NAG FL Interface
f06trf (zuhqr)

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1 Purpose

f06trf performs a QR or RQ factorization (as a sequence of plane rotations) of a complex upper Hessenberg matrix.

2 Specification

Fortran Interface
Subroutine f06trf ( side, n, k1, k2, c, s, a, lda)
Integer, Intent (In) :: n, k1, k2, lda
Real (Kind=nag_wp), Intent (Inout) :: s(*)
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*)
Complex (Kind=nag_wp), Intent (Out) :: c(k2)
Character (1), Intent (In) :: side
C Header Interface
#include <nag.h>
void  f06trf_ (const char *side, const Integer *n, const Integer *k1, const Integer *k2, Complex c[], double s[], Complex a[], const Integer *lda, const Charlen length_side)
The routine may be called by the names f06trf or nagf_blas_zuhqr.

3 Description

f06trf transforms an n×n complex upper Hessenberg matrix H to upper triangular form R by applying a unitary matrix P from the left or the right. H is assumed to have real nonzero subdiagonal elements hk+1,k, for k=k1,,k2-1, only; R has real diagonal elements. P is formed as a sequence of plane rotations in planes k1 to k2.
If side='L', the rotations are applied from the left:
PH=R ,  
where P = D P k2-1 P k1+1 P k1 and D = diag(1,,1,dk2,1,,1) with |dk2|=1.
If side='R', the rotations are applied from the right:
HPH=R ,  
where P = D Pk1 Pk1+1 Pk2-1 and D = diag(1,,1,dk1,1,,1) with |dk1|=1.
In either case, Pk is a rotation in the (k,k+1) plane, chosen to annihilate hk+1,k.
The 2×2 plane rotation part of Pk has the form
( c¯k sk -sk ck )  
with sk real.

4 References


5 Arguments

1: side Character(1) Input
On entry: specifies whether H is operated on from the left or the right.
H is pre-multiplied from the left.
H is post-multiplied from the right.
Constraint: side='L' or 'R'.
2: n Integer Input
On entry: n, the order of the matrix H.
Constraint: n0.
3: k1 Integer Input
4: k2 Integer Input
On entry: the dimension of the array c as declared in the (sub)program from which f06trf is called. The values k1 and k2.
If k1<1 or k2k1 or k2>n, an immediate return is effected.
5: c(k2) Complex (Kind=nag_wp) array Output
On exit: c(k) holds ck, the cosine of the rotation Pk, for k=k1,,k2-1; c(k2) holds dk2, the k2th diagonal element of D, if side='L', or dk1, the k1th diagonal element of D, if side='R'.
6: s(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array s must be at least k2-1.
On entry: the nonzero subdiagonal elements of H: s(k) must hold hk+1,k, for k=k1,,k2-1.
On exit: s(k) holds sk, the sine of the rotation Pk, for k=k1,,k2-1.
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 upper triangular part of the n×n upper Hessenberg matrix H.
On exit: the upper triangular matrix R. The imaginary parts of the diagonal elements are set to zero.
8: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f06trf is called.
Constraint: lda max(1,n) .

6 Error Indicators and Warnings


7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f06trf 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


10 Example