Preface
ix
Target Audience and Background Knowledge This book was written for
students, teachers, engineers, and scientists who know nothing about programming
and numerical methods from before but who seek a minimum of the fundamental
skills required to get started with programming as a tool for solving scientific
and engineering problems. Some knowledge of one- and multivariable calculus
is assumed. The basic programming concepts are presented in Chaps. 1–5 (about
150 pages), before practical applications of these concepts are demonstrated in
important mathematical subjects addressed in the remaining parts of the book
(Chaps. 6–9). Each chapter is followed by a set of exercises that covers a wide range
of application areas, e.g., biology, geology, statistics, physics, and mathematics.
The exercises were particularly designed to bring across important points from the
text.
Learning the very basics of programming should not take long, but as with any
other craft, mastering the skill requires continued and extensive practice. Some
beginning practice is gained through Chaps. 6–9, but the authors strongly emphasize
that this is only a start. Students should continue to practice programming in
subsequent courses, while those who exercise self-study should keep up the learning
process through continued application of the craft. The book is a good starting point
when teaching computer programming as an integrated part of standard university
courses in mathematics and natural science. In our experience, such an integration
is doable and indeed rewarding.
Numerical Methods An overall goal with this book is to motivate computer programming as a very powerful tool for doing mathematics. All examples are related to
mathematics and its use in engineering and science. However, to solve mathematical
problems through computer programming, we need numerical methods. Explaining
basic numerical methods is therefore an integral part of the book. Our choice of
topics is governed by what is most needed in science and engineering, as well as
in the teaching of applied natural science courses. Mathematical models are then
central, with differential equations constituting the most frequent type of models.
Consequently, the numerical focus in this book is on differential equations. As soft
pedagogical starters for the programming of mathematics, we have chosen the topics
of numerical integration and root finding. We remark that the book is deliberately
brief on numerical methods. This is because our focus is on implementing numerical
algorithms, and to develop reliable, working programs, the programmer must be
confident about the basic ideas of the numerical approximations involved.
The Computer Language: Python We have chosen to use the programming
language Python, because this language gives a very compact and readable code
that closely resembles the mathematical recipe for solving the problem at hand.
Python also has a gentle learning curve.
Other computer languages, like Fortran, C, and C++, have a strong position in
science and engineering. During the last two decades, however, there has been a
significant shift in popularity from these compiled languages to more high-level and
easier-to-read languages, for instance, MATLAB, Python, R, Maple, Mathematica,
and IDL. This latter class of languages is computationally less efficient but superior
with respect to overall human problem-solving efficiency. This book emphasizes
ix
Target Audience and Background Knowledge This book was written for
students, teachers, engineers, and scientists who know nothing about programming
and numerical methods from before but who seek a minimum of the fundamental
skills required to get started with programming as a tool for solving scientific
and engineering problems. Some knowledge of one- and multivariable calculus
is assumed. The basic programming concepts are presented in Chaps. 1–5 (about
150 pages), before practical applications of these concepts are demonstrated in
important mathematical subjects addressed in the remaining parts of the book
(Chaps. 6–9). Each chapter is followed by a set of exercises that covers a wide range
of application areas, e.g., biology, geology, statistics, physics, and mathematics.
The exercises were particularly designed to bring across important points from the
text.
Learning the very basics of programming should not take long, but as with any
other craft, mastering the skill requires continued and extensive practice. Some
beginning practice is gained through Chaps. 6–9, but the authors strongly emphasize
that this is only a start. Students should continue to practice programming in
subsequent courses, while those who exercise self-study should keep up the learning
process through continued application of the craft. The book is a good starting point
when teaching computer programming as an integrated part of standard university
courses in mathematics and natural science. In our experience, such an integration
is doable and indeed rewarding.
Numerical Methods An overall goal with this book is to motivate computer programming as a very powerful tool for doing mathematics. All examples are related to
mathematics and its use in engineering and science. However, to solve mathematical
problems through computer programming, we need numerical methods. Explaining
basic numerical methods is therefore an integral part of the book. Our choice of
topics is governed by what is most needed in science and engineering, as well as
in the teaching of applied natural science courses. Mathematical models are then
central, with differential equations constituting the most frequent type of models.
Consequently, the numerical focus in this book is on differential equations. As soft
pedagogical starters for the programming of mathematics, we have chosen the topics
of numerical integration and root finding. We remark that the book is deliberately
brief on numerical methods. This is because our focus is on implementing numerical
algorithms, and to develop reliable, working programs, the programmer must be
confident about the basic ideas of the numerical approximations involved.
The Computer Language: Python We have chosen to use the programming
language Python, because this language gives a very compact and readable code
that closely resembles the mathematical recipe for solving the problem at hand.
Python also has a gentle learning curve.
Other computer languages, like Fortran, C, and C++, have a strong position in
science and engineering. During the last two decades, however, there has been a
significant shift in popularity from these compiled languages to more high-level and
easier-to-read languages, for instance, MATLAB, Python, R, Maple, Mathematica,
and IDL. This latter class of languages is computationally less efficient but superior
with respect to overall human problem-solving efficiency. This book emphasizes