This book provides an introduction to methods for practically solving mathematical problems, such as solving systems of linear equations, determining eigenvalues, approximating and integrating functions, solving nonlinear equations, and the approximate solution of ordinary differential equations.

It consists of three parts:

• Systems of linear equations, eigenvalue problems and optimisation

• Interpolation, quadrature and nonlinear equations

• Initial value problems and Hamiltonian systems

Each of these parts is divided into nine short chapters and corresponds approximately to the scope of a two-hour lecture over one semester. Basic knowledge of linear algebra and analysis as well as elementary programming experience are assumed. Results of analysis are only used in the second and third part of the book. Learning objectives, self-assessment tests and exemplary applications at the end of each chapter are intended to deepen the understanding of the presented material. The last chapters of the book contain extensive collections of exercises, detailed descriptions for programming projects, introductions to the programming languages MATLAB, C++ and Python, compilations of the most important results from linear algebra and analysis, some example programs, a list of further topics as well as detailed literature references. The book is aimed at undergraduate students of mathematics as well as engineering and natural sciences.

The translation of this book was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.