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


SYNOPSIS

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

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

           REAL           A( LDA, * ), TAU( * ), WORK( * )


PURPOSE

       SGEHRD 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  SGEBAL;  otherwise  they
               should be set to 1 and N respectively. See Further
               Details.

       A       (input/output) REAL 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) REAL 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) REAL 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 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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