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NAG Toolbox: nag_lapack_dormrq (f08ck)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dormrq (f08ck) multiplies a general real m by n matrix C by the real orthogonal matrix Q from an RQ factorization computed by nag_lapack_dgerqf (f08ch).

Syntax

[a, c, info] = f08ck(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)
[a, c, info] = nag_lapack_dormrq(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)

Description

nag_lapack_dormrq (f08ck) is intended to be used following a call to nag_lapack_dgerqf (f08ch), which performs an RQ factorization of a real matrix A and represents the orthogonal matrix Q as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC ,   QTC ,   CQ ,   CQT ,  
overwriting the result on C, which may be any real rectangular m by n matrix.
A common application of this function is in solving underdetermined linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Example in nag_lapack_dgerqf (f08ch).

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     side – string (length ≥ 1)
Indicates how Q or QT is to be applied to C.
side='L'
Q or QT is applied to C from the left.
side='R'
Q or QT is applied to C from the right.
Constraint: side='L' or 'R'.
2:     trans – string (length ≥ 1)
Indicates whether Q or QT is to be applied to C.
trans='N'
Q is applied to C.
trans='T'
QT is applied to C.
Constraint: trans='N' or 'T'.
3:     alda: – double array
The first dimension of the array a must be at least max1,k.
The second dimension of the array a must be at least max1,m if side='L' and at least max1,n if side='R'.
The ith row of a must contain the vector which defines the elementary reflector Hi, for i=1,2,,k, as returned by nag_lapack_dgerqf (f08ch).
4:     tau: – double array
The dimension of the array tau must be at least max1,k
taui must contain the scalar factor of the elementary reflector Hi, as returned by nag_lapack_dgerqf (f08ch).
5:     cldc: – double array
The first dimension of the array c must be at least max1,m.
The second dimension of the array c must be at least max1,n.
The m by n matrix C.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the array c.
m, the number of rows of the matrix C.
Constraint: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array c.
n, the number of columns of the matrix C.
Constraint: n0.
3:     k int64int32nag_int scalar
Default: the first dimension of the array a and the dimension of the array tau.
k, the number of elementary reflectors whose product defines the matrix Q.
Constraints:
  • if side='L', m k 0 ;
  • if side='R', n k 0 .

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,k.
The second dimension of the array a will be max1,m if side='L' and at least max1,n if side='R'.
Is modified by nag_lapack_dormrq (f08ck) but restored on exit.
2:     cldc: – double array
The first dimension of the array c will be max1,m.
The second dimension of the array c will be max1,n.
c stores QC or QTC or CQ or CQT as specified by side and trans.
3:     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: side, 2: trans, 3: m, 4: n, 5: k, 6: a, 7: lda, 8: tau, 9: c, 10: ldc, 11: work, 12: lwork, 13: 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 result differs from the exact result by a matrix E such that
E2 = Oε C2  
where ε is the machine precision.

Further Comments

The total number of floating-point operations is approximately 2nk2m-k if side='L' and 2mk2n-k if side='R'.
The complex analogue of this function is nag_lapack_zunmrq (f08cx).

Example

See Example in nag_lapack_dgerqf (f08ch).
function f08ck_example


fprintf('f08ck 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 = [-0.57, -1.28, -0.39,  0.25;
     -1.93,  1.08, -0.31, -2.14;
      2.30,  0.24,  0.40, -0.35;
     -1.93,  0.64, -0.66,  0.08;
      0.15,  0.30,  0.15, -2.13;
     -0.02,  1.03, -1.43,  0.50];
b = [1, 0, -1, 0;
     0, 1, 0, -1];
c = [-1.50; -2.14;  1.23; -0.54; -1.68; 0.82];
d = [0; 0];

% Compute the generalized RQ factorization of (B,A) as
% A = ZRQ, B = TQ
[TQ, taub, ZR, taua, info] = ... 
  f08zf(b, a);

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

% Update c = Z^T*c -> minimize (Ry-c)
[cup, info] = f08ag( ...
		     'Left','Transpose',ZR,taua,c);

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

[y2, info] = f07te( ...
		    '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 = triu(ZR(1:n-p,1:n-p));
[y1, info] = f07te( ...
		    'Upper', 'No transpose', 'Non-unit', R11, c1);

% Contruct y and backtransform for x = Q^Ty
y = [y1;y2];
[~, x, info] = f08ck( ...
		      'Left', 'Transpose', TQ, taub, y);
fprintf('Constrained least squares solution\n');
disp(x);

fprintf('Residuals computed directly\n');
res = a*x - c;
disp(res);
fprintf('Residual norm\n');
disp(norm(res));


f08ck example results

Constrained least squares solution
    0.4890
    0.9975
    0.4890
    0.9975

Residuals computed directly
    0.0030
   -0.0129
   -0.0193
   -0.0084
    0.0012
   -0.0029

Residual norm
    0.0251


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Chapter Contents
Chapter Introduction
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