hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_lapack_zgerqf (f08cv)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zgerqf (f08cv) computes an RQ factorization of a complex m by n matrix A.


[a, tau, info] = f08cv(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_zgerqf(a, 'm', m, 'n', n)


nag_lapack_zgerqf (f08cv) forms the RQ factorization of an arbitrary rectangular real m by n matrix. If mn, the factorization is given by
A = 0 R Q ,  
where R is an m by m lower triangular matrix and Q is an n by n unitary matrix. If m>n the factorization is given by
A =RQ ,  
where R is an m by n upper trapezoidal matrix and Q is again an n by n unitary matrix. In the case where m<n the factorization can be expressed as
A = 0 R Q1 Q2 =RQ2 ,  
where Q1 consists of the first n-m rows of Q and Q2 the remaining m rows.
The matrix Q is not formed explicitly, but is represented as a product of minm,n elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Further Comments).


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


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.

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:     n int64int32nag_int scalar
Default: the second dimension of the array a.
n, the number of columns of the matrix A.
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 R.
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 tau, 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:     tau: – complex array
The dimension of the array tau will be max1,minm,n
The scalar factors of the elementary reflectors.
3:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: 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.


The computed factorization is the exact factorization of a nearby matrix A+E, where
E2 = Oε A2  
and ε is the machine precision.

Further Comments

The total number of floating-point operations is approximately 23m23n-m if mn, or 23n23m-n if m>n.
To form the unitary matrix Q nag_lapack_zgerqf (f08cv) may be followed by a call to nag_lapack_zungrq (f08cw):
[a, info] = f08cw(a, tau, 'k', min(m,n));
but note that the first dimension of the array a must be at least n, which may be larger than was required by nag_lapack_zgerqf (f08cv). When mn, it is often only the first m rows of Q that are required and they may be formed by the call:
[a, info] = f08cw(a, tau);
To apply Q to an arbitrary real rectangular matrix C, nag_lapack_zgerqf (f08cv) may be followed by a call to nag_lapack_zunmrq (f08cx). For example:
[a, c, info] = f08cx('Left','C', a, tau, c);
forms C=QHC, where C is n by p.
The real analogue of this function is nag_lapack_dgerqf (f08ch).


This example finds the minimum norm solution to the underdetermined equations
A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i  
b = -1.35+0.19i 9.41-3.56i -7.57+6.93i .  
The solution is obtained by first obtaining an RQ factorization of the matrix A.
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 f08cv_example

fprintf('f08cv example results\n\n');

% Minimum norm solution of AX = B, m<n
m = 3;
n = 4;
a = [ 0.28 - 0.36i,  0.50 - 0.86i, -0.77 - 0.48i,  1.58 + 0.66i;
     -0.50 - 1.10i, -1.21 + 0.76i, -0.32 - 0.24i, -0.27 - 1.15i;
      0.36 - 0.51i, -0.07 + 1.33i, -0.75 + 0.47i, -0.08 + 1.01i];
b = [ -1.35 + 0.19i; 
       9.41 - 3.56i;
      -7.57 + 6.93i];

% Compute the RQ factorization of A
[rq, tau, info] = f08cv(a);

% RQX = B ==> C = QX = R^-1 B
c = zeros(n, 1);
il = n - m + 1;
[c(il:n,:), info] = f07ts( ...
			   'Upper', 'No transpose','Non-Unit', rq(:,il:n), b);

% QX = C ==> X = Q^H C
[rq, x, info] = f08cx( ...
		       'Left', 'Conjugate Transpose', rq, tau, c);

fprintf('Minimum-norm solution\n');

f08cv example results

Minimum-norm solution
  -2.8501 + 6.4683i
   1.6264 - 0.7799i
   6.9290 + 4.6481i
   1.4048 + 3.2400i

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015