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# LAPACK-3

## man page of LAPACK-3

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

```NAME
LAPACK-3  - reduces 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 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).

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.

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)     March 2011                       LAPACK-3(3)
```