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.


          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

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  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|.