Auto-Validating Numerical Methods

Introduction // 1 Computer arithmetic: 1.1 Positional systems/ 1.2 Floating point numbers / 1.3 Rounding / 1.4 Floating point arithmetic / 1.5 The IEEE standards / 1.6 Examples of floating point computations / 1.7 Computer Lab I // 2 Interval arithmetic: 2.1 Real intervals / 2.2 Real interval arithmetic / 2.3 Extended interval arithmetic / 2.4 Containment sets // 3 Interval analysis: 3.1 Interval functions / 3.2 Centered forms / 3.3 Monotonicity / 3.4 Computer Lab II // 4 Automatic differentiation: 4.1 First-order derivatives / 4.2 Higher-order derivatives / 4.3 Higher-order enclosures / Computer Lab III // 5 Interval analysis in action: 5.1 Root finding / 5.2 Optimization / 5.3 Quadrature / 5.4 Computer Lab IV // 6 Ordinary differential equations: 6.1 A gentle mathematical introduction / 6.2 Simple enclosure methods / 6.3 Higher-order methods / 6.4 Rigorous higher-order examples // Appendices: A Mathematical foundations: A.1 The rational numbers / A.2 What is a real number? / A.3 Completeness / A.4 Fixed point theorems // B Program codes: B.1 IEEE constants / B.2 Changing rounding modes / B.3 IA sample code in C / B.4 IA sample code in C++ // Bibliography