hsein(3)

NAME
hsein - hsein: Hessenberg inverse iteration for eigvec

SYNOPSIS

  Functions

subroutine chsein (side, eigsrc, initv, select, n, h, ldh, w, vl, ldvl, vr, ldvr, mm, m, work, rwork, ifaill, ifailr, info)
CHSEIN
subroutine dhsein (side, eigsrc, initv, select, n, h, ldh, wr, wi, vl, ldvl, vr, ldvr, mm, m, work, ifaill, ifailr, info)
DHSEIN
subroutine shsein (side, eigsrc, initv, select, n, h, ldh, wr, wi, vl, ldvl, vr, ldvr, mm, m, work, ifaill, ifailr, info)
SHSEIN
subroutine zhsein (side, eigsrc, initv, select, n, h, ldh, w, vl, ldvl, vr, ldvr, mm, m, work, rwork, ifaill, ifailr, info)
ZHSEIN

Detailed Description

Function Documentation

  subroutine chsein (character side, character eigsrc, character initv, logical, dimension( * ) select, integer n, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) w, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex, dimension( * ) work, real, dimension( * ) rwork, integer, dimension( * ) ifaill, integer, dimension( * ) ifailr, integer info)

CHSEIN

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.
    !>

Parameters
    SIDE

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

    EIGSRC

    !>          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

    !>          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

    !>          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

    !>          N is INTEGER
    !>          The order of the matrix H.  N >= 0.
    !>

    H

    !>          H is COMPLEX array, dimension (LDH,N)
    !>          The upper Hessenberg matrix H.
    !>          If a NaN is detected in H, the routine will return with INFO=-6.
    !>

    LDH

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

    W

    !>          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

    !>          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

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

    VR

    !>          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

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

    MM

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

    M

    !>          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

    !>          WORK is COMPLEX array, dimension (N*N)
    !>

    RWORK

    !>          RWORK is REAL array, dimension (N)
    !>

    IFAILL

    !>          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

    !>          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

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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|.
    !>

  subroutine dhsein (character side, character eigsrc, character initv, logical, dimension( * ) select, integer n, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision, dimension( * ) work, integer, dimension( * ) ifaill, integer, dimension( * ) ifailr, integer info)

DHSEIN

Purpose:

    !>
    !> DHSEIN 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.
    !>

Parameters
    SIDE

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

    EIGSRC

    !>          EIGSRC is CHARACTER*1
    !>          Specifies the source of eigenvalues supplied in (WR,WI):
    !>          = 'Q': the eigenvalues were found using DHSEQR; 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 DHSEIN 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, DHSEIN must always perform inverse iteration
    !>                 using the whole matrix H.
    !>

    INITV

    !>          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

    !>          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

    !>          N is INTEGER
    !>          The order of the matrix H.  N >= 0.
    !>

    H

    !>          H is DOUBLE PRECISION array, dimension (LDH,N)
    !>          The upper Hessenberg matrix H.
    !>          If a NaN is detected in H, the routine will return with INFO=-6.
    !>

    LDH

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

    WR

    !>          WR is DOUBLE PRECISION array, dimension (N)
    !>

    WI

    !>          WI is DOUBLE PRECISION 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

    !>          VL is DOUBLE PRECISION 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

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

    VR

    !>          VR is DOUBLE PRECISION 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

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

    MM

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

    M

    !>          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

    !>          WORK is DOUBLE PRECISION array, dimension ((N+2)*N)
    !>

    IFAILL

    !>          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

    !>          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

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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|.
    !>

  subroutine shsein (character side, character eigsrc, character initv, logical, dimension( * ) select, integer n, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer, dimension( * ) ifaill, integer, dimension( * ) ifailr, integer info)

SHSEIN

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.
    !>

Parameters
    SIDE

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

    EIGSRC

    !>          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

    !>          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

    !>          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

    !>          N is INTEGER
    !>          The order of the matrix H.  N >= 0.
    !>

    H

    !>          H is REAL array, dimension (LDH,N)
    !>          The upper Hessenberg matrix H.
    !>          If a NaN is detected in H, the routine will return with INFO=-6.
    !>

    LDH

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

    WR

    !>          WR is REAL array, dimension (N)
    !>

    WI

    !>          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

    !>          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

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

    VR

    !>          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

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

    MM

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

    M

    !>          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

    !>          WORK is REAL array, dimension ((N+2)*N)
    !>

    IFAILL

    !>          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

    !>          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

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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|.
    !>

  subroutine zhsein (character side, character eigsrc, character initv, logical, dimension( * ) select, integer n, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( * ) w, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer, dimension( * ) ifaill, integer, dimension( * ) ifailr, integer info)

ZHSEIN

Purpose:

    !>
    !> ZHSEIN 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.
    !>

Parameters
    SIDE

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

    EIGSRC

    !>          EIGSRC is CHARACTER*1
    !>          Specifies the source of eigenvalues supplied in W:
    !>          = 'Q': the eigenvalues were found using ZHSEQR; 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 ZHSEIN 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, ZHSEIN must always perform inverse iteration
    !>                 using the whole matrix H.
    !>

    INITV

    !>          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

    !>          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

    !>          N is INTEGER
    !>          The order of the matrix H.  N >= 0.
    !>

    H

    !>          H is COMPLEX*16 array, dimension (LDH,N)
    !>          The upper Hessenberg matrix H.
    !>          If a NaN is detected in H, the routine will return with INFO=-6.
    !>

    LDH

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

    W

    !>          W is COMPLEX*16 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

    !>          VL is COMPLEX*16 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

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

    VR

    !>          VR is COMPLEX*16 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

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

    MM

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

    M

    !>          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

    !>          WORK is COMPLEX*16 array, dimension (N*N)
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (N)
    !>

    IFAILL

    !>          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

    !>          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

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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|.
    !>

Author

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