DGEHD2  -  reduce a real general matrix A to upper Hessen­
       berg form H by an orthogonal similarity transformation


SYNOPSIS

       SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )

           INTEGER        IHI, ILO, INFO, LDA, N

           DOUBLE         PRECISION A( LDA, * ), TAU( * ),  WORK(
                          * )


PURPOSE

       DGEHD2 reduces a real general matrix A to upper Hessenberg
       form H by an orthogonal 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  DGEBAL;  otherwise  they
               should be set to 1 and N respectively. See Further
               Details.

       A       (input/output) DOUBLE PRECISION 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
               orthogonal 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) DOUBLE PRECISION array, dimension (N-1)
               The  scalar  factors  of the elementary reflectors
               (see Further Details).

       WORK    (workspace) DOUBLE PRECISION 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)


          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 real scalar, and v is a real 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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