How To Solve Large Linear Systems
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How To Solve Large Linear Systems
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Author : Aleksa Srdanov
language : en
Publisher: Universal-Publishers
Release Date : 2019-12-01
How To Solve Large Linear Systems written by Aleksa Srdanov and has been published by Universal-Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-12-01 with Mathematics categories.
Solving the linear equation system n x n can also be a problem for a computer, even when the number of equations and unknowns is relatively small (a few hundred). All existing methods are burdened by at least one of the following problems: 1) Complexity of computation expressed through the number of operations required to be done to obtaining solution; 2) Unrestricted growth of the size of the intermediate result, which causes overflow and underflow problems; 3) Changing the value of some coefficients in the input system, which causes the instability of the solution; 4) Require certain conditions for convergence, etc. In this paper an approximate and exact methods for solving a system of linear equations with an arbitrary number of equations and the same number of unknowns is presented. All the mentioned problems can be avoided by the proposed methods. It is possible to define an algorithm that does not solve the system of equations in the usual mathematical way, but still finds its exact solution in the exact number of steps already defined. The methods consist of simple computations that are not cumulative. At the same time, the number of operations is acceptable even for a relatively large number of equations and unknowns. In addition, the algorithms allows the process to start from an arbitrary initial n-tuple and always leads to the exact solution if it exists.
How To Solve Large Linear Systems
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Author : Aleksa S. Srdanov
language : en
Publisher:
Release Date : 2019
How To Solve Large Linear Systems written by Aleksa S. Srdanov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with Algorithms categories.
"In this paper an approximate and exact methods for solving a system of linear equations with an arbitrary number of equations and the same number of unknowns is presented. All the mentioned problems can be avoided by the proposed methods. It is possible to define an algorithms that does not solve the system of equations in the usual mathematical way, but still finds its exact solution in the exact number of steps already defined. The methods consists of simple computations that are not cumulative. At the same time, the number of operations is acceptable even for a relatively large number of equations and unknowns. In addition, the algorithms allows the process to start from an arbitrary initial n-tuple and always leads to the exact solution if it exists"--
High Performance Computing For Solving Large Sparse Systems Optical Diffraction Tomography As A Case Of Study
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Author : Gloria Ortega López
language : en
Publisher: Universidad Almería
Release Date : 2015-04-14
High Performance Computing For Solving Large Sparse Systems Optical Diffraction Tomography As A Case Of Study written by Gloria Ortega López and has been published by Universidad Almería this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-04-14 with categories.
This thesis, entitled €High Performance Computing for solving large sparse systems. Optical Diffraction Tomography as a case of study€ investigates the computational issues related to the resolution of linear systems of equations which come from the discretization of physical models described by means of Partial Differential Equations (PDEs). These physical models are conceived for the description of the space-temporary behavior of some physical phenomena f(x, y, z, t) in terms of their variations (partial derivative) with respect to the dependent variables of the phenomena. There is a wide variety of discretization methods for PDEs. Two of the most well-known methods are the Finite Difference Method (FDM) and the Finite Element Method (FEM). Both methods result in an algebraic description of the model that can be translated into the approach of a linear system of equations of type (Ax = b), where A is a sparse matrix (a high percentage of zero elements) whose size depends on the required accuracy of the modeled phenomena. This thesis begins with the algebraic description of the model associated with the physical phenomena, and the work herein has been focused on the design of techniques and computational models that allow the resolution of these linear systems of equations. The main interest of this study is specially focused on models which require a high level of discretization and usually generate sparse matrices, A, which have a highly sparse structure and large size. Literature characterizes these types of problems by their high demanding computational requirements (because of their fine degree of discretization) and the sparsity of the matrices involved, suggesting that these kinds of problems can only be solved using High Performance Computing techniques and architectures. One of the main goals of this thesis is the research of the possible alternatives which allow the implementation of routines to solve large and sparse linear systems of equations using High Performance Computing (HPC). The use of massively parallel platforms (GPUs) allows the acceleration of these routines, because they have several advantages for vectorial computation schemes. On the other hand, the use of distributed memory platforms allows the resolution of problems defined by matrices of enormous size. Finally, the combination of both techniques, distributed computation and multi-GPUs, will allow faster resolution of interesting problems in which large and sparse matrices are involved. In this line, one of the goals of this thesis is to supply the scientific community with implementations based on multi-GPU clusters to solve sparse linear systems of equations, which are the key in many scientific computations. The second part of this thesis is focused on a real physical problem of Optical Diffractional Tomography (ODT) based on holographic information. ODT is a non-damaging technique which allows the extraction of the shapes of objects with high accuracy. Therefore, this technique is very suitable to the in vivo study of real specimens, microorganisms, etc., and it also makes the investigation of their dynamics possible. A preliminary physical model based on a bidimensional reconstruction of the seeding particle distribution in fluids was proposed by J. Lobera and J.M. Coupland. However, its high computational cost (in both memory requirements and runtime) made compulsory the use of HPC techniques to extend the implementation to a three dimensional model. In the second part of this thesis, the implementation and validation of this physical model for the case of three dimensional reconstructions is carried out. In such implementation, the resolution of large and sparse linear systems of equations is required. Thus, some of the algebraic routines developed in the first part of the thesis have been used to implement computational strategies capable of solving the problem of 3D reconstruction based on ODT.
Krylov Solvers For Linear Algebraic Systems
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Author : Charles George Broyden
language : en
Publisher: Elsevier
Release Date : 2004-09-08
Krylov Solvers For Linear Algebraic Systems written by Charles George Broyden and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-09-08 with Mathematics categories.
The first four chapters of this book give a comprehensive and unified theory of the Krylov methods. Many of these are shown to be particular examples ofthe block conjugate-gradient algorithm and it is this observation thatpermits the unification of the theory. The two major sub-classes of thosemethods, the Lanczos and the Hestenes-Stiefel, are developed in parallel asnatural generalisations of the Orthodir (GCR) and Orthomin algorithms. Theseare themselves based on Arnoldi's algorithm and a generalised Gram-Schmidtalgorithm and their properties, in particular their stability properties,are determined by the two matrices that define the block conjugate-gradientalgorithm. These are the matrix of coefficients and the preconditioningmatrix.In Chapter 5 the"transpose-free" algorithms based on the conjugate-gradient squared algorithm are presented while Chapter 6 examines the various ways in which the QMR technique has been exploited. Look-ahead methods and general block methods are dealt with in Chapters 7 and 8 while Chapter 9 is devoted to error analysis of two basic algorithms.In Chapter 10 the results of numerical testing of the more important algorithms in their basic forms (i.e. without look-ahead or preconditioning) are presented and these are related to the structure of the algorithms and the general theory. Graphs illustrating the performances of various algorithm/problem combinations are given via a CD-ROM.Chapter 11, by far the longest, gives a survey of preconditioning techniques. These range from the old idea of polynomial preconditioning via SOR and ILU preconditioning to methods like SpAI, AInv and the multigrid methods that were developed specifically for use with parallel computers. Chapter 12 is devoted to dual algorithms like Orthores and the reverse algorithms of Hegedus. Finally certain ancillary matters like reduction to Hessenberg form, Chebychev polynomials and the companion matrix are described in a series of appendices.·comprehensive and unified approach·up-to-date chapter on preconditioners·complete theory of stability·includes dual and reverse methods·comparison of algorithms on CD-ROM·objective assessment of algorithms
Numerical Solution Of Partial Differential Equations
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Author : Gordon D. Smith
language : en
Publisher: Oxford University Press
Release Date : 1985
Numerical Solution Of Partial Differential Equations written by Gordon D. Smith and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with Computers categories.
Substantially revised, this authoritative study covers the standard finite difference methods of parabolic, hyperbolic, and elliptic equations, and includes the concomitant theoretical work on consistency, stability, and convergence. The new edition includes revised and greatly expanded sections on stability based on the Lax-Richtmeyer definition, the application of Pade approximants to systems of ordinary differential equations for parabolic and hyperbolic equations, and a considerably improved presentation of iterative methods. A fast-paced introduction to numerical methods, this will be a useful volume for students of mathematics and engineering, and for postgraduates and professionals who need a clear, concise grounding in this discipline.
Krylov Methods For Nonsymmetric Linear Systems
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Author : Gérard Meurant
language : en
Publisher: Springer Nature
Release Date : 2020-10-02
Krylov Methods For Nonsymmetric Linear Systems written by Gérard Meurant and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-10-02 with Mathematics categories.
This book aims to give an encyclopedic overview of the state-of-the-art of Krylov subspace iterative methods for solving nonsymmetric systems of algebraic linear equations and to study their mathematical properties. Solving systems of algebraic linear equations is among the most frequent problems in scientific computing; it is used in many disciplines such as physics, engineering, chemistry, biology, and several others. Krylov methods have progressively emerged as the iterative methods with the highest efficiency while being very robust for solving large linear systems; they may be expected to remain so, independent of progress in modern computer-related fields such as parallel and high performance computing. The mathematical properties of the methods are described and analyzed along with their behavior in finite precision arithmetic. A number of numerical examples demonstrate the properties and the behavior of the described methods. Also considered are the methods’ implementations and coding as Matlab®-like functions. Methods which became popular recently are considered in the general framework of Q-OR (quasi-orthogonal )/Q-MR (quasi-minimum) residual methods. This book can be useful for both practitioners and for readers who are more interested in theory. Together with a review of the state-of-the-art, it presents a number of recent theoretical results of the authors, some of them unpublished, as well as a few original algorithms. Some of the derived formulas might be useful for the design of possible new methods or for future analysis. For the more applied user, the book gives an up-to-date overview of the majority of the available Krylov methods for nonsymmetric linear systems, including well-known convergence properties and, as we said above, template codes that can serve as the base for more individualized and elaborate implementations.
Computing The Electrical Activity In The Heart
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Author : Joakim Sundnes
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-06-26
Computing The Electrical Activity In The Heart written by Joakim Sundnes and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-06-26 with Mathematics categories.
The heart is a fantastic machine; during a normal lifetime it beats about 2.5 billion times and pumps 200.000 tons of blood through an enormous system of vessels extending 160.000 kilometres throughout the body. For centuries, man has tried to understand how the heart works, but there remain many unsolved problems, problems that have captured the attention of thousands of researchers worldwide. There is, for example, a huge amount of research being devoted to the analysis of single heart cells. Other areas of research include trying to understand how it works as a complete muscle, and how blood ows through the heart. The entire process is extremely complex. The history of bioelectricity can be traced back to the late eighteenth century and the experiments of Luigi Galvani. A century later, in 1887, Augustus Wallers managed to measure the electrical signal generated by the heart at the surface of the body [142]. His dog Jimmy earned a place in history by being the rst to have his heart measured in this way; see Figure 1.1. In 1903 Willem Einthoven [34] - veloped the rst commercial device for recording electrocardiograms (ECGs); see Figure 1.2.
Numerical Methods For Scientists And Engineers
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Author : Zekeriya Altaç
language : en
Publisher: CRC Press
Release Date : 2024-10-15
Numerical Methods For Scientists And Engineers written by Zekeriya Altaç and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-10-15 with Mathematics categories.
Numerical Methods for Scientists and Engineers: With Pseudocodes is designed as a primary textbook for a one-semester course on Numerical Methods for sophomore or junior-level students. It covers the fundamental numerical methods required for scientists and engineers, as well as some advanced topics which are left to the discretion of instructors. The objective of the text is to provide readers with a strong theoretical background on numerical methods encountered in science and engineering, and to explain how to apply these methods to practical, real-world problems. Readers will also learn how to convert numerical algorithms into running computer codes. Features: Numerous pedagogic features including exercises, “pros and cons” boxes for each method discussed, and rigorous highlighting of key topics and ideas Suitable as a primary text for undergraduate courses in numerical methods, but also as a reference to working engineers A Pseudocode approach that makes the book accessible to those with different (or no) coding backgrounds, which does not tie instructors to one particular language over another A dedicated website featuring additional code examples, quizzes, exercises, discussions, and more: https://github.com/zaltac/NumMethodsWPseudoCodes A complete Solution Manual and PowerPoint Presentations are available (free of charge) to instructors at www.routledge.com/9781032754741
Introduction To Methods For Nonlinear Optimization
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Author : Luigi Grippo
language : en
Publisher: Springer Nature
Release Date : 2023-05-27
Introduction To Methods For Nonlinear Optimization written by Luigi Grippo and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-27 with Mathematics categories.
This book has two main objectives: • to provide a concise introduction to nonlinear optimization methods, which can be used as a textbook at a graduate or upper undergraduate level; • to collect and organize selected important topics on optimization algorithms, not easily found in textbooks, which can provide material for advanced courses or can serve as a reference text for self-study and research. The basic material on unconstrained and constrained optimization is organized into two blocks of chapters: • basic theory and optimality conditions • unconstrained and constrained algorithms. These topics are treated in short chapters that contain the most important results in theory and algorithms, in a way that, in the authors’ experience, is suitable for introductory courses. A third block of chapters addresses methods that are of increasing interest for solving difficult optimization problems. Difficulty can be typically due to the high nonlinearity of the objective function, ill-conditioning of the Hessian matrix, lack of information on first-order derivatives, the need to solve large-scale problems. In the book various key subjects are addressed, including: exact penalty functions and exact augmented Lagrangian functions, non monotone methods, decomposition algorithms, derivative free methods for nonlinear equations and optimization problems. The appendices at the end of the book offer a review of the essential mathematical background, including an introduction to convex analysis that can make part of an introductory course.
Numerical Methods For Least Squares Problems Second Edition
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Author : Åke Björck
language : en
Publisher: SIAM
Release Date : 2024-07-05
Numerical Methods For Least Squares Problems Second Edition written by Åke Björck and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-07-05 with Mathematics categories.
The method of least squares, discovered by Gauss in 1795, is a principal tool for reducing the influence of errors when fitting a mathematical model to given observations. Applications arise in many areas of science and engineering. The increased use of automatic data capturing frequently leads to large-scale least squares problems. Such problems can be solved by using recent developments in preconditioned iterative methods and in sparse QR factorization. The first edition of Numerical Methods for Least Squares Problems was the leading reference on the topic for many years. The updated second edition stands out compared to other books on this subject because it provides an in-depth and up-to-date treatment of direct and iterative methods for solving different types of least squares problems and for computing the singular value decomposition. It also is unique because it covers generalized, constrained, and nonlinear least squares problems as well as partial least squares and regularization methods for discrete ill-posed problems. The bibliography of over 1,100 historical and recent references provides a comprehensive survey of past and present research in the field. This book will be of interest to graduate students and researchers in applied mathematics and to researchers working with numerical linear algebra applications.