cgehd2 man page on Scientific

```CGEHD2(1)		 LAPACK routine (version 3.2)		     CGEHD2(1)

NAME
CGEHD2  - 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 )

INTEGER	  IHI, ILO, INFO, LDA, N

COMPLEX	  A( LDA, * ), TAU( * ), WORK( * )

PURPOSE
CGEHD2 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 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 overwrit‐
ten with the upper Hessenberg matrix H, and the elements	 below
the  first  subdiagonal, with the array TAU, represent the uni‐
tary matrix Q as a product of elementary reflectors.  See  Fur‐
ther 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'
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).

LAPACK routine (version 3.2)	 November 2008			     CGEHD2(1)
```
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