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Feasible Computation in Symbolic and Numeric Integration Western University Scholarship@Western Electronic Thesis and Dissertation Repository 12-15-2017 2:00 PM Feasible Computation in Symbolic and Numeric Integration Robert H.C. Moir The University of Western Ontario Supervisor Corless, Robert M. The University of Western Ontario Joint Supervisor Moreno Maza, Marc The University of Western Ontario Graduate Program in Applied Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Robert H.C. Moir 2017 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Numerical Analysis and Computation Commons, and the Numerical Analysis and Scientific Computing Commons Recommended Citation Moir, Robert H.C., "Feasible Computation in Symbolic and Numeric Integration" (2017). Electronic Thesis and Dissertation Repository. 5155. https://ir.lib.uwo.ca/etd/5155 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract Two central concerns in scientific computing are the reliability and efficiency of al- gorithms. We introduce the term feasible computation to describe algorithms that are reliable and efficient given the contextual constraints imposed in practice. The main fo- cus of this dissertation then, is to bring greater clarity to the forms of error introduced in computation and modeling, and in the limited context of symbolic and numeric integra- tion, to contribute to integration algorithms that better account for error while providing results efficiently. Chapter2 considers the problem of spurious discontinuities in the symbolic integration problem, proposing a new method to restore continuity based on a pair of unwinding numbers. Computable conditions for the unwinding numbers are specified, allowing the computation of a variety of continuous integrals. Chapter3 introduces two structure- preserving algorithms for the symbolic-numeric integration of rational functions on exact input. A structured backward and forward error analysis for the algorithms shows that they are a posteriori backward and forward stable, with both forms of error exhibiting tolerance proportionality. Chapter4 identifies the basic logical structure of feasible inference by presenting a logical model of stable approximate inference, illustrated by examples of modeling and numerical integration. In terms of this model it is seen that a necessary condition for the feasibility of methods of abstraction in modeling and complexity reduction in com- putational mathematics is the preservation of inferential structure, in a sense that is made precise. Chapter5 identifies a robust pattern in mathematical sciences of trans- forming problems to make solutions feasible. It is showed that computational complexity reduction methods in computational science involve chains of such transformations. It is argued that the structured and approximate nature of such strategies indicates the need for a \higher-order" model of computation and a new definition of computational complexity. Keywords: symbolic integration, symbolic-numeric integration, unwinding numbers, rational functions, effective validity, effective logic, feasible computation i Co-Authorship Statement Chapter2 was co-authored with Robert Corless and David Jeffrey and will be sub- mitted for publication. Rob Corless supervised the research for this chapter. Chapter3 was co-authored with Robert Corless, Marc Moreno Maza and Ning Xie and will be sub- mitted for publication. Marc Moreno Maza and Robert Corless supervised the research for this chapter, and Marc Moreno Maza supervised the extensive implementation work for this chapter. Chapters4 and5 are both single-authored. Chapter5 has been accepted for publi- cation and chapter4 will be submitted for publication. Robert Corless reviewed earlier drafts of chapters4 and5. Marc Moreno Maza re- viewed and revised portions of an earlier draft of chapter5. ii To my wife and best friend, Lori iii Acknowledgements A great many people have contributed to different aspects of the work leading to this dissertation. Principal among them are my supervisors Rob Corless and Marc Moreno Maza. I would not even have entertained this degree had I not wandered into Rob's office one afternoon, looking for things to do after finishing my PhD in philosophy, leading to him suggesting doing a PhD with him in one year. It ended up taking longer, but the journey led to many excellent opportunities to expand my professional and scholarly horizons. This included becoming the local organizer (with Rob Corless, Chris Smeenk and Nic Fillion) of the first multi-disciplinary conference on Algorithms and Complexity in Mathematics, Epistemology and Science, and included many opportunities to develop the ideas from my philosophy PhD by integrating prior work with my growing knowledge of applied mathematics and computational science, while thinking about new approaches to error handling in symbolic integration. This would not have been possible without the enormously flexible and supportive supervision style of Rob Corless. Taking the graduate course in Distributed and Parallel Systems with Marc Moreno Maza contributed to a rekindling of my love of programming and led to the idea of im- plementing symbolic integration algorithms in C++ using the basic polynomial algebra subprograms (bpas) package developed by Marc Moreno Maza, Ning Xie and others. The project for this course led to my being the first \customer" of bpas, eventually to my joining the project as a developer as my research became intertwined with the project, and now to my continuing the development of bpas as a postdoctoral researcher. The pleasure of working with Marc has allowed me to implement efficient symbolic computa- tion algorithms in the challenging programming context of the CilkPlus branch of C++, and contributed significantly to my noticing the general pattern of inferentially stable transformations that reduce inferential complexity in symbolic computation and pure mathematics. Other teachers of mine that have contributed to making this work possible are my supervisors for my philosophy PhD, Bob Batterman and John Bell, and my medical qigong sifu Bob Youngs. My attempt to harmonize Bob Batterman's philosophy of iv applied mathematics with John Bell's philosophy of pure mathematics has led to the understanding of feasible computation and inference articulated here. Their contributions and insights, as well as those of Rob Corless and Marc Moreno Maza, permeate this dissertation. By teaching me the Daoist principles of good health and longevity, as well as so much more, Bob Youngs has enabled me to find the discipline to maintain good health and to cultivate the energy needed to complete the rigours of this PhD. This work has also benefited from the support of many colleagues, family and friends. Nic Fillion has always supported and stimulated my research as long as I have known him, through friendship and countless excellent conversations over the years. Emerson Doyle has also been an ardent friend and supporter of my philosophical and mathematical ideas for as long as I have known him, providing a challenging interlocutor and excellent feedback on drafts of many of my papers. Conversations with Sona Ghosh, Veromi Arsiradam and Andrew Fenton have also been valuable in more and less subtle ways. As always, I thank my parents for their steady, loving support from the very beginning of my venture into graduate work, up to its completion now. I would also like to thank my examiners for their thoughtful, challenging and pene- trating comments and questions. In particular: I thank David Jeffrey for his challenges to harmonize results with their careful mathematical presentation, and to meet the com- bined interest of the reader and implementer of mathematical algorithms; I thank David Stoutemyer for his support of the project and his very careful reading of the entire dissertation, along with numerous thoughtful comments about how to improve the pre- sentation; and I thank Chris Smeenk for his challenges to justify my view of effective logic as a generalization of traditional deductive logic and to articulate ways in which it stands to have normative force. Most of all I would like to thank my wife Lori, to whom this work is dedicated. She was the first to support the idea of doing a second PhD in applied mathematics, seeing its potential to allow me to more fully develop and integrate my ideas. She has been a consistent source of love and support throughout the process, as well as providing some of the strongest challenges to carefully articulate and justify my thinking, with all of the conceptual benefits that entails. v Contents Abstract i Co-Authorship Statement ii Dedication iii Acknowledgements iv List of Figures viii List of Tables ix List of Abbreviations, Symbols, and Nomenclaturex 1 Introduction: Continuity and Complexity Reduction in Scientific Computing1 1.1 Motivation....................................... 2 1.2 Outline......................................... 4 1.3 Bibliography ..................................... 7 2 An Unwinding Number Pair for Continuous Integrals9 2.1 Introduction...................................... 10 2.2 Unwinding Numbers for Paths on the Riemann Sphere............ 12 2.3 Restoring Continuity for the Logarithm and Arctangent Functions . 17 2.4 Continuous Real & Complex
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