Extrapolation Methods: Theory and Practice.

Abstract

Introduction to the Theory. First Steps. What is an Extrapolation Method? What is an Extrapolation Algorithm? Quasi-linear Sequence Transformations. Sequence Transformations as Ratios of Determinants. Triangular Recursive Schemes. Normal Forms of the Algorithms. Progressive Forms of the Algorithms. Particular Rules of the Algorithms. Accelerability and Non-accelerability. Optimality. Asymptotic Behaviour of Sequences. Scalar Extrapolation Algorithms. The E-algorithm. Richardson Extrapolation Process. The -algorithm. The G-transformation. Rational Extrapolation. Generalizations of the -algorithm. Levin's Transformations. Overholt's Process. -type Algorithms. The Iterated 2 Process. Miscellaneous Algorithms. Special Devices. Error Estimates and Acceleration. Convergence Tests and Acceleration. Construction of Asymptotic Expansions. Construction of Extrapolation Processes. Extraction Procedures. Automatic Selection. Composite Sequence Transformations. Error Control. Contractive Sequence Transformations. Least Squares Extrapolation. Vector Extrapolation Algorithms. The Vector -algorithm. The Topological -algorithm. The Vector E-algorithm. The Recursive Projection Algorithm. The H-algorithm. The Ford-Sidi Algorithms. Miscellaneous Algorithms. Continuous Prediction Algorithms. The Taylor Expansion. Confluent Overholt's process. Confluent -algorithms. Confluent -algorithm. Confluent G-transform. Confluent E-algorithm. -type Confluent Algorithms. Applications. Sequences and Series: Simple Sequences, Double Sequences, Chebyshev and Fourier Series, Continued Fractions, Vector Sequences. Systems of Equations: Linear Systems, Projection Methods, Regularization and Penalty Techniques, Nonlinear Equations, Continuation Methods. Eigenelements: Eigenvalues and eigenvectors, Derivatives of Eigensystems. Integral and Differential Equations: Implicit Runge-Kutta Methods, Boundary Value Problems, Nonlinear Methods, Laplace Transform Inversion, Partial Differential Equations. Interpolation and Approximation. Statistics: The Jackknife, ARMA Models, Monte-Carlo Methods. Integration and Differentiation: Acceleration of Quadrature Formulae, Nonlinear Quadrature Formulae, Cauchy's Principal Values, Infinite Integrals, Multiple Integrals, Numerical Differentiation. Prediction. Software. Programming the Algorithms. Computer Arithmetic. Programs. Bibliography. Index.