# Ubuntu Feisty 7.04 manual page repository

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### NAME

```        ZGEHD2  -  reduce a complex general matrix A to upper Hessenberg form H
by a unitary similarity transformation
```

### SYNOPSIS

```        SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )

INTEGER        IHI, ILO, INFO, LDA, N

COMPLEX*16     A( LDA, * ), TAU( * ), WORK( * )
```

### PURPOSE

```        ZGEHD2 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 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*16 array, dimension (N-1)
The scalar factors of the elementary  reflectors  (see  Further
Details).

WORK    (workspace) COMPLEX*16 array, dimension (N)

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
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 modi‐
fied element of the upper Hessenberg matrix H, and vi denotes  an  ele‐
ment of the vector defining H(i).

```
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