F06TQF (PDF version)
F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy
    9  Example

1  Purpose

F06TQF performs a QR factorization (as a sequence of plane rotations) of a complex upper triangular matrix that has been augmented by a full row.

2  Specification

REAL (KIND=nag_wp)  C(N)
COMPLEX (KIND=nag_wp)  ALPHA, X(*), A(LDA,*), S(N)

3  Description

F06TQF performs the factorization
U αxT =Q R 0
where U and R are n by n complex upper triangular matrices, x is an n-element complex vector, α is a complex scalar, and Q is a complex unitary matrix. If U has real diagonal elements, then so does R.
Q is formed as a sequence of plane rotations
QH = Qn Q2 Q1
where Qk is a rotation in the k,n+1 plane, chosen to annihilate xk.
The 2 by 2 plane rotation part of Qk has the form
ck s-k -sk ck
with ck real.

4  References


5  Parameters

1:     N – INTEGERInput
On entry: n, the order of the matrices U and R.
Constraint: N0.
2:     ALPHA – COMPLEX (KIND=nag_wp)Input
On entry: the scalar α.
3:     X(*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array X must be at least max1, 1+N-1 ×INCX .
On entry: the n-element vector x. xi must be stored in X1+i-1×INCX, for i=1,2,,N.
Intermediate elements of X are not referenced.
On exit: the referenced elements are overwritten by details of the sequence of plane rotations.
4:     INCX – INTEGERInput
On entry: the increment in the subscripts of X between successive elements of x.
Constraint: INCX>0.
5:     A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least N.
On entry: the n by n upper triangular matrix U.
On exit: the upper triangular matrix R.
6:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F06TQF is called.
Constraint: LDA max1,N .
7:     C(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the values ck, the cosines of the rotations Qk, for k=1,2,,n.
8:     S(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: the values sk, the sines of the rotations Qk, for k=1,2,,n.

6  Error Indicators and Warnings


7  Accuracy

Not applicable.

8  Further Comments


9  Example


F06TQF (PDF version)
F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012