NAG FL Interface
f06tpf (zutr1)

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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

Fortran Interface
Subroutine f06tpf ( n, alpha, x, incx, y, incy, a, lda, c, s)
Integer, Intent (In) :: n, incx, incy, lda
Real (Kind=nag_wp), Intent (Out) :: c(n-1)
Complex (Kind=nag_wp), Intent (In) :: alpha, y(*)
Complex (Kind=nag_wp), Intent (Inout) :: x(*), a(lda,*)
Complex (Kind=nag_wp), Intent (Out) :: s(n)
C Header Interface
#include <nag.h>
void  f06tpf_ (const Integer *n, const Complex *alpha, Complex x[], const Integer *incx, const Complex y[], const Integer *incy, Complex a[], const Integer *lda, double c[], Complex s[])
The routine may be called by the names f06tpf or nagf_blas_zutr1.

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×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×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 2×2 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 Arguments

1: n Integer Input
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) array Input/Output
Note: the dimension of the array x must be at least max(1, 1+(n-1) ×incx ) .
On entry: the n-element vector x. xi must be stored in x(1+(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 Integer Input
On entry: the increment in the subscripts of x between successive elements of x.
Constraint: incx>0.
5: y(*) Complex (Kind=nag_wp) array Input
Note: the dimension of the array y must be at least max(1, 1+(n-1) ×incy ) .
On entry: the n-element vector y. yi must be stored in y(1+(i-1)×incy), for i=1,2,,n.
Intermediate elements of y are not referenced.
6: incy Integer Input
On entry: the increment in the subscripts of y between successive elements of y.
Constraint: incy>0.
7: a(lda,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least n.
On entry: the n×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 Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f06tpf is called.
Constraint: lda max(1,n) .
9: c(n-1) Real (Kind=nag_wp) array Output
On exit: the cosines of the rotations Qk, for k=1,2,,n-1.
10: s(n) Complex (Kind=nag_wp) array Output
On exit: the sines of the rotations Qk, for k=1,2,,n-1; s(n) holds dn, the nth diagonal element of D.

6 Error Indicators and Warnings

None.

7 Accuracy

Not applicable.

8 Parallelism and Performance

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.