nag_dorghr (f08nfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C 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 3.2.1.3 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).
Constraints:
  • 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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
NE_INT_3
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.
NE_INTERNAL_ERROR
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  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).

9  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.

9.1  Program Text

Program Text (f08nfce.c)

9.2  Program Data

Program Data (f08nfce.d)

9.3  Program Results

Program Results (f08nfce.r)


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

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