Perturbation Methods Using Backward Error

"Corless and Fillion have written the rare book that bridges mathematical practice, computation, and philosophy. Perturbation Methods Using Backward Error brings new unity and rigor to a venerable subject, showing how backward error analysis can illuminate every corner of approximation — from asymptotics and differential equations to computer algebra and modern numerical methods. Clear, concrete, and historically grounded, it captures both the beauty and the practicality of perturbation theory. Personally, I can't wait to teach from it — and to keep learning from it." -Steven Strogatz, Cornell University Perturbation methods are old but powerful, and they remain in widespread use. Rather than producing numbers or pictures, they yield formulas whose value depends on the skill of the person (or machine!) interpreting them. This unique book presents several classical methods for solving perturbation problems. To ensure a uniform presentation and more reliable, interpretable results, it consistently uses backward error analysis. This provides a systematic way to assess the validity of approximate solutions while encouraging the modeler to examine how small changes in the data or model affect the result. To support this, the book uses the concept of a condition number, familiar from numerical analysis. Perturbation Methods Using Backward Error includes a chapter on the relatively novel renormalization group method, uses computer algebra (via Maple) to ease the computing of symbolic answers, provides solutions to all exercises, and discusses the impact on science of the idea of perturbation.

Robert M. Corless is Emeritus Distinguished University Professor at Western University, an adjunct professor at the Cheriton School of Computer Science at the University of Waterloo, a member of the Rotman Institute of Philosophy, and former scientific director of The Ontario Research Center for Computer Algebra. He is the editor-in-chief of Maple Transactions. In 2019 he was awarded the Ford-Halmos Prize, along with the late Jonathan Borwein, for expository excellence for the paper Gamma and Factorial in the Monthly. Nicolas Fillion is an associate professor of philosophy at Simon Fraser University. His main research contributions are in the philosophy of science and applied mathematics, but his research and teaching include the history of logic, mathematics, and science, formal logic, decision and game theory, critical thinking, philosophy of language, ancient Greek philosophy, and epistemology broadly construed. In 2019 he was awarded the Cormack Award for Excellence in Teaching.