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



NAME
       SHSEIN

SYNOPSIS
       SUBROUTINE SHSEIN (SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, VL,
           LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO)



PURPOSE
            SHSEIN uses inverse iteration to find specified right and/or left
            eigenvectors of a real 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 (WR,WI):
                     = 'Q': the eigenvalues were found using SHSEQR; 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 SHSEIN 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, SHSEIN 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/output)
                     SELECT is LOGICAL array, dimension (N)
                     Specifies the eigenvectors to be computed. To select the
                     real eigenvector corresponding to a real eigenvalue WR(j),
                     SELECT(j) must be set to .TRUE.. To select the complex
                     eigenvector corresponding to a complex eigenvalue
                     (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
                     either SELECT(j) or SELECT(j+1) or both must be set to
                     .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
                     .FALSE..

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

           H         (input)
                     H is REAL 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).

           WR        (input/output)
                     WR is REAL array, dimension (N)

           WI        (input)
                     WI is REAL array, dimension (N)

                     On entry, the real and imaginary parts of the eigenvalues of
                     H; a complex conjugate pair of eigenvalues must be stored in
                     consecutive elements of WR and WI.
                     On exit, WR may have been altered since close eigenvalues
                     are perturbed slightly in searching for independent
                     eigenvectors.

           VL        (input/output)
                     VL is REAL 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(s) 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. A
                     complex eigenvector corresponding to a complex eigenvalue is
                     stored in two consecutive columns, the first holding the real
                     part and the second the imaginary part.
                     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 REAL 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(s) 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. A
                     complex eigenvector corresponding to a complex eigenvalue is
                     stored in two consecutive columns, the first holding the real
                     part and the second the imaginary part.
                     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; each selected real eigenvector
                     occupies one column and each selected complex eigenvector
                     occupies two columns.

           WORK      (output)
                     WORK is REAL array, dimension ((N+2)*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 the i-th and (i+1)th
                     columns of VL hold a complex eigenvector, then IFAILL(i) and
                     IFAILL(i+1) are set to the same value.
                     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 the i-th and (i+1)th
                     columns of VR hold a complex eigenvector, then IFAILR(i) and
                     IFAILR(i+1) are set to the same value.
                     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                      SHSEIN(3)