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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zggqrf (f08zs)

Purpose

nag_lapack_zggqrf (f08zs) computes a generalized QRQR factorization of a complex matrix pair (A,B)(A,B), where AA is an nn by mm matrix and BB is an nn by pp 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 QRQR factorization of an nn by mm matrix AA and an nn by pp matrix BB 
A = QR ,   B = QTZ ,
A =QR ,   B=QTZ ,
where QQ is an nn by nn unitary matrix, ZZ is a pp by pp unitary matrix and RR and TT 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 ​nm; nm-nn(R11R12) ,   if ​n<m,
with R11R11 upper triangular,
T =
{
p − nnn(0T12),   if ​n ≤ p,
pn − p(T11) p T21 ,   if ​n > p,
T = { p-nnn(0T12) ,   if ​np, pn-p(T11) p T21 ,   if ​n>p,
with T12T12 or T21T21 upper triangular.
In particular, if BB is square and nonsingular, the generalized QRQR factorization of AA and BB implicitly gives the QRQR factorization of B1AB-1A as
B1A = ZH (T1R) .
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 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, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,m)max(1,m)
The nn by mm matrix AA.
2:     b(ldb, : :) – complex array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,p)max(1,p)
The nn by pp matrix BB.

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

Input Parameters Omitted from the MATLAB Interface

lda ldb work lwork

Output Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,m)max(1,m)
ldamax (1,n)ldamax(1,n).
The elements on and above the diagonal of the array contain the min (n,m)min(n,m) by mm upper trapezoidal matrix RR (RR is upper triangular if nmnm); the elements below the diagonal, with the array taua, represent the unitary matrix QQ as a product of min (n,m)min(n,m) elementary reflectors (see Section [Representation of orthogonal or unitary matrices] in the F08 Chapter Introduction).
2:     taua(min (n,m)min(n,m)) – complex array
The scalar factors of the elementary reflectors which represent the unitary matrix QQ.
3:     b(ldb, : :) – complex array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,p)max(1,p)
ldbmax (1,n)ldbmax(1,n).
If npnp, the upper triangle of the subarray b( 1 : n , pn + 1 : p ) b 1:n , p-n+1:p contains the nn by nn upper triangular matrix T12T12.
If n > pn>p, the elements on and above the (np)(n-p)th subdiagonal contain the nn by pp upper trapezoidal matrix TT; the remaining elements, with the array taub, represent the unitary matrix ZZ as a product of elementary reflectors (see Section [Representation of orthogonal or unitary matrices] in the F08 Chapter Introduction).
4:     taub(min (n,p)min(n,p)) – complex array
The scalar factors of the elementary reflectors which represent the unitary 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: 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 QRQR 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 unitary matrices QQ and ZZ 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 QQ by another matrix and nag_lapack_zunmrq (f08cx) may be used to multiply ZZ by another matrix.
The real analogue of this function is nag_lapack_dggqrf (f08ze).

Example

function nag_lapack_zggqrf_example
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i;
      -0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i;
      1.08 - 0.28i,  0.2 - 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 = 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_zggqrf(a, b);

% Compute c = (c1) = (Q**H)*d
%             (c2)
[c, info] = nag_lapack_zunmqr('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
  [y(m+p-n+1:p), info] = nag_lapack_ztrtrs('Upper', 'No transpose', 'Non-unit', ...
                               complex(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_ztrtrs('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] = ...
    nag_lapack_zunmrq('Left', 'Conjugate 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
  -0.9846 + 1.9950i   3.9929 - 4.9748i  -3.0026 + 0.9994i

Residual vector
   0.0001 - 0.0005i   0.0011 - 0.0009i   0.0038 - 0.0018i   0.0020 + 0.0030i

Square root of the residual sum of squares
    0.0058


function f08zs_example
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i;
      -0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i;
      1.08 - 0.28i,  0.2 - 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 = 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] = 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
  [y(m+p-n+1:p), info] = f07ts('Upper', 'No transpose', 'Non-unit', ...
                               complex(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] = 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);

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
  -0.9846 + 1.9950i   3.9929 - 4.9748i  -3.0026 + 0.9994i

Residual vector
   0.0001 - 0.0005i   0.0011 - 0.0009i   0.0038 - 0.0018i   0.0020 + 0.0030i

Square root of the residual sum of squares
    0.0058



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Chapter Introduction
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