nag_dgehrd (f08nec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgehrd (f08nec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgehrd (f08nec) reduces a real general matrix to Hessenberg form.

2  Specification

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

3  Description

nag_dgehrd (f08nec) reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: A=QHQT.
The matrix Q is not formed explicitly, but is represented as a product of elementary reflectors (see the f08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Section 8).
The function can take advantage of a previous call to nag_dgebal (f08nhc), which may produce a matrix with the structure:
A11 A12 A13 A22 A23 A33
where A11 and A33 are upper triangular. If so, only the central diagonal block A22, in rows and columns ilo to ihi, needs to be reduced to Hessenberg form (the blocks A12 and A23 will also be affected by the reduction). Therefore the values of ilo and ihi determined by nag_dgebal (f08nhc) can be supplied to the function directly. If nag_dgebal (f08nhc) has not previously been called however, then ilo must be set to 1 and ihi to n.

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 A.
Constraint: n0.
3:     iloIntegerInput
4:     ihiIntegerInput
On entry: if A has been output by nag_dgebal (f08nhc), then ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to 1 and ihi to n.
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.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the n by n general matrix A.
On exit: a is overwritten by the upper Hessenberg matrix H and details of the orthogonal matrix Q.
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]doubleOutput
Note: the dimension, dim, of the array tau must be at least max1,n-1.
On exit: further details of the orthogonal matrix Q.
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 Hessenberg matrix H is exactly similar to a nearby matrix A+E, where
E2 c n ε A2 ,
cn is a modestly increasing function of n, and ε is the machine precision.
The elements of H themselves may be sensitive to small perturbations in A or to rounding errors in the computation, but this does not affect the stability of the eigenvalues, eigenvectors or Schur factorization.

8  Further Comments

The total number of floating point operations is approximately 23q22q+3n, where q=ihi-ilo; if ilo=1 and ihi=n, the number is approximately 103n3.
To form the orthogonal matrix Q nag_dgehrd (f08nec) may be followed by a call to nag_dorghr (f08nfc):
nag_dorghr(order,n,ilo,ihi,&a,pda,tau,&fail)
To apply Q to an m by n real matrix C nag_dgehrd (f08nec) may be followed by a call to nag_dormhr (f08ngc). For example,
nag_dormhr(order,Nag_LeftSide,Nag_NoTrans,m,n,ilo,ihi,&a,pda,
  tau,&c,pdc,&fail)
forms the matrix product QC.
The complex analogue of this function is nag_zgehrd (f08nsc).

9  Example

This example computes the upper Hessenberg form 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 .

9.1  Program Text

Program Text (f08nece.c)

9.2  Program Data

Program Data (f08nece.d)

9.3  Program Results

Program Results (f08nece.r)


nag_dgehrd (f08nec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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