
!!!  This text and the programs in this directory are still under preparation !!!
!!!  and not yet completed                        Rudolf Lohner, Feb. 8, 1994 !!!


This directory contains the progam AWA (for german AnfangsWertAufgabe
= initial value problem) and a small auxiliary tool AWAPLOT.

AWA computes enclosures for the solution of initial value problems.
These enclosures are mathematically safe, i.e. all truncation and
roundoff errors are taken into account.

AWAPLOT is a small tool to generate plot data files from the AWA
output files. The plot data may be used with plot programs such
as GNUPLOT etc. to produce graphical output.


A documentation for AWA is not yet available in electronic form.


Earlier versions of AWA did not have a step size control which now
is built into AWA. Therefore some minor changes have to be observed
with respect to the older version of AWA without step size control :

1) Instead of a constant step size now an initial step size
   h_initial is required on input. If h_initial = 0 then AWA
   tries to determine a suitable initial step size on its own.
   If h_initial is different from zero then this value will be
   used for the computation of the inital step. In certain cases
   the automatic estimation of h_initial by AWA may be unsuccessful
   (resulting in bad enclosures at the beginning already or even in
   a breakdown of the program). Therefore, in these cases it is
   necessary to enter a suitable starting step size h_initial
   manually (trial and error!).

2) After the input of the initial values now two additional parameters
   E_a and E_r are required as input by AWA which are used to control
   the absolute (E_a) and the relative (E_r) accuracy of the solution
   (which does not mean, however, that these values are actually achieved,
   the error control is heuristic only). The choice of E_a and E_r may
   need some experience and may also depend on the problem and the order
   of magnitude of the solution. If in doubt try different values.

3) In the present version the enclosure values of the solution can be
   printed only at those points which were computed as grid points by
   the step size control. E.g. it is not possible with this version to
   generate output on an equidistant grid. Another consequence of this
   is that it is not very meaningful to choose a different value than zero
   for the output option 'out' (i.e. print out every step).
   Future versions of AWA will include a more sophisticated method for
   the choice of output points.
   Caution: The values of t which are printed for the grid points are
            of course not the exact values in general, but they are
            usually rounded on output. Only the internal binary values
            represent the exacly computed grid points.
   However: The end point T_end of an integration block is reached always
            exactly if the value which was entered as T_end is
            representable exactly in the internal binary format.
            Otherwise the internal binary value of T_end will be the
            exact end of the integration.

4) If any uncontrolled roundoff errors should occur then a warning will
   (should?) be printed.

5) Negative step sizes should be allowed as in previous versions. The
   direction in which the integration will proceed is determined automatically
   by the values of the starting and end points T_start and T_end (the sign
   of h_initial is ignored in this case). Negative step sizes are not yet
   tested in this version, however.

6) AWA offers a choice of eight output options ( 0 = print function values
   of solution, ..., 7 = print values of output list ). In the present version
   only options 0 and 7 are implemented. Options 1 ... 6 were available in
   earlier versions and will be available in future versions again, but
   then probably as part of a more general output facility.
   In normal output format (output option 0, enclosures of the solution values)
   an additional third column has been added after the second one (diameters of
   the enclosures). This third column contains upper bounds for the relative
   error of the corresponding enclosure. If, however, the enclosing interval
   contains zero then '---' is printed in this column.



Rudolf Lohner
Institut fuer Angewandte Mathematik
Universitaet Karlsruhe (TH)
Postfach 6980
D-76128 Karlsruhe

Phone: +49 (721) 6084273
Fax  : +49 (721) 385979
email: ae34@rz.uni-karlsruhe.de



AWA is written in PASCAL-XSC a Pascal eXtension for Scientific Computing
which is a powerful language including many valuable features like module
concept, operator overloading, exact scalar products, predefined modules
for interval-, complex-, complex interval- and matrix-vector arithmetic as
well as many self-validating problem solving routines, to mention only few.
 
For more information on PASCAL-XSC (and a similarly, C-XSC) contact:

Michael Neaga
Numerik Software GmbH
P.O.Box 2232
D-76492 Baden-Baden
Germany

Phone: +49 (721) 694217
Fax  : +49 (721) 694418
email: ae18@rz.uni-karlsruhe.de


References on the enclosure method used in AWA :

[1] R. Lohner, Enclosing the Solutions of Ordinary Initial and Boundary Value
    Problems. pp. 255-286 in: E. Kaucher, U. Kulisch, Ch. Ullrich (eds.),
    Computerarithmetic - Scientific Computation and Programming Languages,
    B. G. Teubner, Stuttgart, 1987.

[2] R. Lohner, Einschliessung der Loesung gewoehnlicher Anfangs- und Randwert-
    aufgaben und Anwendungen. Dissertation, University of Karlsruhe, 1988.
 
[3] R. Lohner, Computation of Guaranteed Enclosures for the Solutions of
    Ordinary Initial and Boundary Value Problems. pp. 425-435 in: J. R. Cash,
    I. Gladwell (eds.), Computational Ordinary Differential Equations,
    Clarendon Press, Oxford, 1992.
 
[4] R. Lohner, Einschliessungen bei Anfangs- und Randwertaufgaben gewoehnlicher
    Differentialgleichungen. pp. 183-207 in: U. Kulisch (ed.), Wissenschaft-
    liches Rechnen mit Ergebnisverifikation. Reihe Mathematical Research,
    Vol. 58, Akademie-Verlag, Berlin, 1989, und Vieweg, Braunschweig, 1989.
 
[5] R. Lohner, Praktikum `Einschliessungen bei Differentialgleichungen'.
    pp. 209-223 in: U. Kulisch (ed.), Wissenschaftliches Rechnen mit Ergebnis-
    verifikation. Reihe Mathematical Research, Vol. 58, Akademie-Verlag,
    Berlin, 1989, und Vieweg, Braunschweig, 1989.
 
