Download or read book Iterative Methods for Solving Nonlinear Equations and Systems written by Juan R. Torregrosa and published by MDPI. This book was released on 2019-12-06 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solving nonlinear equations in Banach spaces (real or complex nonlinear equations, nonlinear systems, and nonlinear matrix equations, among others), is a non-trivial task that involves many areas of science and technology. Usually the solution is not directly affordable and require an approach using iterative algorithms. This Special Issue focuses mainly on the design, analysis of convergence, and stability of new schemes for solving nonlinear problems and their application to practical problems. Included papers study the following topics: Methods for finding simple or multiple roots either with or without derivatives, iterative methods for approximating different generalized inverses, real or complex dynamics associated to the rational functions resulting from the application of an iterative method on a polynomial. Additionally, the analysis of the convergence has been carried out by means of different sufficient conditions assuring the local, semilocal, or global convergence. This Special issue has allowed us to present the latest research results in the area of iterative processes for solving nonlinear equations as well as systems and matrix equations. In addition to the theoretical papers, several manuscripts on signal processing, nonlinear integral equations, or partial differential equations, reveal the connection between iterative methods and other branches of science and engineering.
Download or read book Iterative Methods for Linear and Nonlinear Equations written by C. T. Kelley and published by SIAM. This book was released on 1995-01-01 with total page 179 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Iterative Solution of Nonlinear Equations in Several Variables written by J. M. Ortega and published by Elsevier. This book was released on 2014-05-10 with total page 593 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Iterative Solution of Nonlinear Systems of Equations written by R. Ansorge and published by Springer. This book was released on 2006-11-15 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: a
Download or read book Methods for Solving Systems of Nonlinear Equations written by Werner C. Rheinboldt and published by SIAM. This book was released on 1998-01-01 with total page 157 pages. Available in PDF, EPUB and Kindle. Book excerpt: This second edition provides much-needed updates to the original volume. Like the first edition, it emphasizes the ideas behind the algorithms as well as their theoretical foundations and properties, rather than focusing strictly on computational details; at the same time, this new version is now largely self-contained and includes essential proofs. Additions have been made to almost every chapter, including an introduction to the theory of inexact Newton methods, a basic theory of continuation methods in the setting of differentiable manifolds, and an expanded discussion of minimization methods. New information on parametrized equations and continuation incorporates research since the first edition.
Download or read book Iterative Methods for Sparse Linear Systems written by Yousef Saad and published by SIAM. This book was released on 2003-04-01 with total page 537 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- General.
Download or read book Solving Nonlinear Equations with Newton s Method written by C. T. Kelley and published by SIAM. This book was released on 2003-01-01 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Convergence of Iterative Methods Applied to Large Overdetermined Linear and Nonlinear Systems of Equations Using Least Squares written by Charles O. Stearns and published by . This book was released on 1970 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solutions are obtained to large overdetermined systems of equations. Both nonlinear and linear systems are considered. The nonlinear system represents a dipole model of the earth's geomagnetic field, which is generated from spherical harmonic coefficients. This system of 64 unknowns and 1836 equations is solved by a maximum neighborhood method, which is an optimum interpolation between the well known Taylor's series and steepest descent methods. The original given values of the generated field are as large as 60,000 gamma, whereas a rms residual of 27.9 gamma is obtained with 173 iterations. The linear system of equations represents dipole changes required to account for the earth's secular change field which is generated from spherical harmonic coefficients. The dipole parameters computed from the nonlinear model are used as input parameters. The system contains 64 unknowns and 612 equations and is solved using a Chebyshev polynomial iterative method. These results are compared to results obtained by a direct solution of the normal equations of the system and results obtained by a pseudo-inverse method using a modified Gram-Schmidt factorization. Although the latter two methods give smaller rms values than the iterative method, the results of the iterative method are more reasonable in view of known properties of the results. The generated field has a rms value of 45 gamma per year. An rms residual of 2.5 gamma per year was obtained after 25,000 iterations.
Download or read book Iterative Solution of Nonlinear Systems of Equations written by and published by . This book was released on 1982 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Iterative Solution of Large Sparse Systems of Equations written by Wolfgang Hackbusch and published by Springer. This book was released on 1993-12-13 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt: C. F. GauS in a letter from Dec. 26, 1823 to Gerling: 3c~ empfe~le 3~nen biegen IDlobu9 aur 9tac~a~mung. ec~werlic~ werben eie ie wieber bi reet eliminiren, wenigftens nic~t, wenn eie me~r als 2 Unbefannte ~aben. :Da9 inbirecte 93erfa~ren 109st sic~ ~alb im ec~lafe ausfii~ren, ober man fann wo~renb be9gelben an anbere :Dinge benfen. [CO F. GauS: Werke vol. 9, Gottingen, p. 280, 1903] What difference exists between solving large and small systems of equations? The standard methods well-known to any student oflinear algebra are appli cable to all systems, whether large or small. The necessary amount of work, however, increases dramatically with the size, so one has to search for algo rithms that most efficiently and accurately solve systems of 1000, 10,000, or even one million equations. The choice of algorithms depends on the special properties the matrices in practice have. An important class of large systems arises from the discretisation of partial differential equations. In this case, the matrices are sparse (i. e. , they contain mostly zeros) and well-suited to iterative algorithms. Because of the background in partial differential equa tions, this book is closely connected with the author's Theory and Numerical Treatment of Elliptic Differential Equations, whose English translation has also been published by Springer-Verlag. This book grew out of a series of lectures given by the author at the Christian-Albrecht University of Kiel to students of mathematics.
Download or read book Iterative Methods and Preconditioners for Systems of Linear Equations written by Gabriele Ciaramella and published by SIAM. This book was released on 2022-02-08 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative methods use successive approximations to obtain more accurate solutions. This book gives an introduction to iterative methods and preconditioning for solving discretized elliptic partial differential equations and optimal control problems governed by the Laplace equation, for which the use of matrix-free procedures is crucial. All methods are explained and analyzed starting from the historical ideas of the inventors, which are often quoted from their seminal works. Iterative Methods and Preconditioners for Systems of Linear Equations grew out of a set of lecture notes that were improved and enriched over time, resulting in a clear focus for the teaching methodology, which derives complete convergence estimates for all methods, illustrates and provides MATLAB codes for all methods, and studies and tests all preconditioners first as stationary iterative solvers. This textbook is appropriate for undergraduate and graduate students who want an overview or deeper understanding of iterative methods. Its focus on both analysis and numerical experiments allows the material to be taught with very little preparation, since all the arguments are self-contained, and makes it appropriate for self-study as well. It can be used in courses on iterative methods, Krylov methods and preconditioners, and numerical optimal control. Scientists and engineers interested in new topics and applications will also find the text useful.
Download or read book Iterative Methods for Linear Systems written by Maxim A. Olshanskii and published by SIAM. This book was released on 2014-07-21 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative Methods for Linear Systems?offers a mathematically rigorous introduction to fundamental iterative methods for systems of linear algebraic equations. The book distinguishes itself from other texts on the topic by providing a straightforward yet comprehensive analysis of the Krylov subspace methods, approaching the development and analysis of algorithms from various algorithmic and mathematical perspectives, and going beyond the standard description of iterative methods by connecting them in a natural way to the idea of preconditioning.??
Download or read book Iterative Solution of Nonlinear Systems of Equations written by R. Ansorge and published by . This book was released on 2014-01-15 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Iterative Methods for Solving Linear Systems written by Anne Greenbaum and published by SIAM. This book was released on 1997-01-01 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Computing -- Numerical Analysis.
Download or read book Iterative Methods for Sparse Linear Systems written by Yousef Saad and published by SIAM. This book was released on 2003-01-01 with total page 546 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the first edition of this book was published in 1996, tremendous progress has been made in the scientific and engineering disciplines regarding the use of iterative methods for linear systems. The size and complexity of the new generation of linear and nonlinear systems arising in typical applications has grown. Solving the three-dimensional models of these problems using direct solvers is no longer effective. At the same time, parallel computing has penetrated these application areas as it became less expensive and standardized. Iterative methods are easier than direct solvers to implement on parallel computers but require approaches and solution algorithms that are different from classical methods. Iterative Methods for Sparse Linear Systems, Second Edition gives an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations. These equations can number in the millions and are sparse in the sense that each involves only a small number of unknowns. The methods described are iterative, i.e., they provide sequences of approximations that will converge to the solution.
Download or read book Iterative Solution of Large Linear Systems written by David M. Young and published by Elsevier. This book was released on 2014-05-10 with total page 599 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iterative Solution of Large Linear Systems describes the systematic development of a substantial portion of the theory of iterative methods for solving large linear systems, with emphasis on practical techniques. The focal point of the book is an analysis of the convergence properties of the successive overrelaxation (SOR) method as applied to a linear system where the matrix is "consistently ordered". Comprised of 18 chapters, this volume begins by showing how the solution of a certain partial differential equation by finite difference methods leads to a large linear system with a sparse matrix. The next chapter reviews matrix theory and the properties of matrices, as well as several theorems of matrix theory without proof. A number of iterative methods, including the SOR method, are then considered. Convergence theorems are also given for various iterative methods under certain assumptions on the matrix A of the system. Subsequent chapters deal with the eigenvalues of the SOR method for consistently ordered matrices; the optimum relaxation factor; nonstationary linear iterative methods; and semi-iterative methods. This book will be of interest to students and practitioners in the fields of computer science and applied mathematics.
Download or read book Programming for Computations MATLAB Octave written by Svein Linge and published by Springer. This book was released on 2016-08-01 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
