F06TPF (PDF version)
F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F06TPF

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
    10  Example

1  Purpose

F06TPF performs a QR factorization (as a sequence of plane rotations) of a complex upper triangular matrix that has been modified by a rank-1 update.

2  Specification

SUBROUTINE F06TPF ( N, ALPHA, X, INCX, Y, INCY, A, LDA, C, S)
INTEGER  N, INCX, INCY, LDA
REAL (KIND=nag_wp)  C(N-1)
COMPLEX (KIND=nag_wp)  ALPHA, X(*), Y(*), A(LDA,*), S(N)

3  Description

F06TPF performs a QR factorization of an upper triangular matrix which has been modified by a rank-1 update:
αxyT + U=QR  
where U and R are n by n complex upper triangular matrices with real diagonal elements, x and y are n-element complex vectors, α is a complex scalar, and Q is an n by n complex unitary matrix.
Q is formed as the product of two sequences of plane rotations and a unitary diagonal matrix D:
QH = DQn-1 Q2 Q1 P1 P2 Pn-1  
where
The 2 by 2 plane rotation part of Pk or Qk has the form
ck s-k -sk ck  
with ck real. The tangents of the rotations Pk are returned in the array X; the cosines and sines of these rotations can be recovered by calling F06BCF. The cosines and sines of the rotations Qk are returned directly in the arrays C and S.

4  References

None.

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:     Y* – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array Y must be at least max1, 1+N-1 ×INCY .
On entry: the n-element vector y. yi must be stored in Y1+i-1×INCY, for i=1,2,,N.
Intermediate elements of Y are not referenced.
6:     INCY – INTEGERInput
On entry: the increment in the subscripts of Y between successive elements of y.
Constraint: INCY>0.
7:     ALDA* – 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. The imaginary parts of the diagonal elements must be zero.
On exit: the upper triangular matrix R. The imaginary parts of the diagonal elements must be zero.
8:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F06TPF is called.
Constraint: LDA max1,N .
9:     CN-1 – REAL (KIND=nag_wp) arrayOutput
On exit: the cosines of the rotations Qk, for k=1,2,,n-1.
10:   SN – COMPLEX (KIND=nag_wp) arrayOutput
On exit: the sines of the rotations Qk, for k=1,2,,n-1; Sn holds dn, the nth diagonal element of D.

6  Error Indicators and Warnings

None.

7  Accuracy

Not applicable.

8  Parallelism and Performance

F06TPF is not threaded by NAG in any implementation.
F06TPF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

None.

10  Example

None.

F06TPF (PDF version)
F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

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