shsein(3S)

NAME
     SHSEIN - use inverse iteration to find specified right and/or left
     eigenvectors of a real upper Hessenberg matrix H

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

         CHARACTER      EIGSRC, INITV, SIDE

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

         LOGICAL        SELECT( * )

         INTEGER        IFAILL( * ), IFAILR( * )

         REAL           H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ),
                        WORK( * ), WR( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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) 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 (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) 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) 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

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

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

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

     WR      (input/output) REAL array, dimension (N)
             WI      (input) 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) 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) INTEGER
             The leading dimension of the array VL.  LDVL >= max(1,N) if SIDE
             = 'L' or 'B'; LDVL >= 1 otherwise.
     VR      (input/output) 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) INTEGER
             The leading dimension of the array VR.  LDVR >= max(1,N) if SIDE
             = 'R' or 'B'; LDVR >= 1 otherwise.

     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; each selected real eigenvector occupies
             one column and each selected complex eigenvector occupies two
             columns.

     WORK    (workspace) REAL array, dimension ((N+2)*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 (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) 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) 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|.


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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