The fourth, revised and expanded edition of this standard textbook continues to consistently follow the approach of guiding the reader from a solid theoretical foundation directly to practically proven methods - from derivation through analysis to implementation issues. This makes the book appealing to both mathematicians and natural scientists as well as engineers. The textbook is suitable as a companion for lectures for students as well as for self-study for professionals in the natural sciences. It only requires basic knowledge of analysis (corresponding to the Higher Mathematics lecture for physicists and engineers) and numerical mathematics (introductory lecture).