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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_matop_complex_trapez_rq (f01rg)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_matop_complex_trapez_rq (f01rg) reduces the complex m by n (mn) upper trapezoidal matrix A to upper triangular form by means of unitary transformations.


[a, theta, ifail] = f01rg(a, 'm', m, 'n', n)
[a, theta, ifail] = nag_matop_complex_trapez_rq(a, 'm', m, 'n', n)


The m by nmn upper trapezoidal matrix A given by
A= U X ,  
where U is an m by m upper triangular matrix, is factorized as
A= R 0 PH,  
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
Pk= I 0 0 Tk ,  
Tk=I-γkukukH, uk= ζk 0 zk cr ,  
γk is a scalar for which Reγk=1.0, ζk is a real scalar and zk is an n-m element vector. γk, ζk and zk are chosen to annihilate the elements of the kth row of X and to make the diagonal elements of R real.
The scalar γk and the vector uk are returned in the kth element of the array theta and in the kth row of a, such that θk, given by
is in thetak and the elements of zk are in akm+1,,akn. The elements of R are returned in the upper triangular part of a.
For further information on this factorization and its use see Section 6.5 of Golub and Van Loan (1996).


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 upper trapezoidal 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 n-m upper trapezoidal part of a 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 Tk=I then thetak=0.0; if
Tk= α 0 0 I ,  Reα<0.0  
then thetak=α, otherwise thetak contains θk as described in Description and Reθk 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
R 0 PH=A+E,  
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×m2n-m.


This example reduces the 3 by 4 matrix
2.4 0.8+0.8i -1.4+0.6i 3.0-1.0i 0.0 1.6i+0.0 0.8+0.3i 0.4+0.5i 0.0 0.0i+0.0 1.0i+0.0 2.0-1.0i  
to upper triangular form.
function f01rg_example

fprintf('f01rg example results\n\n');

a = [ 2.4,     0.8 + 0.8i, -1.4 + 0.6i,  3   - i;
      0 + 0i,  1.6 + 0i,    0.8 + 0.3i,  0.4 + 0.5i;
      0 + 0i,  0   + 0i,    1   + 0i,    2   - i];

[RQ, theta, ifail] = f01rg(a);

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

f01rg example results

RQ Factorization of A
Vector theta
   1.2924 + 0.0000i   1.3861 + 0.0000i   1.1867 + 0.0000i

Matrix A after factorization (R in left-hand upper triangle
  -3.5808 + 0.0000i   0.2533 - 0.9059i  -2.2862 - 0.6532i   0.5120 + 0.2601i
   0.0000 + 0.0000i  -1.7369 + 0.0000i  -0.4491 - 0.6940i  -0.2544 - 0.1187i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -2.4495 + 0.0000i   0.6880 + 0.3440i

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Chapter Introduction
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