chseqr man page on Scientific

```CHSEQR(1)	      LAPACK driver routine (version 3.2)	     CHSEQR(1)

NAME
CHSEQR  - CHSEQR compute the eigenvalues of a Hessenberg matrix H  and,
optionally, the matrices T and Z from the Schur decomposition  H = Z  T
Z**H, where T is an upper triangular matrix (the	 Schur form), and Z is
the unitary matrix of Schur vectors

SYNOPSIS
SUBROUTINE CHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W,  Z,  LDZ,	 WORK,
LWORK, INFO )

INTEGER	  IHI, ILO, INFO, LDH, LDZ, LWORK, N

CHARACTER	  COMPZ, JOB

COMPLEX	  H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE
CHSEQR computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.
Optionally Z may be postmultiplied into an input unitary
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.

ARGUMENTS
JOB   (input) CHARACTER*1
= 'E':  compute eigenvalues only;
=	'S':  compute eigenvalues and the Schur form T.	 COMPZ (input)
CHARACTER*1
= 'N':  no Schur vectors are computed;
= 'I':  Z is initialized to the unit matrix and the matrix	 Z  of
Schur vectors of H is returned; = 'V':  Z must contain an unitary
matrix Q on entry, and the product Q*Z is returned.

N     (input) INTEGER
The order of the matrix H.	 N .GE. 0.

ILO   (input) INTEGER
IHI   (input) INTEGER It is assumed that H is already upper  tri‐
angular  in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
normally set by a previous call to CGEBAL,	 and  then  passed  to
CGEHRD  when the matrix output by CGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set	to  1  and  N  respec‐
tively.   If  N.GT.0,  then 1.LE.ILO.LE.IHI.LE.N.	If N = 0, then
ILO = 1 and IHI = 0.

H     (input/output) COMPLEX array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.  On exit, if  INFO  =  0
and  JOB = 'S', H contains the upper triangular matrix T from the
Schur decomposition (the Schur form). If INFO = 0 and JOB =  'E',
the  contents of H are unspecified on exit.  (The output value of
H when INFO.GT.0 is given under the description of	 INFO  below.)
Unlike earlier versions of CHSEQR, this subroutine may explicitly
H(i,j) = 0 for i.GT.j and j = 1, 2,  ...  ILO-1  or  j  =	IHI+1,
IHI+2, ... N.

LDH   (input) INTEGER
The leading dimension of the array H. LDH .GE. max(1,N).

W	(output) COMPLEX array, dimension (N)
The  computed  eigenvalues.  If JOB = 'S', the eigenvalues are
stored in the same order as on the diagonal of the Schur  form
returned in H, with W(i) = H(i,i).

Z     (input/output) COMPLEX array, dimension (LDZ,N)
If	 COMPZ = 'N', Z is not referenced.  If COMPZ = 'I', on entry Z
need not be set and on exit, if INFO = 0, Z contains the  unitary
matrix  Z	of the Schur vectors of H.  If COMPZ = 'V', on entry Z
must contain an N-by-N matrix Q, which is assumed to be equal  to
the  unit	matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On
exit, if INFO = 0, Z contains Q*Z.	 Normally  Q  is  the  unitary
matrix  generated by CUNGHR after the call to CGEHRD which formed
the Hessenberg matrix H. (The output value of Z when INFO.GT.0 is
given under the description of INFO below.)

LDZ   (input) INTEGER
The  leading dimension of the array Z.  if COMPZ = 'I' or COMPZ =
'V', then LDZ.GE.MAX(1,N).	 Otherwize, LDZ.GE.1.

WORK  (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns an estimate of the  optimal
value  for	 LWORK.	  LWORK	 (input)  INTEGER The dimension of the
array WORK.  LWORK .GE. max(1,N) is sufficient and delivers  very
good  and sometimes optimal performance.  However, LWORK as large
as 11*N may be required for  optimal  performance.	  A  workspace
query is recommended to determine the optimal workspace size.  If
LWORK = -1, then CHSEQR does a workspace query.   In  this	 case,
CHSEQR  checks  the  input	 parameters  and estimates the optimal
workspace size for the given values of N, ILO and IHI.  The esti‐
mate  is  returned in WORK(1).  No error message related to LWORK
is issued by XERBLA.  Neither H nor Z are accessed.

INFO  (output) INTEGER
=	0:  successful exit
value
the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR and WI contain
those  eigenvalues which have been successfully computed.	(Fail‐
ures are rare.)  If INFO .GT. 0 and JOB = 'E', then on exit,  the
remaining	unconverged  eigenvalues  are the eigen- values of the
upper Hessenberg matrix rows and columns ILO through INFO of  the
final,  output  value of H.  If INFO .GT. 0 and JOB   = 'S', then
on exit

(*)  (initial value of H)*U  = U*(final value of H)
where U is a unitary matrix.  The final value of  H is upper  Hes‐
senberg and triangular in rows and columns INFO+1 through IHI.  If
INFO .GT. 0 and COMPZ = 'V', then on exit (final value  of	Z)   =
(initial  value  of	 Z)*U  where  U	 is  the unitary matrix in (*)
(regard- less of the value of JOB.)	 If INFO .GT. 0	 and  COMPZ  =
'I',  then	on exit (final value of Z)  = U where U is the unitary
matrix in (*) (regard- less of the value of JOB.)  If INFO .GT.  0
and COMPZ = 'N', then Z is not accessed.

LAPACK driver routine (version 3November 2008			     CHSEQR(1)
```
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