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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zggqrf (f08zs)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zggqrf (f08zs) computes a generalized QR factorization of a complex matrix pair A,B, where A is an n by m matrix and B is an n by p matrix.

Syntax

[a, taua, b, taub, info] = f08zs(a, b, 'n', n, 'm', m, 'p', p)
[a, taua, b, taub, info] = nag_lapack_zggqrf(a, b, 'n', n, 'm', m, 'p', p)

Description

nag_lapack_zggqrf (f08zs) forms the generalized QR factorization of an n by m matrix A and an n by p matrix B 
A =QR ,   B=QTZ ,  
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 = mmR11n-m0() ,   if ​nm; nm-nnR11R12() ,   if ​n<m,  
with R11 upper triangular,
T = p-nnn0T12() ,   if ​np, pn-pT11pT21() ,   if ​n>p,  
with T12 or T21 upper triangular.
In particular, if B is square and nonsingular, the generalized QR factorization of A and B implicitly gives the QR factorization of B-1A as
B-1A= ZH 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 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,n.
The second dimension of the array a must be at least max1,m.
The n by m matrix A.
2:     bldb: – complex array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,p.
The n by p matrix 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, the number of rows of the matrices A and B.
Constraint: n0.
2:     m int64int32nag_int scalar
Default: the second dimension of the array a.
m, the number of columns of the matrix A.
Constraint: m0.
3:     p int64int32nag_int scalar
Default: the second dimension of the array b.
p, the number of columns of the matrix B.
Constraint: p0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,m.
The elements on and above the diagonal of the array contain the minn,m by m upper trapezoidal matrix R (R is upper triangular if nm); the elements below the diagonal, with the array taua, represent the unitary matrix Q as a product of minn,m elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2:     tauaminn,m – 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,n.
The second dimension of the array b will be max1,p.
If np, the upper triangle of the subarray b1:np-n+1:p contains the n by n upper triangular matrix T12.
If n>p, the elements on and above the n-pth subdiagonal contain the n by p upper trapezoidal matrix T; the remaining elements, 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:     taubminn,p – 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: 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 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_zungqr (f08at) and nag_lapack_zungrq (f08cw) respectively. nag_lapack_zunmqr (f08au) may be used to multiply Q by another matrix and nag_lapack_zunmrq (f08cx) may be used to multiply Z by another matrix.
The real analogue of this function is nag_lapack_dggqrf (f08ze).

Example

This example solves the general Gauss–Markov linear model problem
minx y2   subject to   d=Ax+By  
where
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.98+1.98i -1.20+0.19i -0.66+0.42i 0.62-0.46i 1.01+0.02i 0.63-0.17i 1.08-0.28i 0.20-0.12i -0.07+1.23i ,  
B = 0.5-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 0.0i+0.0 5.0-4.0i  
and
d = 6.00-0.40i -5.27+0.90i 2.72-2.13i -1.30-2.80i .  
The solution is obtained by first computing a generalized QR 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 f08zs_example


fprintf('f08zs example results\n\n');

a = [ 0.96 - 0.81i, -0.03 + 0.96i, -0.91 + 2.06i;
     -0.98 + 1.98i, -1.20 + 0.19i, -0.66 + 0.42i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i;
      1.08 - 0.28i,  0.20 - 0.12i, -0.07 + 1.23i];
b = [ 0.5 - 1i,  0 + 0i,  0 + 0i,  0 + 0i;
      0   + 0i,  1 - 2i,  0 + 0i,  0 + 0i;
      0   + 0i,  0 + 0i,  2 - 3i,  0 + 0i;
      0   + 0i,  0 + 0i,  0 + 0i,  5 - 4i];
d = [ 6.00 - 0.40i;
     -5.27 + 0.90i;
      2.72 - 2.13i;
     -1.30 - 2.80i];
[n,m] = size(a);
p = size(b,2);

% 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] = f08zs( ...
                                  a, b);

% Compute c = (c1) = (Q**H)*d
%             (c2)
[c, info] = f08au( ...
                   'Left', 'Conjugate 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
  indy = m+p-n+1:p;
  [y(indy), info] = f07ts( ...
                           'Upper', 'No transpose', 'Non-unit', ...
                           complex(b(m+1:n, indy)), c(m+1:n));

  % Compute estimate of the square root of the residual sum of squares
  % norm(y) = norm(w2)
  rnorm = norm(y(indy), 1);

  % Form c1 - T12*w2 in c
  c = c - b(:,indy)*y(indy);
end

% Solve R*x = c1 - T12*w2 for x
[c(1:m), info] = f07ts( ...
                        'Upper', 'No transpose', 'Non-unit', a(1:m,:), c(1:m));

% Compute y = (Z^H)*w
[b(max(1, n-p+1):n,:), y, info] = ...
f08cx( ...
       'Left', 'Conjugate Transpose', b(max(1, n-p+1):n,:), taub, y);

disp('Generalized least squares solution');
disp(c(1:m));

ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
                 'Gen', ' ', y, 'B', '1P,E10.2', 'Residual vector', ...
                 'N', 'N', ncols, indent);

fprintf('\nSquare root of the residual sum of squares\n%11.2e\n', ...
        rnorm);


f08zs example results

Generalized least squares solution
  -0.9846 + 1.9950i
   3.9929 - 4.9748i
  -3.0026 + 0.9994i

 Residual vector
   (  1.26E-04, -4.66E-04)
   (  1.11E-03, -8.61E-04)
   (  3.84E-03, -1.82E-03)
   (  2.03E-03,  3.02E-03)

Square root of the residual sum of squares
   5.79e-03

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Chapter Introduction
NAG Toolbox

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