Oε B2 ,
and εε is the machine precision.

Further Comments

The orthogonal matrices QQ and ZZ 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 QQ by another matrix and nag_lapack_dormqr (f08ag) may be used to multiply ZZ by another matrix.
The complex analogue of this function is nag_lapack_zggrqf (f08zt).

Example

function nag_lapack_dggrqf_example
a = [1, 0, -1, 0;
     0, 1, 0, -1];
b = [-0.57, -1.28, -0.39, 0.25;
     -1.93, 1.08, -0.31, -2.14;
     2.3, 0.24, 0.4, -0.35;
     -1.93, 0.64, -0.66, 0.08;
     0.15, 0.3, 0.15, -2.13;
     -0.02, 1.03, -1.43, 0.5];
c= [-1.50; -2.14; 1.23; -0.54; -1.68; 0.82];
d = [0; 0];

p = 6;
n = 4;
m = 2;

% Compute the generalized RQ factorization of (B,A) as
% A = (0 R12)*Q,   B = Z*(T11 T12 T13)*Q, where R12, T11 and T22
%                        ( 0  T22 T23)
% are upper triangular
[a, taua, b, taub, info] = nag_lapack_dggrqf(a, b);

% Compute (f1) = (Z^T)*c
%         (f2)
[f, info] = nag_lapack_dormqr('Left', 'Transpose', b, taub, c);

% Putting Q*x = (y1), solve R12*w = d for w, storing result in d
%               (w )
[d, info] = nag_lapack_dtrtrs('Upper', 'No transpose', 'Non-unit', a(:, n-m+1:n), d);

% Form f1 - T1*w, T1 = (T12 T13), in f
f = f - b(:, n-m+1:n)*d;

% Solve T11*y1 = f1 - T1*w for y1
[f(1:n-m), info] = nag_lapack_dtrtrs('Upper', 'No transpose', 'Non-unit', ...
                         b(1:n-m, 1:n-m), f(1:n-m));

% Compute x = (Q^T)*y
[a, x, info] = nag_lapack_dormrq('Left', 'Transpose', a, taua, [f(1:n-m);d]);

% Putting w = (y2), form f2 - T22*y2 - T23*y3
%             (y3)
% d = T22*y2
d = triu(b(n-m+1:n,n-m+1:n))*d;

% f = f2 - T22*y2
for i=1:min(p,n)-n+m
 f(n-m+i) = f(n-m+i) - d(i);
end

if p < n
  % f = f2 - T22*y2 - T23*y3
  f(n-m+1:p) = f(n-m+1) - b(n-m+1,p+1:n)*d(p-n+m+1:m);
end

% Compute estimate of the square root of the residual sum of
% squares norm(r) = norm(f2 - T22*y2 - T23*y3)
rnorm = norm(f(n-m+1:p));

fprintf('\nConstrained least squares solution\n');
disp(transpose(x(1:n)));

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

Constrained least squares solution
    0.4890    0.9975    0.4890    0.9975

Square root of the residual sum of squares
    0.0251


function f08zf_example
a = [1, 0, -1, 0;
     0, 1, 0, -1];
b = [-0.57, -1.28, -0.39, 0.25;
     -1.93, 1.08, -0.31, -2.14;
     2.3, 0.24, 0.4, -0.35;
     -1.93, 0.64, -0.66, 0.08;
     0.15, 0.3, 0.15, -2.13;
     -0.02, 1.03, -1.43, 0.5];
c= [-1.50; -2.14; 1.23; -0.54; -1.68; 0.82];
d = [0; 0];

p = 6;
n = 4;
m = 2;

% Compute the generalized RQ factorization of (B,A) as
% A = (0 R12)*Q,   B = Z*(T11 T12 T13)*Q, where R12, T11 and T22
%                        ( 0  T22 T23)
% are upper triangular
[a, taua, b, taub, info] = f08zf(a, b);

% Compute (f1) = (Z^T)*c
%         (f2)
[f, info] = f08ag('Left', 'Transpose', b, taub, c);

% Putting Q*x = (y1), solve R12*w = d for w, storing result in d
%               (w )
[d, info] = f07te('Upper', 'No transpose', 'Non-unit', a(:, n-m+1:n), d);

% Form f1 - T1*w, T1 = (T12 T13), in f
f = f - b(:, n-m+1:n)*d;

% Solve T11*y1 = f1 - T1*w for y1
[f(1:n-m), info] = f07te('Upper', 'No transpose', 'Non-unit', ...
                         b(1:n-m, 1:n-m), f(1:n-m));

% Compute x = (Q^T)*y
[a, x, info] = f08ck('Left', 'Transpose', a, taua, [f(1:n-m);d]);

% Putting w = (y2), form f2 - T22*y2 - T23*y3
%             (y3)
% d = T22*y2
d = triu(b(n-m+1:n,n-m+1:n))*d;

% f = f2 - T22*y2
for i=1:min(p,n)-n+m
 f(n-m+i) = f(n-m+i) - d(i);
end

if p < n
  % f = f2 - T22*y2 - T23*y3
  f(n-m+1:p) = f(n-m+1) - b(n-m+1,p+1:n)*d(p-n+m+1:m);
end

% Compute estimate of the square root of the residual sum of
% squares norm(r) = norm(f2 - T22*y2 - T23*y3)
rnorm = norm(f(n-m+1:p));

fprintf('\nConstrained least squares solution\n');
disp(transpose(x(1:n)));

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

Constrained least squares solution
    0.4890    0.9975    0.4890    0.9975

Square root of the residual sum of squares
    0.0251



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

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