Computational Integration is the first book in over 10 years dedicated to a comprehensive discussion of computational integration methods and the fundamental mathematical principles they are based on. It gives special coverage of many recent developments, such as parallel integration algorithms, that have not yet been coherently presented in any other textbook. It also attempts to bridge the gap between theoretical knowledge about and practical application of computational integration by providing references to and descriptions of numerous relevant software products. This survey covers a wide range of topics fundamental to calculating integrals on computer systems and discusses both the theoretical and computational aspects of numerical and symbolic methods. It includes extensive sections on one- and multidimensional integration formulas, like polynomial, number-theoretic, and pseudorandom formulas, and deals with issues concerning the construction of numerical integration algorithms, including error estimation, discretization refinement, and convergence acceleration. Individual chapters are dedicated to parallel integration algorithms, the assessment of numerical integration software, and numerical methods tailored to particular kinds of integration problems involving, for instance, oscillatory or singular integrands.
Preface Chapter 1: Mathematical Foundations of Integration. Riemann Integrals Improper Integrals Cauchy Principal Value Integrals Hadamard Finite Part Integrals Curve and Surface Integrals Chapter 2: Computational Integration in Practice. Computational Statistics Integral Transforms Finite Element Methods Boundary Integral Methods Chapter 3: Fundamentals of Computational Integration. Integration Problems Problem Settings Integration Methods The Conditioning of Integration Problems Software for Computational Integration The Preprocessing of Integrals The Postprocessing of Integrals Chapter 4: Symbolic Integration. Representations and Operations in Algebraic Computation The Problem of Symbolic Integration Integration of Rational Functions Integration of Elementary Functions Integration of Nonelementary Functions Definite Integration Symbolic Methods for Preprocessing Integration Problems Chapter 5: Univariate Integration Formulas. Construction of Quadrature Formulas Simple Interpolatory Quadrature Formulas Compound Quadrature Formulas Chapter 6: Multivariate Integration Formulas. Construction of Cubature Formulas Polynomial Formulas Number-Theoretic Formulas Pseudorandom Formulas Lattice Rules Miscellaneous Formulas Chapter 7: Methods for Special Integration Problems. Oscillatory Integrals on Bounded Regions Integrals on Unbounded Regions Weakly Singular Integrals Cauchy Singular Integrals Finite Part Integrals Chapter 8: Integration Algorithms. Error Estimation Discretization Refinement Special Features of Integration Algorithms Chapter 9: Parallel Numerical Integration. Parallelism in Integration Algorithms Parallelization Schemes for Integration Algorithms Practical Parallelization of Integration Algorithms Chapter 10: Assessment of Numerical Integration Software. Assessment Criteria Assessment Techniques Bibliography Author Index Software Index Subject Index.
