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Author: Philip J. Davis Publisher: Academic Press ISBN: 1483264289 Category : Mathematics Languages : en Pages : 626
Book Description
Methods of Numerical Integration, Second Edition describes the theoretical and practical aspects of major methods of numerical integration. Numerical integration is the study of how the numerical value of an integral can be found. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. A chapter describes the approximate integration in two or more dimensions. The final chapter looks into the goals and processes of automatic integration, with particular attention to the application of Tschebyscheff polynomials. This book will be of great value to theoreticians and computer programmers.
Author: Philip J. Davis Publisher: Academic Press ISBN: 1483264289 Category : Mathematics Languages : en Pages : 626
Book Description
Methods of Numerical Integration, Second Edition describes the theoretical and practical aspects of major methods of numerical integration. Numerical integration is the study of how the numerical value of an integral can be found. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. A chapter describes the approximate integration in two or more dimensions. The final chapter looks into the goals and processes of automatic integration, with particular attention to the application of Tschebyscheff polynomials. This book will be of great value to theoreticians and computer programmers.
Author: Sergio Blanes Publisher: CRC Press ISBN: 1315354861 Category : Mathematics Languages : en Pages : 218
Book Description
Discover How Geometric Integrators Preserve the Main Qualitative Properties of Continuous Dynamical Systems A Concise Introduction to Geometric Numerical Integration presents the main themes, techniques, and applications of geometric integrators for researchers in mathematics, physics, astronomy, and chemistry who are already familiar with numerical tools for solving differential equations. It also offers a bridge from traditional training in the numerical analysis of differential equations to understanding recent, advanced research literature on numerical geometric integration. The book first examines high-order classical integration methods from the structure preservation point of view. It then illustrates how to construct high-order integrators via the composition of basic low-order methods and analyzes the idea of splitting. It next reviews symplectic integrators constructed directly from the theory of generating functions as well as the important category of variational integrators. The authors also explain the relationship between the preservation of the geometric properties of a numerical method and the observed favorable error propagation in long-time integration. The book concludes with an analysis of the applicability of splitting and composition methods to certain classes of partial differential equations, such as the Schrödinger equation and other evolution equations. The motivation of geometric numerical integration is not only to develop numerical methods with improved qualitative behavior but also to provide more accurate long-time integration results than those obtained by general-purpose algorithms. Accessible to researchers and post-graduate students from diverse backgrounds, this introductory book gets readers up to speed on the ideas, methods, and applications of this field. Readers can reproduce the figures and results given in the text using the MATLAB® programs and model files available online.
Author: Gwynne Evans Publisher: ISBN: Category : Mathematics Languages : en Pages : 350
Book Description
Offers the quadrature user a selection of the most effective algorithms in each of the main areas of the subject. Topics range from Simpson's rule and Gaussian quadrature to recent research on irregular oscillatory and singular quadrature. A full set of test examples is given and implemented for each method discussed, demonstrating its practical limitations.
Author: Ernst Hairer Publisher: Springer Science & Business Media ISBN: 3662050188 Category : Mathematics Languages : en Pages : 526
Book Description
This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. 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 numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.
Author: G.N. Milstein Publisher: Springer Science & Business Media ISBN: 9401584559 Category : Computers Languages : en Pages : 178
Book Description
This book is devoted to mean-square and weak approximations of solutions of stochastic differential equations (SDE). These approximations represent two fundamental aspects in the contemporary theory of SDE. Firstly, the construction of numerical methods for such systems is important as the solutions provided serve as characteristics for a number of mathematical physics problems. Secondly, the employment of probability representations together with a Monte Carlo method allows us to reduce the solution of complex multidimensional problems of mathematical physics to the integration of stochastic equations. Along with a general theory of numerical integrations of such systems, both in the mean-square and the weak sense, a number of concrete and sufficiently constructive numerical schemes are considered. Various applications and particularly the approximate calculation of Wiener integrals are also dealt with. This book is of interest to graduate students in the mathematical, physical and engineering sciences, and to specialists whose work involves differential equations, mathematical physics, numerical mathematics, the theory of random processes, estimation and control theory.
Author: Helmut Brass Publisher: American Mathematical Soc. ISBN: 0821853619 Category : Mathematics Languages : en Pages : 376
Book Description
Every book on numerical analysis covers methods for the approximate calculation of definite integrals. The authors of this book provide a complementary treatment of the topic by presenting a coherent theory of quadrature methods that encompasses many deep and elegant results as well as a large number of interesting (solved and open) problems. The inclusion of the word ``theory'' in the title highlights the authors' emphasis on analytical questions, such as the existence and structure of quadrature methods and selection criteria based on strict error bounds for quadrature rules. Systematic analyses of this kind rely on certain properties of the integrand, called ``co-observations,'' which form the central organizing principle for the authors' theory, and distinguish their book from other texts on numerical integration. A wide variety of co-observations are examined, as a detailed understanding of these is useful for solving problems in practical contexts. While quadrature theory is often viewed as a branch of numerical analysis, its influence extends much further. It has been the starting point of many far-reaching generalizations in various directions, as well as a testing ground for new ideas and concepts. The material in this book should be accessible to anyone who has taken the standard undergraduate courses in linear algebra, advanced calculus, and real analysis.
Author: Anthony Ralston Publisher: Courier Corporation ISBN: 9780486414546 Category : Mathematics Languages : en Pages : 644
Book Description
Outstanding text, oriented toward computer solutions, stresses errors in methods and computational efficiency. Problems — some strictly mathematical, others requiring a computer — appear at the end of each chapter.
Author: Anne Greenbaum Publisher: Princeton University Press ISBN: 1400842670 Category : Mathematics Languages : en Pages : 471
Book Description
A rigorous and comprehensive introduction to numerical analysis Numerical Methods provides a clear and concise exploration of standard numerical analysis topics, as well as nontraditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. Filled with appealing examples that will motivate students, the textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering. Exercises use MATLAB and promote understanding of computational results. The book gives instructors the flexibility to emphasize different aspects—design, analysis, or computer implementation—of numerical algorithms, depending on the background and interests of students. Designed for upper-division undergraduates in mathematics or computer science classes, the textbook assumes that students have prior knowledge of linear algebra and calculus, although these topics are reviewed in the text. Short discussions of the history of numerical methods are interspersed throughout the chapters. The book also includes polynomial interpolation at Chebyshev points, use of the MATLAB package Chebfun, and a section on the fast Fourier transform. Supplementary materials are available online. Clear and concise exposition of standard numerical analysis topics Explores nontraditional topics, such as mathematical modeling and Monte Carlo methods Covers modern applications, including information retrieval and animation, and classical applications from physics and engineering Promotes understanding of computational results through MATLAB exercises Provides flexibility so instructors can emphasize mathematical or applied/computational aspects of numerical methods or a combination Includes recent results on polynomial interpolation at Chebyshev points and use of the MATLAB package Chebfun Short discussions of the history of numerical methods interspersed throughout Supplementary materials available online
Author: Philip J. Davis
Author: Philip J. Davis
Author: Sergio Blanes
Author: Gwynne Evans
Author: Ernst Hairer
Author: Philip J. Davis
Author: G.N. Milstein
Author: Helmut Brass
Author: Anthony Ralston
Author: Forman S. Acton
Author: Anne Greenbaum