NAG CL Interface
f08nfc (dorghr)

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1 Purpose

f08nfc generates the real orthogonal matrix Q which was determined by f08nec when reducing a real general matrix A to Hessenberg form.

2 Specification

#include <nag.h>
void  f08nfc (Nag_OrderType order, Integer n, Integer ilo, Integer ihi, double a[], Integer pda, const double tau[], NagError *fail)
The function may be called by the names: f08nfc, nag_lapackeig_dorghr or nag_dorghr.

3 Description

f08nfc is intended to be used following a call to f08nec, which reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: A=QHQT. f08nec represents the matrix Q as a product of ihi-ilo elementary reflectors. Here ilo and ihi are values determined by f08nhc when balancing the matrix; if the matrix has not been balanced, ilo=1 and ihi=n.
This function 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: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: n Integer Input
On entry: n, the order of the matrix Q.
Constraint: n0.
3: ilo Integer Input
4: ihi Integer Input
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to f08nec.
  • if n>0, 1 ilo ihi n ;
  • if n=0, ilo=1 and ihi=0.
5: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
On entry: details of the vectors which define the elementary reflectors, as returned by f08nec.
On exit: the n×n orthogonal matrix Q.
6: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
7: tau[dim] const double Input
Note: the dimension, dim, of the array tau must be at least max(1,n-1).
On entry: further details of the elementary reflectors, as returned by f08nec.
8: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, n=value, ilo=value and ihi=value.
Constraint: if n>0, 1 ilo ihi n ;
if n=0, ilo=1 and ihi=0.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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.
f08nfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nfc 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 function. 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 function is f08ntc.

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 f08nec. The program then calls f08nfc to form Q, and passes this matrix to f08pec which computes the Schur factorization of A.

10.1 Program Text

Program Text (f08nfce.c)

10.2 Program Data

Program Data (f08nfce.d)

10.3 Program Results

Program Results (f08nfce.r)