# Computer-Assisted Proofs and Symbolic Computations

Serdica Journal of Computing (2010)

- Volume: 4, Issue: 1, page 73-84
- ISSN: 1312-6555

## Access Full Article

top## Abstract

top## How to cite

topKrämer, Walter. "Computer-Assisted Proofs and Symbolic Computations." Serdica Journal of Computing 4.1 (2010): 73-84. <http://eudml.org/doc/11376>.

@article{Krämer2010,

abstract = {We discuss some main points of computer-assisted proofs based
on reliable numerical computations. Such so-called self-validating numerical
methods in combination with exact symbolic manipulations result in very
powerful mathematical software tools. These tools allow proving mathematical
statements (existence of a fixed point, of a solution of an ODE, of
a zero of a continuous function, of a global minimum within a given range,
etc.) using a digital computer. To validate the assertions of the underlying
theorems fast finite precision arithmetic is used. The results are absolutely
rigorous.
To demonstrate the power of reliable symbolic-numeric computations we
investigate in some details the verification of very long periodic orbits of
chaotic dynamical systems. The verification is done directly in Maple, e.g.
using the Maple Power Tool intpakX or, more efficiently, using the C++
class library C-XSC.* This work is partially supported by DFG: KR1612/7-1.},

author = {Krämer, Walter},

journal = {Serdica Journal of Computing},

keywords = {Computer-Assisted Proofs; Symbolic Computations; Self-Validating Numerical Methods; Dynamical System; Verified Periodic Orbit; IntpakX; C-XSC; computer-assisted proofs; symbolic computation; self-validating numerical methods; dynamical system; verified periodic orbit; intpakX; C-XSC; mathematical software},

language = {eng},

number = {1},

pages = {73-84},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Computer-Assisted Proofs and Symbolic Computations},

url = {http://eudml.org/doc/11376},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Krämer, Walter

TI - Computer-Assisted Proofs and Symbolic Computations

JO - Serdica Journal of Computing

PY - 2010

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 4

IS - 1

SP - 73

EP - 84

AB - We discuss some main points of computer-assisted proofs based
on reliable numerical computations. Such so-called self-validating numerical
methods in combination with exact symbolic manipulations result in very
powerful mathematical software tools. These tools allow proving mathematical
statements (existence of a fixed point, of a solution of an ODE, of
a zero of a continuous function, of a global minimum within a given range,
etc.) using a digital computer. To validate the assertions of the underlying
theorems fast finite precision arithmetic is used. The results are absolutely
rigorous.
To demonstrate the power of reliable symbolic-numeric computations we
investigate in some details the verification of very long periodic orbits of
chaotic dynamical systems. The verification is done directly in Maple, e.g.
using the Maple Power Tool intpakX or, more efficiently, using the C++
class library C-XSC.* This work is partially supported by DFG: KR1612/7-1.

LA - eng

KW - Computer-Assisted Proofs; Symbolic Computations; Self-Validating Numerical Methods; Dynamical System; Verified Periodic Orbit; IntpakX; C-XSC; computer-assisted proofs; symbolic computation; self-validating numerical methods; dynamical system; verified periodic orbit; intpakX; C-XSC; mathematical software

UR - http://eudml.org/doc/11376

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.