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NAME
     sgehrd - reduce a real general matrix A to upper  Hessenberg
     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( LWORK )



     #include <sunperf.h>

     void sgehrd(int n, int ilo, int ihi,  float  *sa,  int  lda,
               float *tau, int *info) ;

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.

     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  example,
     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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