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Solving Nonlinear Equations with Newton s Method

Solving Nonlinear Equations with Newton s Method Author C. T. Kelley
ISBN-10 0898718899
Release 2003
Pages 104
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This book on Newton's method is a user-oriented guide to algorithms and implementation. In just over 100 pages, it shows, via algorithms in pseudocode, in MATLAB, and with several examples, how one can choose an appropriate Newton-type method for a given problem, diagnose problems, and write an efficient solver or apply one written by others. It contains trouble-shooting guides to the major algorithms, their most common failure modes, and the likely causes of failure. It also includes many worked-out examples (available on the SIAM website) in pseudocode and a collection of MATLAB codes, allowing readers to experiment with the algorithms easily and implement them in other languages.

Solving Nonlinear Equations with Newton s Method

Solving Nonlinear Equations with Newton s Method Author C. T. Kelley
ISBN-10 9780898715460
Release 2003-01-01
Pages 104
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Contains trouble-shooting guides to the major algorithms for Newton's method, their common failure modes, and the likely causes of failure.

Solving Nonlinear Equations with Newton s Method

Solving Nonlinear Equations with Newton s Method Author C. T. Kelley
ISBN-10 0898715466
Release 2003-01
Pages 104
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Contains trouble-shooting guides to the major algorithms for Newton's method, their common failure modes, and the likely causes of failure.

Iterative Methods for Linear and Nonlinear Equations

Iterative Methods for Linear and Nonlinear Equations Author C. T. Kelley
ISBN-10 1611970946
Release 1995
Pages 166
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Linear and nonlinear systems of equations are the basis for many, if not most, of the models of phenomena in science and engineering, and their efficient numerical solution is critical to progress in these areas. This is the first book to be published on nonlinear equations since the mid-1980s. Although it stresses recent developments in this area, such as Newton-Krylov methods, considerable material on linear equations has been incorporated. This book focuses on a small number of methods and treats them in depth. The author provides a complete analysis of the conjugate gradient and generalized minimum residual iterations as well as recent advances including Newton-Krylov methods, incorporation of inexactness and noise into the analysis, new proofs and implementations of Broyden's method, and globalization of inexact Newton methods. Examples, methods, and algorithmic choices are based on applications to infinite dimensional problems such as partial differential equations and integral equations. The analysis and proof techniques are constructed with the infinite dimensional setting in mind and the computational examples and exercises are based on the MATLAB environment.

Optimization and Nonlinear Equations

Optimization and Nonlinear Equations Author L. T. Watson
ISBN-10 0444505997
Release 2001
Pages 371
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/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! In one of the papers in this collection, the remark that "nothing at all takes place in the universe in which some rule of maximum of minimum does not appear" is attributed to no less an authority than Euler. Simplifying the syntax a little, we might paraphrase this as Everything is an optimization problem. While this might be something of an overstatement, the element of exaggeration is certainly reduced if we consider the extended form: Everything is an optimization problem or a system of equations. This observation, even if only partly true, stands as a fitting testimonial to the importance of the work covered by this volume. Since the 1960s, much effort has gone into the development and application of numerical algorithms for solving problems in the two areas of optimization and systems of equations. As a result, many different ideas have been proposed for dealing efficiently with (for example) severe nonlinearities and/or very large numbers of variables. Libraries of powerful software now embody the most successful of these ideas, and one objective of this volume is to assist potential users in choosing appropriate software for the problems they need to solve. More generally, however, these collected review articles are intended to provide both researchers and practitioners with snapshots of the 'state-of-the-art' with regard to algorithms for particular classes of problem. These snapshots are meant to have the virtues of immediacy through the inclusion of very recent ideas, but they also have sufficient depth of field to show how ideas have developed and how today's research questions have grown out of previous solution attempts. The most efficient methods for local optimization, both unconstrained and constrained, are still derived from the classical Newton approach. As well as dealing in depth with the various classical, or neo-classical, approaches, the selection of papers on optimization in this volume ensures that newer ideas are also well represented. Solving nonlinear algebraic systems of equations is closely related to optimization. The two are not completely equivalent, however, and usually something is lost in the translation. Algorithms for nonlinear equations can be roughly classified as locally convergent or globally convergent. The characterization is not perfect. Locally convergent algorithms include Newton's method, modern quasi-Newton variants of Newton's method, and trust region methods. All of these approaches are well represented in this volume.

Numerical Methods for Unconstrained Optimization and Nonlinear Equations

Numerical Methods for Unconstrained Optimization and Nonlinear Equations Author J. E. Dennis, Jr.
ISBN-10 1611971209
Release 1996-12-01
Pages 378
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This book has become the standard for a complete, state-of-the-art description of the methods for unconstrained optimization and systems of nonlinear equations. Originally published in 1983, it provides information needed to understand both the theory and the practice of these methods and provides pseudocode for the problems. The algorithms covered are all based on Newton's method or "quasi-Newton" methods, and the heart of the book is the material on computational methods for multidimensional unconstrained optimization and nonlinear equation problems. The republication of this book by SIAM is driven by a continuing demand for specific and sound advice on how to solve real problems. The level of presentation is consistent throughout, with a good mix of examples and theory, making it a valuable text at both the graduate and undergraduate level. It has been praised as excellent for courses with approximately the same name as the book title and would also be useful as a supplemental text for a nonlinear programming or a numerical analysis course. Many exercises are provided to illustrate and develop the ideas in the text. A large appendix provides a mechanism for class projects and a reference for readers who want the details of the algorithms. Practitioners may use this book for self-study and reference. For complete understanding, readers should have a background in calculus and linear algebra. The book does contain background material in multivariable calculus and numerical linear algebra.

Iterative Methods for Optimization

Iterative Methods for Optimization Author C. T. Kelley
ISBN-10 161197092X
Release 1999
Pages 180
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This book presents a carefully selected group of methods for unconstrained and bound constrained optimization problems and analyzes them in depth both theoretically and algorithmically. It focuses on clarity in algorithmic description and analysis rather than generality, and while it provides pointers to the literature for the most general theoretical results and robust software, the author thinks it is more important that readers have a complete understanding of special cases that convey essential ideas. A companion to Kelley's book, Iterative Methods for Linear and Nonlinear Equations (SIAM, 1995), this book contains many exercises and examples and can be used as a text, a tutorial for self-study, or a reference. Iterative Methods for Optimization does more than cover traditional gradient-based optimization: it is the first book to treat sampling methods, including the Hooke-Jeeves, implicit filtering, MDS, and Nelder-Mead schemes in a unified way, and also the first book to make connections between sampling methods and the traditional gradient-methods. Each of the main algorithms in the text is described in pseudocode, and a collection of MATLAB codes is available. Thus, readers can experiment with the algorithms in an easy way as well as implement them in other languages.

Isaac Newton

Isaac Newton Author
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Isaac Newton has been writing in one form or another for most of life. You can find so many inspiration from Isaac Newton also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Isaac Newton book for free.

Hierarchical Matrices

Hierarchical Matrices Author Mario Bebendorf
ISBN-10 9783540771470
Release 2008-06-25
Pages 296
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Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.

Numerical Analysis

Numerical Analysis Author Walter Gautschi
ISBN-10 9780817682590
Release 2011-12-07
Pages 588
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Revised and updated, this second edition of Walter Gautschi's successful Numerical Analysis explores computational methods for problems arising in the areas of classical analysis, approximation theory, and ordinary differential equations, among others. Topics included in the book are presented with a view toward stressing basic principles and maintaining simplicity and teachability as far as possible, while subjects requiring a higher level of technicality are referenced in detailed bibliographic notes at the end of each chapter. Readers are thus given the guidance and opportunity to pursue advanced modern topics in more depth. Along with updated references, new biographical notes, and enhanced notational clarity, this second edition includes the expansion of an already large collection of exercises and assignments, both the kind that deal with theoretical and practical aspects of the subject and those requiring machine computation and the use of mathematical software. Perhaps most notably, the edition also comes with a complete solutions manual, carefully developed and polished by the author, which will serve as an exceptionally valuable resource for instructors.

Iterative Solution of Nonlinear Equations in Several Variables

Iterative Solution of Nonlinear Equations in Several Variables Author J. M. Ortega
ISBN-10 9781483276724
Release 2014-05-10
Pages 592
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Computer Science and Applied Mathematics: Iterative Solution of Nonlinear Equations in Several Variables presents a survey of the basic theoretical results about nonlinear equations in n dimensions and analysis of the major iterative methods for their numerical solution. This book discusses the gradient mappings and minimization, contractions and the continuation property, and degree of a mapping. The general iterative and minimization methods, rates of convergence, and one-step stationary and multistep methods are also elaborated. This text likewise covers the contractions and nonlinear majorants, convergence under partial ordering, and convergence of minimization methods. This publication is a good reference for specialists and readers with an extensive functional analysis background.

Domain Decomposition Methods in Science and Engineering XXII

Domain Decomposition Methods in Science and Engineering XXII Author Thomas Dickopf
ISBN-10 9783319188270
Release 2016-03-11
Pages 647
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These are the proceedings of the 22nd International Conference on Domain Decomposition Methods, which was held in Lugano, Switzerland. With 172 participants from over 24 countries, this conference continued a long-standing tradition of internationally oriented meetings on Domain Decomposition Methods. The book features a well-balanced mix of established and new topics, such as the manifold theory of Schwarz Methods, Isogeometric Analysis, Discontinuous Galerkin Methods, exploitation of modern HPC architectures and industrial applications. As the conference program reflects, the growing capabilities in terms of theory and available hardware allow increasingly complex non-linear and multi-physics simulations, confirming the tremendous potential and flexibility of the domain decomposition concept.

Programming for Computations Python

Programming for Computations   Python Author Svein Linge
ISBN-10 9783319324289
Release 2016-07-25
Pages 232
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This book presents computer programming as a key method for solving mathematical problems. There are two versions of the book, one for MATLAB and one for Python. The book was inspired by the Springer book TCSE 6: A Primer on Scientific Programming with Python (by Langtangen), but the style is more accessible and concise, in keeping with the needs of engineering students. The book outlines the shortest possible path from no previous experience with programming to a set of skills that allows the students to write simple programs for solving common mathematical problems with numerical methods in engineering and science courses. The emphasis is on generic algorithms, clean design of programs, use of functions, and automatic tests for verification.

Haar Wavelets

Haar Wavelets Author Ülo Lepik
ISBN-10 9783319042954
Release 2014-01-09
Pages 207
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This is the first book to present a systematic review of applications of the Haar wavelet method for solving Calculus and Structural Mechanics problems. Haar wavelet-based solutions for a wide range of problems, such as various differential and integral equations, fractional equations, optimal control theory, buckling, bending and vibrations of elastic beams are considered. Numerical examples demonstrating the efficiency and accuracy of the Haar method are provided for all solutions.

Fundamentals of Matrix Computations

Fundamentals of Matrix Computations Author David S. Watkins
ISBN-10 9780471461678
Release 2004-08-27
Pages 640
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A significantly revised and improved introduction to a critical aspect of scientific computation Matrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understanding of the topic is essential. Fundamentals of Matrix Computations, Second Edition explains matrix computations and the accompanying theory clearly and in detail, along with useful insights. This Second Edition of a popular text has now been revised and improved to appeal to the needs of practicing scientists and graduate and advanced undergraduate students. New to this edition is the use of MATLAB for many of the exercises and examples, although the Fortran exercises in the First Edition have been kept for those who want to use them. This new edition includes: * Numerous examples and exercises on applications including electrical circuits, elasticity (mass-spring systems), and simple partial differential equations * Early introduction of the singular value decomposition * A new chapter on iterative methods, including the powerful preconditioned conjugate-gradient method for solving symmetric, positive definite systems * An introduction to new methods for solving large, sparse eigenvalue problems including the popular implicitly-restarted Arnoldi and Jacobi-Davidson methods With in-depth discussions of such other topics as modern componentwise error analysis, reorthogonalization, and rank-one updates of the QR decomposition, Fundamentals of Matrix Computations, Second Edition will prove to be a versatile companion to novice and practicing mathematicians who seek mastery of matrix computation.

Using Additional Information in Streaming Algorithms

Using Additional Information in Streaming Algorithms Author Raffael Buff
ISBN-10 9783961160426
Release 2016-10-04
Pages 125
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Streaming problems are algorithmic problems that are mainly characterized by their massive input streams. Because of these data streams, the algorithms for these problems are forced to be space-efficient, as the input stream length generally exceeds the available storage. In this thesis, the two streaming problems most frequent item and number of distinct items are studied in detail relating to their algorithmic complexities, and it is compared whether the verification of solution hypotheses has lower algorithmic complexity than computing a solution from the data stream. For this analysis, we introduce some concepts to prove space complexity lower bounds for an approximative setting and for hypothesis verification. For the most frequent item problem which consists in identifying the item which has the highest occurrence within the data stream, we can prove a linear space complexity lower bound for the deterministic and probabilistic setting. This implies that, in practice, this streaming problem cannot be solved in a satisfactory way since every algorithm has to exceed any reasonable storage limit. For some settings, the upper and lower bounds are almost tight, which implies that we have designed an almost optimal algorithm. Even for small approximation ratios, we can prove a linear lower bound, but not for larger ones. Nevertheless, we are not able to design an algorithm that solves the most frequent item problem space-efficiently for large approximation ratios. Furthermore, if we want to verify whether a hypothesis of the highest frequency count is true or not, we get exactly the same space complexity lower bounds, which leads to the conclusion that we are likely not able to profit from a stated hypothesis. The number of distinct items problem counts all different elements of the input stream. If we want to solve this problem exactly (in a deterministic or probabilistic setting) or approximately with a deterministic algorithm, we require once again linear storage size which is tight to the upper bound. However, for the approximative and probabilistic setting, we can enhance an already known space-efficient algorithm such that it is usable for arbitrarily small approximation ratios and arbitrarily good success probabilities. The hypothesis verification leads once again to the same lower bounds. However, there are some streaming problems that are able to profit from additional information such as hypotheses, as e.g., the median problem.

Data Assimilation Methods Algorithms and Applications

Data Assimilation  Methods  Algorithms  and Applications Author Mark Asch
ISBN-10 9781611974539
Release 2016-12-29
Pages 306
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Data assimilation is an approach that combines observations and model output, with the objective of improving the latter. This book places data assimilation into the broader context of inverse problems and the theory, methods, and algorithms that are used for their solution. It provides a framework for, and insight into, the inverse problem nature of data assimilation, emphasizing ?why? and not just ?how.? Methods and diagnostics are emphasized, enabling readers to readily apply them to their own field of study. Readers will find a comprehensive guide that is accessible to nonexperts; numerous examples and diverse applications from a broad range of domains, including geophysics and geophysical flows, environmental acoustics, medical imaging, mechanical and biomedical engineering, economics and finance, and traffic control and urban planning; and the latest methods for advanced data assimilation, combining variational and statistical approaches.