CHSEIN(3)      LAPACK routine of NEC Numeric Library Collection      CHSEIN(3)



NAME
       CHSEIN

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



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

           EIGSRC    (input)
                     EIGSRC is 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 containing
                            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 diagonal blocks.  In this
                            case, CHSEIN must always perform inverse iteration
                            using the whole matrix H.

           INITV     (input)
                     INITV is CHARACTER*1
                     = 'N': no initial vectors are supplied;
                     = 'U': user-supplied initial vectors are stored in the arrays
                            VL and/or VR.

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

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

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

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

           W         (input/output)
                     W is 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)
                     VL is 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 eigenvectors
                     specified by SELECT will be stored consecutively in the
                     columns of VL, in the same order as their eigenvalues.
                     If SIDE = 'R', VL is not referenced.

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

           VR        (input/output)
                     VR is 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 eigenvectors
                     specified by SELECT will be stored consecutively in the
                     columns of VR, in the same order as their eigenvalues.
                     If SIDE = 'L', VR is not referenced.

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

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

           M         (output)
                     M is 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      (output)
                     WORK is COMPLEX array, dimension (N*N)

           RWORK     (output)
                     RWORK is REAL array, dimension (N)

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

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

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal 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|.



LAPACK routine                  31 October 2017                      CHSEIN(3)