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Walter Gautschi has written extensively on topics ranging from special functions, quadrature and orthogonal polynomials to difference and differential equations, software implementations, and the history of mathematics. He is world renowned for his pioneering work in numerical analysis and constructive orthogonal polynomials, including a definitive textbook in the former, and a monograph in the latter area.   This three-volume set, Walter Gautschi: Selected Works with Commentaries, is a compilation of Gautschi’s most influential papers and includes commentaries by leading experts. The work begins with a detailed biographical section and ends with a section commemorating Walter’s prematurely deceased twin brother. This title will appeal to graduate students and researchers in numerical analysis, as well as to historians of science.   Selected Works with Commentaries, Vol. 1 Numerical Conditioning Special Functions Interpolation and Approximation   Selected Works with Commentaries, Vol. 2 Orthogonal Polynomials on the Real Line Orthogonal Polynomials on the Semicircle Chebyshev Quadrature Kronrod and Other Quadratures Gauss-type Quadrature   Selected Works with Commentaries, Vol. 3 Linear Difference Equations Ordinary Differential Equations Software History and Biography Miscellanea Works of Werner Gautschi.

Numerical integration. --- Orthogonal polynomials. --- Gautschi, Walter. --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Mathematics. --- Computer science --- Approximation theory. --- Differential equations. --- Numerical analysis. --- History. --- Numerical Analysis. --- Mathematics of Computing. --- Approximations and Expansions. --- History of Mathematical Sciences. --- Ordinary Differential Equations. --- Definite integrals --- Interpolation --- Numerical analysis --- Computer science. --- Differential Equations. --- 517.91 Differential equations --- Differential equations --- Math --- Science --- Informatics --- Mathematical analysis --- Computer science—Mathematics. --- Annals --- Auxiliary sciences of history --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Computer mathematics --- Electronic data processing --- Mathematics --- Chebyshev approximation. --- Gaussian quadrature formulas. --- Mathematicians.

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Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules have increased in popularity, with many fruitful applications in mathematical practice. These rules require nodes with good uniform distribution properties, and digital nets and sequences in the sense of Niederreiter are known to be excellent candidates. Besides the classical theory, the book contains chapters on reproducing kernel Hilbert spaces and weighted integration, duality theory for digital nets, polynomial lattice rules, the newest constructions by Niederreiter and Xing and many more. The authors present an accessible introduction to the subject based mainly on material taught in undergraduate courses with numerous examples, exercises and illustrations.

Monte Carlo method. --- Nets (Mathematics) --- Sequences (Mathematics) --- Numerical integration. --- Digital filters (Mathematics) --- Data smoothing filters --- Filters, Digital (Mathematics) --- Linear digital filters (Mathematics) --- Linear filters (Mathematics) --- Numerical filters --- Smoothing filters (Mathematics) --- Digital electronics --- Filters (Mathematics) --- Fourier transformations --- Functional analysis --- Numerical analysis --- Numerical calculations --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Moore-Smith convergence --- Net equations --- Net methods (Mathematics) --- Convergence --- Set theory --- Topology --- Artificial sampling --- Model sampling --- Monte Carlo simulation --- Monte Carlo simulation method --- Stochastic sampling --- Games of chance (Mathematics) --- Mathematical models --- Stochastic processes

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This monograph is a comprehensive treatment of the theoretical and computational aspects of numerical integration. The authors give a unique overview of the topic by bringing into line many recent research results not yet presented coherently; the extensive bibliography lists 268 items. Particular emphasis is given to the potential parallelism of numerical integration problems and to utilizing it by means of dynamic load distribution techniques. The book discusses the basics and provides methodologies for producing efficient and reliable software for numerical integration on advanced computer systems. The book addresses researchers, graduate students, and computational scientists.

Numerical analysis --- Computer. Automation --- Numerical integration --- -519.6 --- 681.3*G14 --- 681.3*G4 --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Data processing --- Computational mathematics. Numerical analysis. Computer programming --- Quadrature and numerical differentiation: adaptive quadrature; equal intervalintegration; error analysis; finite difference methods; gaussian quadrature; iterated methods; multiple quadrature --- Mathematical software: algorithm analysis; certification and testing; efficiency; portability; reliability and robustness; verification --- 681.3*G4 Mathematical software: algorithm analysis; certification and testing; efficiency; portability; reliability and robustness; verification --- 681.3*G14 Quadrature and numerical differentiation: adaptive quadrature; equal intervalintegration; error analysis; finite difference methods; gaussian quadrature; iterated methods; multiple quadrature --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- 519.6 --- Numerical integration - Data processing. --- Information theory. --- Numerical analysis. --- Algorithms. --- Theory of Computation. --- Numerical Analysis. --- Algorism --- Algebra --- Arithmetic --- Mathematical analysis --- Communication theory --- Communication --- Cybernetics --- Foundations

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Numerical integration --- 519.6 --- 681.3*G14 --- 681.3*G14 Quadrature and numerical differentiation: adaptive quadrature equal intervalintegration error analysis finite difference methods gaussian quadrature iterated methods multiple quadrature --- Quadrature and numerical differentiation: adaptive quadrature equal intervalintegration error analysis finite difference methods gaussian quadrature iterated methods multiple quadrature --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- 681.3*G14 Quadrature and numerical differentiation: adaptive quadrature; equal intervalintegration; error analysis; finite difference methods; gaussian quadrature; iterated methods; multiple quadrature --- Quadrature and numerical differentiation: adaptive quadrature; equal intervalintegration; error analysis; finite difference methods; gaussian quadrature; iterated methods; multiple quadrature --- Numerical integration. --- Integration numerique

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Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.

Numerical integration. --- Hamiltonian systems. --- Differential equations --- Numerical solutions. --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Numerical analysis. --- Biomathematics. --- Physics. --- Numerical Analysis. --- Analysis. --- Theoretical, Mathematical and Computational Physics. --- Mathematical Methods in Physics. --- Numerical and Computational Physics. --- Mathematical and Computational Biology. --- 517.91 Differential equations --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- Integration, Numerical --- Mechanical quadrature --- Quadrature, Mechanical --- Definite integrals --- Interpolation --- Numerical analysis --- Global analysis (Mathematics). --- Mathematical physics. --- Numerical and Computational Physics, Simulation. --- Physical mathematics --- Physics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis --- Mathematics --- Biology --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- 517.1 Mathematical analysis --- Hamiltonian systems --- Differential equations - Numerical solutions --- 517.91 --- Numerical integration --- 519.62 --- 681.3*G17 --- 681.3*G17 Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- Ordinary differential equations: boundary value problems; convergence and stability; error analysis; initial value problems; multistep methods; single step methods; stiff equations (Numerical analysis) --- 519.62 Numerical methods for solution of ordinary differential equations --- Numerical methods for solution of ordinary differential equations --- Numerical solutions