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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_matop_complex_gen_rq (f01rj)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_matop_complex_gen_rq (f01rj) finds the RQ factorization of the complex m by n (mn), matrix A, so that A is reduced to upper triangular form by means of unitary transformations from the right.


[a, theta, ifail] = f01rj(a, 'm', m, 'n', n)
[a, theta, ifail] = nag_matop_complex_gen_rq(a, 'm', m, 'n', n)


The m by n matrix A is factorized as
A= R 0 PH when ​m<n, A=RPH when ​m=n,  
where P is an n by n unitary matrix and R is an m by m upper triangular matrix.
P is given as a sequence of Householder transformation matrices
the m-k+1th transformation matrix, Pk, being used to introduce zeros into the kth row of A. Pk has the form
uk= wk ζk 0 zk .  
γk is a scalar for which Reγk=1.0, ζk is a real scalar, wk is a k-1 element vector and zk is an n-m element vector. γk and uk are chosen to annihilate the elements in the kth row of A.
The scalar γk and the vector uk are returned in the kth element of theta and in the kth row of a, such that θk, given by
is in thetak, the elements of wk are in ak1,,akk-1 and the elements of zk are in akm+1,,akn. The elements of R are returned in the upper triangular part of a.


Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford


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 leading m by n part of the array a must contain the matrix to be factorized.

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.
When m=0 then an immediate return is effected.
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: nm.

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.
The m by m upper triangular part of a will contain the upper triangular matrix R, and the m by m strictly lower triangular part of a and the m by n-m rectangular part of a to the right of the upper triangular part will contain details of the factorization as described in Description.
2:     thetam – complex array
thetak contains the scalar θk for the m-k+1th transformation. If Pk=I then thetak=0.0; if
Tk= I 0 0 0 α 0 0 0 I ,  Reα<0.0  
then thetak=α, otherwise thetak contains θk as described in Description and θk is always in the range 1.0,2.0.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,m<0,
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The computed factors R and P satisfy the relation
Ecε A,  
ε is the machine precision (see nag_machine_precision (x02aj)), c is a modest function of m and n, and . denotes the spectral (two) norm.

Further Comments

The approximate number of floating-point operations is given by 8×m23n-m/3.
The first k rows of the unitary matrix PH can be obtained by calling nag_matop_complex_gen_rq_formq (f01rk), which overwrites the k rows of PH on the first k rows of the array a. PH is obtained by the call:
[a, ifail] = f01qk('Separate', m, k, a, theta);


This example obtains the RQ factorization of the 3 by 5 matrix
A= -0.5i 0.4-0.3i 0.4i+0.0 0.3-0.4i 0.3i -0.5-1.5i 0.9-1.3i -0.4-0.4i 0.1-0.7i 0.3-0.3i -1.0-1.0i 0.2-1.4i 1.8i+0.0 0.0i+0.0 -2.4i .  
function f01rj_example

fprintf('f01rj example results\n\n');

a = [  0   - 0.5i,  0.4 - 0.3i,  0.4 + 0i,    0.3 + 0.4i,  0   + 0.3i;
      -0.5 - 1.5i,  0.9 - 1.3i, -0.4 - 0.4i,  0.1 - 0.7i,  0.3 - 0.3i;
      -1   - i,     0.2 - 1.4i,  1.8 + 0i,    0   + 0i,    0   - 2.4i];

[RQ, theta, ifail] = f01rj(a);

disp('RQ Factorization of A');
disp('Vector theta');
disp('Matrix A after factorization (R in left-hand upper triangle');

f01rj example results

RQ Factorization of A
Vector theta
   1.0387 + 0.1006i   1.1810 - 0.3809i   1.2244 + 0.0000i

Matrix A after factorization (R in left-hand upper triangle
   0.7878 + 0.0000i  -0.2549 - 0.4006i  -0.2774 - 0.2774i  -0.2850 + 0.5586i   0.1154 + 0.7031i
   0.0396 + 0.5222i  -2.1122 + 0.0000i  -1.1094 - 0.5547i   0.1283 + 0.2317i   0.0790 - 0.0361i
  -0.2265 + 0.2265i   0.0453 + 0.3171i  -3.6056 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.5436i

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

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