Mathematical Analysis and the Mathematics of Computation

Authors: Werner Römisch, Thomas Zeugmann



Many interdisciplinary applications, including support vector machines, kernel-based learning methods, pattern recognition, statistical learning theory, computer graphics, and approximation theory

Rich field experienced enormous development in recent decades, comprising numerical analysis and computational discrete mathematics

Book content tested in lectures, seminars, and discussions at Humboldt University and other institutions over decades

This book is a comprehensive, unifying introduction to the field of mathematical analysis and the mathematics of computing. It develops the relevant theory at a modern level and it directly relates modern mathematical ideas to their diverse applications.

The authors develop the whole theory. Starting with a simple axiom system for the real numbers, they then lay the foundations, developing the theory, exemplifying where it's applicable, in turn motivating further development of the theory. They progress from sets, structures, and numbers to metric spaces, continuous functions in metric spaces, linear normed spaces and linear mappings; and then differential calculus and its applications, the integral calculus, the gamma function, and linear integral operators. They then present important aspects of approximation theory, including numerical integration. The remaining parts of the book are devoted to ordinary differential equations, the discretization of operator equations, and numerical solutions of ordinary differential equations.

This textbook contains many exercises of varying degrees of difficulty, suitable for self-study, and at the end of each chapter the authors present more advanced problems that shed light on interesting features, suitable for classroom seminars or study groups. It will be valuable for undergraduate and graduate students in mathematics, computer science, and related fields such as engineering. This is a rich field that has experienced enormous development in recent decades, and the book will also act as a reference for graduate students and practitioners who require a deeper understanding of the methodologies, techniques, and foundations.

Table of contents (13 chapters, 703 pages)

Front Matter

Sets, Structures, Numbers

Metric Spaces

Continuous Functions in Metric Spaces

Linear Normed Spaces, Linear Operators

The Differential Calculus

Applications of the Differential Calculus

The Integral Calculus

Linear Integral Operators

Inner Product Spaces

Approximative Representation of Functions

Ordinary Differential Equations

Discretization of Operator Equations

Numerical Solution of Ordinary Differential Equations

Back Matter