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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zggrqf (f08zt)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zggrqf (f08zt) computes a generalized RQ factorization of a complex matrix pair A,B, where A is an m by n matrix and B is a p by n matrix.

Syntax

[a, taua, b, taub, info] = f08zt(a, b, 'm', m, 'p', p, 'n', n)
[a, taua, b, taub, info] = nag_lapack_zggrqf(a, b, 'm', m, 'p', p, 'n', n)

Description

nag_lapack_zggrqf (f08zt) 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 unitary matrix, Z is a p by p unitary matrix and R and T are of the form
R = n-mmm0R12() ;   if ​ mn , nm-nR11nR21() ;   if ​ m>n ,  
with R12 or R21 upper triangular,
T = nnT11p-n0() ;   if ​ pn , pn-ppT11T12() ;   if ​ p<n ,  
with T11 upper triangular.
In particular, if B is square and nonsingular, the generalized RQ factorization of A and B implicitly gives the RQ factorization of AB-1 as
AB-1= R T-1 ZH .  

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:     alda: – complex array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,n.
The m by n matrix A.
2:     bldb: – complex array
The first dimension of the array b must be at least max1,p.
The second dimension of the array b must be at least max1,n.
The p by n matrix B.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the array a.
m, the number of rows of the matrix A.
Constraint: m0.
2:     p int64int32nag_int scalar
Default: the first dimension of the array b.
p, the number of rows of the matrix B.
Constraint: p0.
3:     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: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,m.
The second dimension of the array a will be max1,n.
If mn, the upper triangle of the subarray a1:mn-m+1:n contains the m by m upper triangular matrix R12.
If mn, the elements on and above the m-nth subdiagonal contain the m by n upper trapezoidal matrix R; the remaining elements, with the array taua, represent the unitary matrix Q as a product of minm,n elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2:     tauaminm,n – complex array
The scalar factors of the elementary reflectors which represent the unitary matrix Q.
3:     bldb: – complex array
The first dimension of the array b will be max1,p.
The second dimension of the array b will be max1,n.
The elements on and above the diagonal of the array contain the minp,n by n upper trapezoidal matrix T (T is upper triangular if pn); the elements below the diagonal, with the array taub, represent the unitary matrix Z as a product of elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
4:     taubminp,n – complex array
The scalar factors of the elementary reflectors which represent the unitary matrix Z.
5:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info=-i
If 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 A+E and B+F, where
E2 = Oε A2   and   F2= Oε B2 ,  
and ε is the machine precision.

Further Comments

The unitary matrices Q and Z may be formed explicitly by calls to nag_lapack_zungrq (f08cw) and nag_lapack_zungqr (f08at) respectively. nag_lapack_zunmrq (f08cx) may be used to multiply Q by another matrix and nag_lapack_zunmqr (f08au) may be used to multiply Z by another matrix.
The real analogue of this function is nag_lapack_dggrqf (f08zf).

Example

This example solves the least squares problem
minimize x c-Ax2   subject to   Bx=d  
where
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 = 1 0 -1 0 0 1 0 -1 ,   c= -2.54+0.09i 1.65-2.26i -2.11-3.96i 1.82+3.30i -6.41+3.77i 2.07+0.66i   and   d= 0 0 .  
The constraints Bx=d correspond to x1=x3 and x2=x4.
The solution is obtained by first obtaining a generalized RQ factorization of the matrix pair A,B. The example illustrates the general solution process, although the above data corresponds to a simple weighted least squares problem.
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 f08zt_example


fprintf('f08zt 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 = [ 0.96-0.81i -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 = complex([ 1 0 -1  0;
              0 1  0 -1]); 
c = [-2.54+0.09i;
      1.65-2.26i;
     -2.11-3.96i;
      1.82+3.30i;
     -6.41+3.77i;
      2.07+0.66i];
d = complex([0;0]);

% Compute the generalized RQ factorization of (B,A) as
% A = ZRQ, B = TQ
[TQ, taub, ZR, taua, info] = f08zt(b, a);

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

% Update c = Z^H*c -> minimize (Ry-c)
[cup, info] = f08au( ...
                     'Left','Conjugate Transpose',ZR,taua,c);

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

[y2, info] = f07ts( ...
                    '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 = complex(triu(ZR(1:n-p,1:n-p)));
[y1, info] = f07ts( ...
                    'Upper', 'No transpose', 'Non-unit', R11, c1);

% Contruct y and backtransform for x = Q^Hy
y = [y1;y2];
[~, x, info] = f08cx( ...
                      'Left', 'Conjugate Transpose', TQ, taub, y);

disp('Constrained least squares solution');
disp(x);

res = a*x - c;
fprintf('Square root of the residual sum of squares\n%11.2e\n', ...
        norm(res));


f08zt example results

Constrained least squares solution
   1.0874 - 1.9621i
  -0.7409 + 3.7297i
   1.0874 - 1.9621i
  -0.7409 + 3.7297i

Square root of the residual sum of squares
   1.59e-01

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Chapter Introduction
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