zgehrd(3P)

NAME

zgehrd - reduce a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation

SYNOPSIS

SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )

INTEGER IHI, ILO, INFO, LDA, LWORK, N

COMPLEX∗16 A( LDA, ∗ ), TAU( ∗ ), WORK( LWORK )

#include <sunperf.h>

void zgehrd(int n, int ilo, int ihi, doublecomplex ∗za, int lda, doublecomplex ∗tau, int ∗info) ;

PURPOSE

ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q’ ∗ A ∗ Q = H .

ARGUMENTS

N (input) INTEGER
The order of the matrix A. N >= 0.

ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to ZGEBAL; otherwise they should be set to 1 and N respectively. See Further Details.

A (input/output) COMPLEX∗16 array, dimension (LDA,N)
On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N).

TAU (output) COMPLEX∗16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.

WORK (workspace/output) COMPLEX∗16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,N). For optimum performance LWORK >= N∗NB, where NB is the optimal blocksize.

INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors

   Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

   H(i) = I - tau ∗ v ∗ v’

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

on entry,                        on exit,

( a   a   a   a   a   a   a )    ( a   a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      a   h   h   h   h   a )
(     a   a   a   a   a   a )    (      h   h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2 h   h   h   h   h )
(     a   a   a   a   a   a )    (      v2 v3 h   h   h   h )
(     a   a   a   a   a   a )    (      v2 v3 v4 h   h   h )
(                         a )    (                          a )

where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).