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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

  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|.