C++ Toolbox for verified computing

This book offers a general discussion on arithmetic and computational reliability, analytical mathematics and verification techniques, algorithms, and (most importantly) actual C++ implementations. In each chapter, examples, exercises, and numerical results demonstrate the application of the routines presented. The book introduces many computational verification techniques. It is not assumed that the reader has any prior formal knowledge of numerical verfication or any familiarity with interval analysis. The necessary concepts are introduced.

Part I of this book contains introductory chapters on the features of the c-xsc library, a C++ class library for eXtended Scientific Computing, and on the basics and terminology of interval arithmetic. Part II addresses one-dimensional problems: evaluation of polynomials and general arithmetic expressions, nonlinear root-finding, automatic differentiation, and optimization. Part 111 presents routines to solve multi-dimensional problems such as linear and nonlinear systems of equations, linear and global optimization, and automatic differentiation for gradients, Hessians, and jacobians.

Some of the subjects that the book covers in detail are not usually found in standard numerical analysis texts. This book is intended primarily as a reference text for anyone wishing to apply, modify, or develop routines to obtain mathematically certain and reliable results. It can also be used as a textbook for an advanced course in scientific computation with automatic result verification. Last but not least this book is particularly useful for those programmers who have already worked with the PASCALxsc edition but have only little knowledge in the C++ language. For those readers the book may be a source of inspiration when switching from PASCAL tO C++.

Table of Contents

• Introduction
• Preliminaries
• The Features of C-XSC
• Mathematical Preliminaries
• One -Dimensional Problems
• Evaluation of Polynomials
• Automatic Differentiation
• Nonlinear Equations in One Variable
• Global Optimization
• Evaluation of Arithmetic Expression
• Zeros of Complex Polynomials
• Multi -Dimensional Problems
• Linear Systems of Equations
• Automatic Differentiation for Gradients, Jacobians, and Hessians
• Nonlinear Systems of Equations
• Global Optimization
• Appendix
• Index