## Automatic validation of numerical solutions

### Ole Stauning

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## Abstract

This thesis is concerned with ``Automatic Validation of Numerical Solutions''.
The basic theory of interval analysis and self-validating methods is introduced.
The mean value enclosure is applied to discrete mappings for obtaining
narrow enclosures of the iterates when applying these mappings with intervals as
initial values. A modification of the mean value enclosure of discrete
mappings is considered, namely the extended mean value enclosure which in most
cases
leads to even better enclosures. These methods have previously been described in
connection with discretizing solutions of ordinary differential equations,
but in this thesis, we describe how to use the methods for enclosing iterates
of discrete mappings, and then later use them for discretizing solutions of
ordinary differential equations.
The theory of automatic differentiation is introduced, and three methods
for obtaining derivatives are described: The forward, the backward, and the
Taylor expansion methods. The three methods have been implemented in the C++
program packages FADBAD/TADIFF. Some examples showing how to use the three metho
ds
are presented. A feature of FADBAD/TADIFF not present in other automatic
differentiation packages is the possiblility to combine the three methods
in an extremely flexible way. We examine some applications where this
flexibility is very useful.

A method for Taylor expanding solutions of ordinary differential equations
is presented, and a method for obtaining interval enclosures of the
truncation errors incurred, when truncating these Taylor series expansions is
described.
By combining the forward method and the Taylor expansion method, it is
possible to implement the (extended) mean value enclosure of a truncated Taylor
series expansion with enclosures of the truncation errors. A C++ program package
ADIODES, using this method has been developed. (ADIODES is an abbreviation of ``
Automatic Differentiation Interval Ordinary Differential
Equation Solver'').

ADIODES is used to prove existence and uniqueness of periodic solutions to
specific ordinary differential equations occuring in dynamical systems
theory. These proofs of existence and uniqueness are difficult or impossible to
obtain using other known methods. Also, a method for solving boundary value
problems is described.

Finally a method for enclosing solutions to a class of integral equations
is described. This method is based on the mean value enclosure of an integral
operator and uses interval Bernstein polynomials for enclosing the solution. Two
numerical examples are given, using two orders of approximation and using
different numbers of discretization points.

## IMM ph.d thesis 36, 1997

*Last modified Jan 19, 1998**
For further information, please contact, Finn Kuno
Christensen, IMM, Bldg. 321, DTU *

Phone: (+45) 4588 1433. Fax: (+45) 4588
2673, E-mail: fkc@imm.dtu.dk

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