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.

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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  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|.