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Chebyshev iteration

About: Chebyshev iteration is a research topic. Over the lifetime, 1942 publications have been published within this topic receiving 40815 citations.


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Book
01 Aug 1997
TL;DR: The symmetric Eigenproblem and singular value decomposition and the Iterative methods for linear systems Bibliography Index.
Abstract: Preface 1. Introduction 2. Linear equation solving 3. Linear least squares problems 4. Nonsymmetric Eigenvalue problems 5. The symmetric Eigenproblem and singular value decomposition 6. Iterative methods for linear systems 7. Iterative methods for Eigenvalue problems Bibliography Index.

2,983 citations

Book
01 Jan 1987
TL;DR: Preface How to Get the Software How to get the Software Part I.
Abstract: Preface How to Get the Software Part I. Linear Equations. 1. Basic Concepts and Stationary Iterative Methods 2. Conjugate Gradient Iteration 3. GMRES Iteration Part II. Nonlinear Equations. 4. Basic Concepts and Fixed Point Iteration 5. Newton's Method 6. Inexact Newton Methods 7. Broyden's Method 8. Global Convergence Bibliography Index.

2,531 citations

Book
01 Jan 1982
TL;DR: One-point iteration functions with memory have been studied extensively in the literature as discussed by the authors, where it is shown that one-point iterators with memory achieve linear and superlinear convergence with respect to a fixed-point problem.
Abstract: General Preliminaries: 1.1 Introduction 1.2 Basic concepts and notations General Theorems on Iteration Functions: 2.1 The solution of a fixed-point problem 2.2 Linear and superlinear convergence 2.3 The iteration calculus The Mathematics of Difference Relations: 3.1 Convergence of difference inequalities 3.2 A theorem on the solutions of certain inhomogeneous difference equations 3.3 On the roots of certain indicial equations 3.4 The asymptotic behavior of the solutions of certain difference equations Interpolatory Iteration Functions: 4.1 Interpolation and the solution of equations 4.2 The order of interpolatory iteration functions 4.3 Examples One-Point Iteration Functions: 5.1 The basic sequence $E_s$ 5.2 Rational approximations to $E_s$ 5.3 A basic sequence of iteration functions generated by direct interpolation 5.4 The fundamental theorem of one-point iteration functions 5.5 The coefficients of the error series of $E_s$ One-Point Iteration Functions With Memory: 6.1 Interpolatory iteration functions 6.2 Derivative-estimated one-point iteration functions with memory 6.3 Discussion of one-point iteration functions with memory Multiple Roots: 7.1 Introduction 7.2 The order of $E_s$ 7.3 The basic sequence $\scr{E}_s$ 7.4 The coefficients of the error series of $\scr{E}_s$ 7.5 Iteration functions generated by direct interpolation 7.6 One-point iteration functions with memory 7.7 Some general results 7.8 An iteration function of incommensurate order Multipoint Iteration Functions: 8.1 The advantages of multipoint iteration functions 8.2 A new interpolation problem 8.3 Recursively formed iteration functions 8.4 Multipoint iteration functions generated by derivative estimation 8.5 Multipoint iteration functions generated by composition 8.6 Multipoint iteration functions with memory Multipoint Iteration Functions: Continuation: 9.1 Introduction 9.2 Multipoint iteration functions of type 1 9.3 Multipoint iteration functions of type 2 9.4 Discussion of criteria for the selection of an iteration function Iteration Functions Which Require No Evaluation of Derivatives: 10.1 Introduction 10.2 Interpolatory iteration functions 10.3 Some additional iteration functions Systems of Equations: 11.1 Introduction 11.2 The generation of vector-valued iteration functions by inverse interpolation 11.3 Error estimates for some vector-valued iteration functions 11.4 Vector-valued iteration functions which require no derivative evaluations A Compilation of Iteration Functions: 12.1 Introduction 12.2 One-point iteration functions 12.3 One-point iteration functions with memory 12.4 Multiple roots 12.5 Multipoint iteration functions 12.6 Multipoint iteration functions with memory 12.7 Systems of equations Appendices: A. Interpolation B. On the $j$th derivative of the inverse function C. Significant figures and computational efficiency D. Acceleration of convergence E. Numerical examples F. Areas for future research Bibliography Index.

1,938 citations

Journal ArticleDOI
TL;DR: In this paper, a method for numerical integration of a well-behaved function over a finite range of argument is described, which consists essentially of expanding the integrand in a series of Chebyshev polynomials, and integrating this series term by term.
Abstract: A new method for the numerical integration of a "well-behaved" function over a finite range of argument is described. It consists essentially of expanding the integrand in a series of Chebyshev polynomials, and integrating this series term by term. Illustrative examples are given, and the method is compared with the most commonly-used alternatives, namelySimpson's rule and the method ofGauss.

919 citations

01 Jan 1952
TL;DR: In this paper, the authors adopt the general principles of the previous investigation to the specific demands that arise if we are not interested in the complete analysis of a matrix but only in the more special problem of obtaining the solution of a given set of linear equations.
Abstract: In an earlier publication [14] a method was described which generated the eigenvalues and eigenvectors of a matrix by a successive algorithm based on minimizations by least squares. The advantage of this method consists in the fact that the successive iterations are constantly employed with maximum efficiency which guarantees fastest convergence for a given number of iterations. Moreover, with the proper care the accumulation of rounding errors can be avoided. The resulting high precision is of great advantage if the separation of closely bunched eigenvalues and eigenvectors is demanded [16]. It was pointed out in [14, p. 256] that the inversion of a matrix, and thus the solution of simultaneous systems of linear equations, is contained in the general procedure as a special case. However, in view of the great importance associated with the solution of large systems of linear equations, this problem deserved more than passing attention. It is the purpose of the present discussion to adopt the general principles of the previous investigation to the specific demands that arise if we are not interested in the complete analysis of a matrix but only in the more special problem of obtaining the solution of a given set of linear equations

782 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202318
202228
20212
20204
20196
201825