====================================================================
dgehrd examples
====================================================================

====================================================================
Man Page Details
====================================================================

NAME
       DGEHRD  -  Reduce a real general matrix A to upper Hessenberg form H by
       an orthogonal similarity transformation

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

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

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

PURPOSE
       DGEHRD 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 subdiagonal of A are overwrit-
               ten with the upper Hessenberg 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).  Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.

       WORK    (workspace/output) DOUBLE PRECISION 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.

               If  LWORK  = -1, then a workspace query is assumed; 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 illegal value.

FURTHER DETAILS
       The matrix Q is  represented  as  a  product  of  (ihi-ilo)  elementary
       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 modi-
       fied element of the upper Hessenberg matrix H, and vi denotes  an  ele-
       ment of the vector defining H(i).

       See Quintana-Orti and Van  de  Geijn (2005).

