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CGEHD2

Section: LAPACK routine (version 3.3.1) (1)
Updated: April 2011
Index

NAME

LAPACK-3 - reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation

SYNOPSIS

SUBROUTINE CGEHD2(: N, ILO, IHI, A, LDA, TAU, WORK, INFO )
: INTEGERIHI, ILO, INFO, LDA, N
: COMPLEXA( LDA, * ), TAU( * ), WORK( * )

PURPOSE

CGEHD2 reduces a complex general matrix A to upper Hessenberg form Hby a unitary similarity transformation: Q**H * 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 CGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
A (input/output) COMPLEX 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 array, dimension (N-1): The scalar factors of the elementary reflectors (see Further
Details).
WORK (workspace) COMPLEX array, dimension (N)
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**H

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).

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