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# zgehrd

```
NAME
ZGEHRD - reduce a complex general matrix A to upper Hessen-
berg 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 )

PURPOSE
ZGEHRD reduces a complex general matrix A to upper Hessen-
berg 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 pre-
vious call to ZGEBAL; otherwise they should be set
to 1 and N respectively. See Further Details. If N >
0,

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 subdiago-
nal of A are overwritten with the upper Hessenberg
matrix H, and the elements below the first subdiago-
nal, with the array TAU, represent the unitary
matrix Q as a product of elementary reflectors. See
Further Details.  LDA     (input) INTEGER The lead-
ing 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) 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) ele-
mentary 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).
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