NAG Library Routine Document

f06qrf (duhqr)

1
Purpose

f06qrf performs a QR or RQ factorization (as a sequence of plane rotations) of a real upper Hessenberg matrix.

2
Specification

Fortran Interface
Subroutine f06qrf ( side, n, k1, k2, c, s, a, lda)
Integer, Intent (In):: n, k1, k2, lda
Real (Kind=nag_wp), Intent (Inout):: c(k2-1), s(k2-1), a(lda,*)
Character (1), Intent (In):: side
C Header Interface
#include <nagmk26.h>
void  f06qrf_ (const char *side, const Integer *n, const Integer *k1, const Integer *k2, double c[], double s[], double a[], const Integer *lda, const Charlen length_side)

3
Description

f06qrf transforms an n by n real upper Hessenberg matrix H to upper triangular form R by applying an orthogonal matrix P from the left or the right. H is assumed to have nonzero subdiagonal elements hk+1,k, for k=k1,,k2-1, only. 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 = P k2-1 P k1+1 P k1 .
If side='R', the rotations are applied from the right:
HPT=R ,  
where P = Pk1 Pk1+1 Pk2-1 .
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
ck sk -sk ck .  

4
References

None.

5
Arguments

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 values k1 and k2.
If k1<1 or k2k1 or k2>n, an immediate return is effected.
5:     ck2-1 – Real (Kind=nag_wp) arrayOutput
On exit: ck holds ck, the cosine of the rotation Pk, for k=k1,,k2-1.
6:     sk2-1 – Real (Kind=nag_wp) arrayInput/Output
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:     alda* – Real (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.
8:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06qrf is called.
Constraint: lda max1,n .

6
Error Indicators and Warnings

None.

7
Accuracy

Not applicable.

8
Parallelism and Performance

f06qrf is not threaded in any implementation.

9
Further Comments

None.

10
Example

None.