# cgehrd.f

Section: LAPACK (3)
Updated: Tue Nov 14 2017
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cgehrd.f

## SYNOPSIS

### Functions/Subroutines

subroutine cgehrd (N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
CGEHRD

## Function/Subroutine Documentation

### subroutine cgehrd (integer N, integer ILO, integer IHI, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TAU, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CGEHRD

Purpose:

``` CGEHRD reduces a complex general matrix A to upper Hessenberg form H by
an unitary similarity transformation:  Q**H * A * Q = H .
```

Parameters:

N

```          N is INTEGER
The order of the matrix A.  N >= 0.
```

ILO

```          ILO is INTEGER
```

IHI

```          IHI is 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.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
```

A

```          A is 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

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).
```

TAU

```          TAU is COMPLEX 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

```          WORK is COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
```

LWORK

```          LWORK is INTEGER
The length of the array WORK.  LWORK >= max(1,N).
For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

December 2016

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).

This file is a slight modification of LAPACK-3.0's DGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See DLAHR2.)
```

Definition at line 169 of file cgehrd.f.

## Author

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