Purpose:  1) Can be used to verify orders of convergence of 
=======      the provided palindromic schemes.
          2) Can be used to solve a special kind of ODE with very high 
             accuracy: y'=f(y), where f(y) is at most quadratic. 
             In fact, OneStep.f and allS1.f are written for any system 
             of this kind. But f(y) and f'(y) must be provided. For this 
             particular example they are provided by fexjex.f.

                                        Ren-Cang Li, June 1, 1996
                                        na.rcli@na-net.ornl.gov

Files in this directory:
=======================

      main.f     ---  Main program 1 for the Lorenz Strange Attractor.
                      It runs with a particular scheme specified 
                      manually at run time and with a fixed stepsize.
                      It outputs y(t) at each time step.
                      (Read main.f for detail.)

      mainHs.f   ---  Main program 2 for the Lorenz Strange Attractor.
                      It runs with a particular scheme specified 
                      manually at run time and with several different
                      stepsizes, each of which is half of the previous
                      run. It outputs errors for each component of computed
                      y(t) at t=1 for each different stepsize. The output
                      can be used to verified orders of convergence
                      of palindromic schemes.
                      (Read mainHs.f for detail.)
                      
     Makefile    ---  Make file.

     OneStep.f   ---  Provide a reflexive update formula for the above
                      mentioned systems.

     allS1.f     ---  Call OneStep.f repeatedly to yield any particular
                      palindromic scheme.

     Results3    ---  Output of mainHs with WHICH = 3, running on IBM RS6000/590.

     The following routines are modified routines with the same names  
     --- --------- -------- --- -------- -------- ---- --- ---- -----
     from LAPACK and BLAS:
     ---- ------ --- ----

       dgemm.f dger.f  dgesv.f dgetf2.f dgetrf.f dgetrs.f dlaswp.f
       dscal.f dswap.f dtrsm.f fexjex.f idamax.f ilaenv.f lsame.f 
       xerbla.f

Files s* define 15 Palindromic Schemes:
===== == ====== == =========== =======

      s3odr4    s5odr4   s5odr4a   s7odr6    s9odr6a   s9odr6b 
     s15odr8   s17odr8a s17odr8b s31odr10a s31odr10b s33odr10a 
     s33odr10b s33odr10c
