nag_dorghr (f08nfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dorghr (f08nfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dorghr (Nag_OrderType order, Integer n, Integer ilo, Integer ihi, double a[], Integer pda, const double tau[], NagError *fail)

3  Description

nag_dorghr (f08nfc) is intended to be used following a call to nag_dgehrd (f08nec), which reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: A=QHQT. nag_dgehrd (f08nec) represents the matrix Q as a product of ihi-ilo elementary reflectors. Here ilo and ihi are values determined by nag_dgebal (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:     orderNag_OrderTypeInput
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 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     nIntegerInput
On entry: n, the order of the matrix Q.
Constraint: n0.
3:     iloIntegerInput
4:     ihiIntegerInput
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to nag_dgehrd (f08nec).
  • if n>0, 1 ilo ihi n ;
  • if n=0, ilo=1 and ihi=0.
5:     a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgehrd (f08nec).
On exit: the n by n orthogonal matrix Q.
If order=Nag_ColMajor, the i,jth element of the matrix is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, the i,jth element of the matrix is stored in a[i-1×pda+j-1].
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
7:     tau[dim]const doubleInput
Note: the dimension, dim, of the array tau must be at least max1,n-1.
On entry: further details of the elementary reflectors, as returned by nag_dgehrd (f08nec).
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
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: pdamax1,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.

7  Accuracy

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

8  Parallelism and Performance

nag_dorghr (f08nfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dorghr (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 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 nag_zunghr (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 nag_dgehrd (f08nec). The program then calls nag_dorghr (f08nfc) to form Q, and passes this matrix to nag_dhseqr (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)

nag_dorghr (f08nfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014