If mathematical modeling is the process of turning real phenomena into mathematical abstractions, then numerical computation is largely about the transformation from abstract mathematics to concrete reality. Many science and engineering disciplines have long benefited from the tremendous value of the correspondence between quantitative information and mathematical manipulation. -from the Preface Fundamentals of Numerical Computation is an advanced undergraduate-level introduction to the mathematics and use of algorithms for the fundamental problems of numerical computation: linear algebra, finding roots, approximating data and functions, and solving differential equations. The book is organized with simpler methods in the first half and more advanced methods in the second half, allowing use for either a single course or a sequence of two courses. The authors take readers from basic to advanced methods, illustrating them with over 200 self-contained MATLAB functions and examples designed for those with no prior MATLAB experience. Although the text provides many examples, exercises, and illustrations, the aim of the authors is not to provide a cookbook per se, but rather an exploration of the principles of cooking.
 Professors Driscoll and Braun have developed an online resource that includes well-tested materials related to every chapter. Among these materials are lecture-related slides and videos, ideas for student projects, laboratory exercises, computational examples and scripts, and all the functions presented in the book.
 The authors describe their goals and intentions for the book in specific terms: Our guiding point of view is that the student users of the book are far more likely to apply and remix fundamental computational algorithms than to reinvent them. We want to prepare them for more advanced courses in numerical analysis, but we place a higher priority on conveying foundational skills in scientific computation. Accordingly, we emphasize knowing how to cast a problem into a form that can be coded and solved, tradeoffs between different methods for a problem, and assessing the correctness and convergence of the results. We do not, however, make much of an issue of optimizing the speed of implementations, except when orders of magnitude are in play, nor do we discuss parallelism.
 Above all, we emphasize linear algebra. Linear algebra is the lingua franca of scientific computing a fact in no small measure established and affirmed by MATLAB over more than three decades."
 Audience: Fundamentals of Numerical Computation is intended for advanced undergraduates in math, applied math, engineering, or science disciplines, as well as for researchers and professionals looking for an introduction to a subject they missed or overlooked in their education.
 Contents: Prologue: Two Imperfect Ideas; Chapter 1: Numbers, Problems, and Algorithms; Chapter 2: Square Linear Systems; Chapter 3: Overdetermined Linear Systems; Chapter 4: Roots of Nonlinear Equations; Chapter 5: Piecewise Interpolation and Calculus; Chapter 6: Initial-Value Problems for ODEs; Chapter 7: Matrix Analysis; Chapter 8: Krylov Methods in Linear Algebra; Chapter 9: Global Function Approximation; Chapter 10: Boundary-Value Problems; Chapter 11: Diffusion Equations; Chapter 12: Advection Equations; Chapter 13: Two-Dimensional Problems; Appendix A: Review of Linear Algebra.

Fundamentals of Numerical Computation

Science & Math

Tobin A. Driscoll

RIchard J. Braun

Mathematics

SIAM-Society for Industrial & Applied Mathematics

Exploring ODEs is a textbook of ordinary differential equations for advanced undergraduates, graduate students, scientists, and engineers. It is unlike other books in this field in that each concept is illustrated numerically via a few lines of Chebfun code. There are about 400 computer-generated figures in all, and Appendix B presents 100 more examples as templates for further exploration.
 Contents: 1. Introduction; 2. First-order scalar linear ODEs; 3. First-order scalar nonlinear ODEs; 4. Second-order equations and damped oscillations; 5. Boundary-value problems; 6. Eigenvalues of linear BVPs; 7. Variable coefficients and adjoints; 8. Resonance; 9. Second-order equations in the phase plane; 10. Systems of equations 11. The fundamental existence theorem; 12. Random functions and random ODEs; 13. Chaos; 14. Linear systems and linearization; 15. Stable and unstable fixed points; 16. Multiple solutions of nonlinear BVPs; 17. Bifurcation; 18. Continuation and path-following; 19. Periodic ODEs; 20. Boundary and interior layers; 21. Into the complex plane; 22. Time-dependent PDEs; Appendix A: Chebfun and its ODE algorithms; Appendix B: 100 more examples.

Exploring ODEs

Lloyd N. Trefethen

Ásgeir Birkisson

SIAM: Society for Industrial and Applied Mathematics

Julia is an open-source and fast-growing programming language for scientific computing that offers clarity and ease of use for beginners but also speed and power for advanced applications. Fundamentals of Numerical Computation: Julia Edition provides a complete solution for teaching Julia in the context of numerical methods. It introduces the mathematics and use of algorithms for the fundamental problems of numerical computation: linear algebra, finding roots, approximating data and functions, and solving differential equations. A clear progression from simple to more advanced methods allows for use in either a one-semester course or a two-semester sequence. The book includes <ul> <li>more than 40 functions and 160 examples fully coded in Julia and available for download,</li> <li>online supplemental content including tested source materials for student projects and in-class labs related to every chapter, and</li> <li>over 600 exercises, evenly split between mathematical and computational work, and solutions to most exercises for instructors.</li> </ul>

Fundamentals of Numerical Computation: Julia Edition

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