NAG FL Interface
f08nff (dorghr)

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1 Purpose

f08nff generates the real orthogonal matrix Q which was determined by f08nef when reducing a real general matrix A to Hessenberg form.

2 Specification

Fortran Interface
Subroutine f08nff ( n, ilo, ihi, a, lda, tau, work, lwork, info)
Integer, Intent (In) :: n, ilo, ihi, lda, lwork
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (In) :: tau(*)
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*)
Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
C Header Interface
#include <nag.h>
void  f08nff_ (const Integer *n, const Integer *ilo, const Integer *ihi, double a[], const Integer *lda, const double tau[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08nff, nagf_lapackeig_dorghr or its LAPACK name dorghr.

3 Description

f08nff is intended to be used following a call to f08nef, which reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: A=QHQT. f08nef represents the matrix Q as a product of ihi-ilo elementary reflectors. Here ilo and ihi are values determined by f08nhf when balancing the matrix; if the matrix has not been balanced, ilo=1 and ihi=n.
This routine may be used to generate Q explicitly as a square matrix. Q has the structure:
Q = ( I 0 0 0 Q22 0 0 0 I )  
where Q22 occupies rows and columns ilo to ihi.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix Q.
Constraint: n0.
2: ilo Integer Input
3: ihi Integer Input
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to f08nef.
  • if n>0, 1 ilo ihi n ;
  • if n=0, ilo=1 and ihi=0.
4: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: details of the vectors which define the elementary reflectors, as returned by f08nef.
On exit: the n×n orthogonal matrix Q.
5: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08nff is called.
Constraint: ldamax(1,n).
6: tau(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array tau must be at least max(1,n-1).
On entry: further details of the elementary reflectors, as returned by f08nef.
7: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
8: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08nff is called, unless lwork=−1, in which case a workspace query is assumed and the routine only calculates the optimal dimension of work (using the formula given below).
Suggested value: for optimal performance lwork should be at least (ihi-ilo)×nb, where nb is the block size.
Constraint: lworkmax(1,ihi-ilo) or lwork=−1.
9: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that
E2 = O(ε) ,  
where ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08nff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nff 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

The total number of floating-point operations is approximately 43q3, where q=ihi-ilo.
The complex analogue of this routine is f08ntf.

10 Example

This example computes the Schur factorization of the matrix A, where
A = ( 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 ) .  
Here A is general and must first be reduced to Hessenberg form by f08nef. The program then calls f08nff to form Q, and passes this matrix to f08pef which computes the Schur factorization of A.

10.1 Program Text

Program Text (f08nffe.f90)

10.2 Program Data

Program Data (f08nffe.d)

10.3 Program Results

Program Results (f08nffe.r)