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SHSEQR(l)					)					SHSEQR(l)

NAME
       SHSEQR  - compute the eigenvalues of a real upper Hessenberg matrix H and, optionally, the
       matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-tri-
       angular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors

SYNOPSIS
       SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO )

	   CHARACTER	  COMPZ, JOB

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

	   REAL 	  H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z( LDZ, * )

PURPOSE
       SHSEQR  computes  the eigenvalues of a real upper Hessenberg matrix H and, optionally, the
       matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-tri-
       angular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors.  Option-
       ally Z may be postmultiplied into an input orthogonal 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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

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 orthogonal matrix Q on entry, and the prod-
	       uct Q*Z is returned.

       N       (input) INTEGER
	       The order of the matrix H.  N >= 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 SGEBAL, and then passed to SGEHRD when the matrix output by SGEBAL  is  reduced
	       to  Hessenberg  form. Otherwise ILO and IHI should be set to 1 and N respectively.
	       1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

       H       (input/output) REAL array, dimension (LDH,N)
	       On entry, the upper Hessenberg matrix H.  On exit, if JOB = 'S',  H  contains  the
	       upper  quasi-triangular	matrix	T  from the Schur decomposition (the Schur form);
	       2-by-2 diagonal blocks (corresponding to complex conjugate pairs  of  eigenvalues)
	       are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
	       If JOB = 'E', the contents of H are unspecified on exit.

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

       WR      (output) REAL array, dimension (N)
	       WI      (output) REAL array, dimension (N) The real and imaginary  parts,  respec-
	       tively,	of the computed eigenvalues. If two eigenvalues are computed as a complex
	       conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th
	       and  (i+1)th,  with  WI(i)  > 0 and WI(i+1) < 0. If JOB = 'S', the eigenvalues are
	       stored in the same order as on the diagonal of the Schur form returned in H,  with
	       WR(i)  =  H(i,i)  and,  if  H(i:i+1,i:i+1)  is  a  2-by-2  diagonal block, WI(i) =
	       sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

       Z       (input/output) REAL array, dimension (LDZ,N)
	       If COMPZ = 'N': Z is not referenced.
	       If COMPZ = 'I': on entry, Z need not be set, and on exit, Z contains the  orthogo-
	       nal  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 Z contains Q*Z.  Normally Q is the orthogo-
	       nal matrix generated by SORGHR after the call to SGEHRD which formed  the  Hessen-
	       berg matrix H.

       LDZ     (input) INTEGER
	       The  leading dimension of the array Z.  LDZ >= max(1,N) if COMPZ = 'I' or 'V'; LDZ
	       >= 1 otherwise.

       WORK    (workspace/output) REAL array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The dimension of the array WORK.  LWORK >= max(1,N).

	       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    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value
	       > 0:  if INFO = i, SHSEQR failed to compute all of the eigenvalues in a	total  of
	       30*(IHI-ILO+1)  iterations;  elements 1:ilo-1 and i+1:n of WR and WI contain those
	       eigenvalues which have been successfully computed.

LAPACK version 3.0			   15 June 2000 				SHSEQR(l)
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