ZGEHD2  -  reduce a complex general matrix A to upper Hes­
       senberg form H by a unitary similarity transformation


SYNOPSIS

       SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )

           INTEGER        IHI, ILO, INFO, LDA, N

           COMPLEX*16     A( LDA, * ), TAU( * ), WORK( * )


PURPOSE

       ZGEHD2 reduces a complex general matrix A to upper Hessen­
       berg form H by a unitary similarity transformation: Q' * A
       * Q = H .


ARGUMENTS

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

       ILO     (input) INTEGER
               IHI     (input) INTEGER It is assumed  that  A  is
               already  upper  triangular  in  rows  and  columns
               1:ILO-1 and IHI+1:N. ILO and IHI are normally  set
               by  a  previous  call  to  ZGEBAL;  otherwise they
               should be set to 1 and N respectively. See Further
               Details.

       A       (input/output) COMPLEX*16 array, dimension (LDA,N)
               On entry, the n by n general matrix to be reduced.
               On exit, the upper triangle and the first subdiag­
               onal of A are overwritten with the  upper  Hessen­
               berg  matrix  H,  and the elements below the first
               subdiagonal, with the  array  TAU,  represent  the
               unitary  matrix  Q  as  a  product  of  elementary
               reflectors. See Further Details.  LDA      (input)
               INTEGER The leading dimension of the array A.  LDA
               >= max(1,N).

       TAU     (output) COMPLEX*16 array, dimension (N-1)
               The scalar factors of  the  elementary  reflectors
               (see Further Details).

       WORK    (workspace) COMPLEX*16 array, dimension (N)

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value.


FURTHER DETAILS

       The matrix Q is represented as a product of (ihi-ilo) ele­
       mentary reflectors


       Each H(i) has the form

          H(i) = I - tau * v * v'

       where  tau  is a complex scalar, and v is a complex vector
       with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi)
       is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

       The  contents  of A are illustrated by the following exam­
       ple, with n = 7, ilo = 2 and ihi = 6:

       on entry,                        on exit,

       ( a   a   a   a   a   a   a )    (  a   a   h   h   h    h
       a  ) (     a   a   a   a   a   a )    (      a   h   h   h
       h   a ) (     a   a   a   a   a   a )    (      h   h    h
       h    h   h ) (     a   a   a   a   a   a )    (      v2  h
       h   h   h   h ) (     a   a   a   a   a   a )    (      v2
       v3   h    h    h    h ) (     a   a   a   a   a   a )    (
       v2  v3  v4  h   h   h ) (                         a )    (
       a )

       where  a  denotes  an  element of the original matrix A, h
       denotes a modified element of the upper Hessenberg  matrix
       H,  and vi denotes an element of the vector defining H(i).


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