Ebook: Computer Aided Proofs in Analysis

This IMA Volume in Mathematics and its Applications COMPUTER AIDED PROOFS IN ANALYSIS is based on the proceedings of an IMA Participating Institutions (PI) Conference held at the University of Cincinnati in April 1989. Each year the 19 Participating Institutions select, through a competitive process, several conferences proposals from the PIs, for partial funding. This conference brought together leading figures in a number of fields who were interested in finding exact answers to problems in analysis through computer methods. We thank Kenneth Meyer and Dieter Schmidt for organizing the meeting and editing the proceedings. A vner Friedman Willard Miller, Jr. PREFACE Since the dawn of the computer revolution the vast majority of scientific compu­ tation has dealt with finding approximate solutions of equations. However, during this time there has been a small cadre seeking precise solutions of equations and rigorous proofs of mathematical results. For example, number theory and combina­ torics have a long history of computer-assisted proofs; such methods are now well established in these fields. In analysis the use of computers to obtain exact results has been fragmented into several schools.

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This volume is the proceedings of a scientific conference on the use of computers to do exact mathematics which was held at the University of Cincinnati, March 22-25, 1989. One group of papers deals with the use of general algebraic processors like Macsyma, Reduce, Scratchpad etc. to do precise computations in bifurcation analysis and related areas of analysis. Another group of papers deals with the development of the software to solve equations exactly, carry of explicit integrations etc. for these general software packages. A third group discusses the use of interval arithmetic algorithms and software to give rigorous proofs of mathematical theorems and give precise estimates of stabililty regions.

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This volume is the proceedings of a scientific conference on the use of computers to do exact mathematics which was held at the University of Cincinnati, March 22-25, 1989. One group of papers deals with the use of general algebraic processors like Macsyma, Reduce, Scratchpad etc. to do precise computations in bifurcation analysis and related areas of analysis. Another group of papers deals with the development of the software to solve equations exactly, carry of explicit integrations etc. for these general software packages. A third group discusses the use of interval arithmetic algorithms and software to give rigorous proofs of mathematical theorems and give precise estimates of stabililty regions.
Content:
Front Matter....Pages i-xvi
The Conversion of a High Order Programming Language from Floating-Point Arithmetic to Range Arithmetic....Pages 1-4
Sylvester’s form of the Resultant and the Matrix-Triangularization Subresultant PRS Method....Pages 5-11
Computing the Tsirelson Space Norm....Pages 12-21
Floating-Point Systems for Theorem Proving....Pages 22-32
Computer Algebra and Indefinite Integrals....Pages 33-42
A Computer-Assisted Approach to Small-Divisors Problems Arising in Hamiltonian Mechanics....Pages 43-51
On a Computer Algebra Aided Proof in Bifurcation Theory....Pages 52-70
MACSYMA Program to Implement Averaging Using Elliptic Functions....Pages 71-89
Validated Anti-Derivatives....Pages 90-96
A Toolbox for Nonlinear Dynamics....Pages 97-115
Computer Assisted Proofs of Stability of Matter....Pages 116-126
Accurate Strategies for K.A.M. Bounds and Their Implementation....Pages 127-146
A Software Tool for Analysis in Function Spaces....Pages 147-167
Equation Solving by Symbolic Computation....Pages 168-174
Deciding a Class of Euclidean Geometry Theorems with Buchberger’s Algorithm....Pages 175-189
Lie Transform Tutorial — II....Pages 190-210
Interval Tools for Computer Aided Proofs in Analysis....Pages 211-216
Tools for Mathematical Computation....Pages 217-228
Shadowing Trajectories of Dynamical Systems....Pages 229-234
Transformation to Versal Normal Form....Pages 235-240
Computer Assisted Lower Bounds for Atomic Energies....Pages 241-251

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This volume is the proceedings of a scientific conference on the use of computers to do exact mathematics which was held at the University of Cincinnati, March 22-25, 1989. One group of papers deals with the use of general algebraic processors like Macsyma, Reduce, Scratchpad etc. to do precise computations in bifurcation analysis and related areas of analysis. Another group of papers deals with the development of the software to solve equations exactly, carry of explicit integrations etc. for these general software packages. A third group discusses the use of interval arithmetic algorithms and software to give rigorous proofs of mathematical theorems and give precise estimates of stabililty regions.
Content:
Front Matter....Pages i-xvi
The Conversion of a High Order Programming Language from Floating-Point Arithmetic to Range Arithmetic....Pages 1-4
Sylvester’s form of the Resultant and the Matrix-Triangularization Subresultant PRS Method....Pages 5-11
Computing the Tsirelson Space Norm....Pages 12-21
Floating-Point Systems for Theorem Proving....Pages 22-32
Computer Algebra and Indefinite Integrals....Pages 33-42
A Computer-Assisted Approach to Small-Divisors Problems Arising in Hamiltonian Mechanics....Pages 43-51
On a Computer Algebra Aided Proof in Bifurcation Theory....Pages 52-70
MACSYMA Program to Implement Averaging Using Elliptic Functions....Pages 71-89
Validated Anti-Derivatives....Pages 90-96
A Toolbox for Nonlinear Dynamics....Pages 97-115
Computer Assisted Proofs of Stability of Matter....Pages 116-126
Accurate Strategies for K.A.M. Bounds and Their Implementation....Pages 127-146
A Software Tool for Analysis in Function Spaces....Pages 147-167
Equation Solving by Symbolic Computation....Pages 168-174
Deciding a Class of Euclidean Geometry Theorems with Buchberger’s Algorithm....Pages 175-189
Lie Transform Tutorial — II....Pages 190-210
Interval Tools for Computer Aided Proofs in Analysis....Pages 211-216
Tools for Mathematical Computation....Pages 217-228
Shadowing Trajectories of Dynamical Systems....Pages 229-234
Transformation to Versal Normal Form....Pages 235-240
Computer Assisted Lower Bounds for Atomic Energies....Pages 241-251
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