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


SYNOPSIS

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

           INTEGER        IHI, ILO, INFO, LDA, LWORK, N

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


PURPOSE

       ZGEHRD 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).  Elements   1:ILO-1   and
               IHI:N-1 of TAU are set to zero.

       WORK    (workspace/output) COMPLEX*16 array, dimension
               (LWORK)
               On exit, if INFO = 0, WORK(1) returns the  optimal
               LWORK.

       LWORK   (input) INTEGER
               The  length of the array WORK.  LWORK >= max(1,N).
               For optimum performance LWORK >= N*NB, where NB is
               the optimal blocksize.

               the routine only calculates the  optimal  size  of
               the  WORK  array,  returns this value as the first
               entry of the WORK  array,  and  no  error  message
               related to LWORK is issued by XERBLA.

       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

          Q = H(ilo) H(ilo+1) . . . H(ihi-1).

       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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