F06TRF (PDF version)
F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F06TRF

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

F06TRF performs a QR or RQ factorization (as a sequence of plane rotations) of a complex upper Hessenberg matrix.

2  Specification

SUBROUTINE F06TRF ( SIDE, N, K1, K2, C, S, A, LDA)
INTEGER  N, K1, K2, LDA
REAL (KIND=nag_wp)  S(*)
COMPLEX (KIND=nag_wp)  C(K2), A(LDA,*)
CHARACTER(1)  SIDE

3  Description

F06TRF transforms an n by 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 = diag1,,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 by 2 plane rotation part of Pk has the form
c-k sk -sk ck
with sk real.

4  References

None.

5  Parameters

1:     SIDE – CHARACTER(1)Input
On entry: specifies whether H is operated on from the left or the right.
SIDE='L'
H is pre-multiplied from the left.
SIDE='R'
H is post-multiplied from the right.
Constraint: SIDE='L' or 'R'.
2:     N – INTEGERInput
On entry: n, the order of the matrix H.
Constraint: N0.
3:     K1 – INTEGERInput
4:     K2 – INTEGERInput
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) arrayOutput
On exit: Ck holds ck, the cosine of the rotation Pk, for k=k1,,k2-1; Ck2 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) arrayInput/Output
Note: the dimension of the array S must be at least K2-K1.
On entry: the nonzero subdiagonal elements of H: Sk must hold hk+1,k, for k=k1,,k2-1.
On exit: Sk holds sk, the sine of the rotation Pk, for k=k1,,k2-1.
7:     A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least N.
On entry: the upper triangular part of the n by 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 – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F06TRF is called.
Constraint: LDA max1,N .

6  Error Indicators and Warnings

None.

7  Accuracy

Not applicable.

8  Further Comments

None.

9  Example

None.

F06TRF (PDF version)
F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

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