Arildsen Thomas - Applied scientific computing with Python

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This book represents a modern approach to teaching numerical methodsor scientific computingto students with a broad range of primary interests. The underlying mathematical content is not new, of course, but the focus on applications and models (or modeling) is. Todays mathematics or computer science students have a strong desire to see the relevance of what they are studying in a practical way. This is even more true for students of other STEM disciplines in their mathematics and computing classes.

In an introductory text such as this, it is difficult to give complete detail on the application of the methods to real-world engineering, or economics, or physical science, or social science applications but it is important to connect those dots. Throughout the book, we emphasize applications and include opportunities for both problem- and project-based learning through examples, exercises, and projects drawn from practical applications. Thus, the book provides a self-contained answer to the common question why do I need to learn this?

That is a question which we believe any student has a right both to ask and to expect a reasonable and credible answer, hence our focus on Applied Scientific Computing. The intention is that this book is suitable for an introductory first course in scientific computing for students across a range of major fields of study. Therefore, we make no pretense at a fully rigorous development of all the state-of-the-art methods and their analyses. The intention is to give a basic understanding of the need for, and methods of, scientific computing for different types of problems. In all cases, there is sufficient mathematical justification and practical evidence to provide motivation for the reader.

Any text such as this needs to provide practical guidance on coding the methods, too. Coding a linear system solver, for example, certainly helps the students understanding of the practical issues, and limitations, of its use. Typically students find introductory numerical methods more difficult than their professors expect. Part of the reason is that students are expected to combine skills from different parts of their experiencedifferent branches of mathematics, programming and other computer science skills, and some insight in applications fields. Arguably, these are independent skill sets and so the probabilities of success multiply.

Our choice of Python for the computer language is based on the desire to minimize the overhead of learning a high-level language or of the intricacies (and cost) of a specific application package. The coding examples here are intended to be relatively easily readable; they are not intended to be professional-level software. The Python code is there to facilitate the learning and understanding of the basic material, rather than being an objective in itself.

Turning briefly to the content of the book, we gave considerable thought to the ordering of material and believe that the order we have is one good way to organize a course, though of course every instructor will have his/her own ideas on that. The chapters are largely independent and so using them in a different order will not be problematic.

We begin with a brief background chapter that simply introduces the main topics application and modeling, Python programming, sources of error. The latter is exemplified at this point with simple series expansions which should be familiar from calculus but need nothing more. These series expansions also demonstrate that the most obvious approach that a beginning student might adopt will not always be practical. As such it serves to motivate the need for more careful analysis of problems and their solutions. Chapter is still somewhat foundational with its focus on number representation and errors. The impact of how numbers are represented in a computer, and the effects of rounding and truncation errors recur in discussing almost any computational solution to any real-life problem.

From Chap. itself the focus is on numerical calculus. We put this before interpolation to reflect the students familiarity with the basic concepts and likely exposure to at least some simple approaches. These are treated without any explicit reference to finite difference schemes.

Chapters are devoted to linear and then nonlinear equations. Systems of linear equations are another topic with which students are somewhat familiar, at least in low dimension, and we build on that knowledge to develop more robust methods. The nonlinear equation chapter also builds on prior knowledge and is motivated by practical applications. It concludes with a brief treatment of the multivariable Newtons method.

The final two chapters are on interpolation and the numerical solution of differential equations. Polynomial interpolation is based mostly on a divided difference approach which then leads naturally to splines. Differential equations start from Eulers method, and then Runge Kutta, for initial value problems. Shooting methods for two-point boundary value problems are also covered, tying in many ideas from the previous chapters.