nag_zgehrd (f08nsc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zgehrd (f08nsc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgehrd (f08nsc) reduces a complex general matrix to Hessenberg form.

2  Specification

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

3  Description

nag_zgehrd (f08nsc) reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: A=QHQH. H has real subdiagonal elements.
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 9).
The function can take advantage of a previous call to nag_zgebal (f08nvc), 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_zgebal (f08nvc) can be supplied to the function directly. If nag_zgebal (f08nvc) 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_zgebal (f08nvc), 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]ComplexInput/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 unitary matrix Q. The subdiagonal elements of H are real.
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]ComplexOutput
Note: the dimension, dim, of the array tau must be at least max1,n-1.
On exit: further details of the unitary 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  Parallelism and Performance

nag_zgehrd (f08nsc) is not threaded by NAG in any implementation.
nag_zgehrd (f08nsc) 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 real floating-point operations is approximately 83q22q+3n, where q=ihi-ilo; if ilo=1 and ihi=n, the number is approximately 403n3.
To form the unitary matrix Q nag_zgehrd (f08nsc) may be followed by a call to nag_zunghr (f08ntc):
nag_zunghr(order,n,ilo,ihi,&a,pda,tau,&fail)
To apply Q to an m by n complex matrix C nag_zgehrd (f08nsc) may be followed by a call to nag_zunmhr (f08nuc). For example,
nag_zunmhr(order,Nag_LeftSide,Nag_NoTrans,m,n,ilo,ihi,&a,pda,
  tau,&c,pdc,&fail)
forms the matrix product QC.
The real analogue of this function is nag_dgehrd (f08nec).

10  Example

This example computes the upper Hessenberg form of the matrix A, where
A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .

10.1  Program Text

Program Text (f08nsce.c)

10.2  Program Data

Program Data (f08nsce.d)

10.3  Program Results

Program Results (f08nsce.r)


nag_zgehrd (f08nsc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

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