CHSEIN  -  use  inverse  iteration to find specified right
       and/or left eigenvectors of  a  complex  upper  Hessenberg
       matrix H


SYNOPSIS

       SUBROUTINE CHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH,
                          W, VL, LDVL, VR,  LDVR,  MM,  M,  WORK,
                          RWORK, IFAILL, IFAILR, INFO )

           CHARACTER      EIGSRC, INITV, SIDE

           INTEGER        INFO, LDH, LDVL, LDVR, M, MM, N

           LOGICAL        SELECT( * )

           INTEGER        IFAILL( * ), IFAILR( * )

           REAL           RWORK( * )

           COMPLEX        H( LDH, * ), VL( LDVL, * ), VR( LDVR, *
                          ), W( * ), WORK( * )


PURPOSE

       CHSEIN uses inverse  iteration  to  find  specified  right
       and/or  left  eigenvectors  of  a complex upper Hessenberg
       matrix H.  The right eigenvector x and the left  eigenvec­
       tor y of the matrix H corresponding to an eigenvalue w are
       defined by:

                    H * x = w * x,     y**h * H = w * y**h

       where y**h denotes the conjugate transpose of  the  vector
       y.


ARGUMENTS

       SIDE    (input) CHARACTER*1
               = 'R': compute right eigenvectors only;
               = 'L': compute left eigenvectors only;
               = 'B': compute both right and left eigenvectors.

       EIGSRC  (input) CHARACTER*1
               Specifies the source of eigenvalues supplied in W:
               = 'Q': the eigenvalues were  found  using  CHSEQR;
               thus,  if  H has zero subdiagonal elements, and so
               is block-triangular, then the j-th eigenvalue  can
               be  assumed  to be an eigenvalue of the block con­
               taining the j-th row/column.  This property allows
               CHSEIN  to  perform  inverse iteration on just one
               diagonal block.  = 'N': no assumptions are made on
               the  correspondence between eigenvalues and diago­
               nal blocks.  In this case, CHSEIN must always per­
               form inverse iteration using the whole matrix H.

               = 'N': no initial vectors are supplied;
               = 'U': user-supplied initial vectors are stored in
               the arrays VL and/or VR.

       SELECT  (input) LOGICAL array, dimension (N)
               Specifies the  eigenvectors  to  be  computed.  To
               select the eigenvector corresponding to the eigen­
               value W(j), SELECT(j) must be set to .TRUE..

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

       H       (input) COMPLEX array, dimension (LDH,N)
               The upper Hessenberg matrix H.

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

       W       (input/output) COMPLEX array, dimension (N)
               On entry, the eigenvalues of H.  On exit, the real
               parts of W  may  have  been  altered  since  close
               eigenvalues  are  perturbed  slightly in searching
               for independent eigenvectors.

       VL      (input/output) COMPLEX array, dimension (LDVL,MM)
               On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL
               must  contain  starting  vectors  for  the inverse
               iteration for the left eigenvectors; the  starting
               vector  for  each  eigenvector must be in the same
               column in which the eigenvector  will  be  stored.
               On  exit, if SIDE = 'L' or 'B', the left eigenvec­
               tors specified by SELECT will be  stored  consecu­
               tively  in the columns of VL, in the same order as
               their eigenvalues.  If SIDE = 'R', VL is not  ref­
               erenced.

       LDVL    (input) INTEGER
               The  leading  dimension  of the array VL.  LDVL >=
               max(1,N) if SIDE = 'L' or 'B'; LDVL  >=  1  other­
               wise.

       VR      (input/output) COMPLEX array, dimension (LDVR,MM)
               On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR
               must contain  starting  vectors  for  the  inverse
               iteration for the right eigenvectors; the starting
               vector for each eigenvector must be  in  the  same
               column  in  which  the eigenvector will be stored.
               On exit, if SIDE = 'R' or 'B', the right eigenvec­
               tors  specified  by SELECT will be stored consecu­
               tively in the columns of VR, in the same order  as
               their  eigenvalues.  If SIDE = 'L', VR is not ref­
               erenced.

               The leading dimension of the array  VR.   LDVR  >=
               max(1,N)  if  SIDE  = 'R' or 'B'; LDVR >= 1 other­
               wise.

       MM      (input) INTEGER
               The number of columns in the arrays VL and/or  VR.
               MM >= M.

       M       (output) INTEGER
               The  number  of columns in the arrays VL and/or VR
               required to store the eigenvectors (=  the  number
               of .TRUE. elements in SELECT).

       WORK    (workspace) COMPLEX array, dimension (N*N)

       RWORK   (workspace) REAL array, dimension (N)

       IFAILL  (output) INTEGER array, dimension (MM)
               If  SIDE  =  'L'  or 'B', IFAILL(i) = j > 0 if the
               left eigenvector in the i-th column of VL  (corre­
               sponding  to  the  eigenvalue w(j)) failed to con­
               verge; IFAILL(i) = 0 if the eigenvector  converged
               satisfactorily.  If SIDE = 'R', IFAILL is not ref­
               erenced.

       IFAILR  (output) INTEGER array, dimension (MM)
               If SIDE = 'R' or 'B', IFAILR(i) = j  >  0  if  the
               right eigenvector in the i-th column of VR (corre­
               sponding to the eigenvalue w(j))  failed  to  con­
               verge;  IFAILR(i) = 0 if the eigenvector converged
               satisfactorily.  If SIDE = 'L', IFAILR is not ref­
               erenced.

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value
               > 0:  if INFO = i, i is the number of eigenvectors
               which  failed  to  converge; see IFAILL and IFAILR
               for further details.


FURTHER DETAILS

       Each eigenvector is normalized  so  that  the  element  of
       largest magnitude has magnitude 1; here the magnitude of a
       complex number (x,y) is taken to be |x|+|y|.


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