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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: ZHSEQR compute the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur

```NAME
LAPACK-3  -  ZHSEQR  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 ZHSEQR( 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*16     H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE
ZHSEQR 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 triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to ZGEBAL, and then passed to ZGEHRD
when the matrix output by ZGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively.  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*16 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 ZHSEQR, 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*16 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*16 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 ZUNGHR
after the call to ZGEHRD 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*16 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 ZHSEQR does a workspace query.
In this case, ZHSEQR checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI.  The estimate 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
.LT. 0:  if INFO = -i, the i-th argument had an illegal
value
.GT. 0:  if INFO = i, ZHSEQR failed to compute all of
the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed.  (Failures 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 Hessenberg 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 computational routine (verMarch32011)                      LAPACK-3(3)
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