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Fixed-point iteration

About: Fixed-point iteration is a research topic. Over the lifetime, 3003 publications have been published within this topic receiving 61245 citations.


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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
02 Jan 2016
TL;DR: In this paper, the authors present an ALGOLGOL-based approach for the inclusion of complex Zeros of polynomials of a function of one real variable in a system of linear systems of equations.
Abstract: Preface to the English Edition. Preface to the German Edition. Real Interval Arithmetic. Further Concepts and Properties. Interval Evaluation and Range of Real Functions. Machine Interval Arithmetic. Complex Interval Arithmetic. Metric, Absolute, Value, and Width in. Inclusion of Zeros of a Function of One Real Variable. Methods for the Simultaneous Inclusion of Real Zeros of Polynomials. Methods for the Simultaneous Inclusion of Complex Zeros of Polynomials. Interval Matrix Operations. Fixed Point Iteration for Nonlinear Systems of Equations. Systems of Linear Equations Amenable to Interation. Optimality of the Symmetric Single Step Method with Taking Intersection after Every Component. On the Feasibility of the Gaussian Algorithm for Systems of Equations with Intervals as Coefficients. Hansen's Method. The Procedure of Kupermann and Hansen. Ireation Methods for the Inclusion of the Inverse Matrix and for Triangular Decompositions. Newton-like Methods for Nonlinear Systems of Equations. Newton-like Methods without Matrix Inversions. Newton-like Methods for Particular Systems of Nonlinear Equations. Newton-like Total step and Single Step Methods. Appendix A. The Order of Convergence of Iteration Methods in vn(Ic) and Mmn(iC) ). Appendix B. Realizations of Machine Interval Arithmetics in ALGOL 60. Appendix C. ALGOL Procedures. Bibliography. Index of Notation. Subject Index.

2,054 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
01 Jan 1974
TL;DR: In this article, it was shown that a certain sequence of points which is iteratively defined converges always to a fixed point of a lipschitzian pseudo-contractive map.
Abstract: The following result is shown. If T is a lipschitzian pseudo-contractive map of a compact convex subset E of a Hilbert space into itself and xl is any point in E, then a certain mean value sequence defined by Xn+1 = anT[f3.Tx. + (1fl.)x.] + (1ocn)xn converges strongly to a fixed point of T, where {cn} and {fl3} are sequences of positive numbers that satisfy some conditions. It was recently shown in [1] that a mean value iteration method is available to find a fixed point of a strictly pseudo-contractive map. In this paper we shall prove that a certain sequence of points which is iteratively defined converges always to a fixed point of a lipschitzian pseudo-contractive map. For the definitions of a strictly pseudo-contractive map and a pseudo-contractive map in a Hilbert space, see, for example, [3]. THEOREM. If E is a convex compact subset of a Hilbert space H, T is a lipschitzian pseudo-contractive map from E into itself and x1 is any point in E, then the sequence {xn}nL' 1 converges strongly to a fixed point of T, where xn is defined iteratively for each positive integer n by ( 1 ) xn+1 = Xn T[/ TXn + (1 /3n)xn] + (1 (n)Xn, where {cxj=L~ and {/}?n?l are sequences of positive numbers that satisfy the following three conditions: (2) 0 < O? n < /3n ? 1 for allpositive integers n, (3) lim /n = 0,

1,289 citations

Journal ArticleDOI
TL;DR: A fixed point algorithm for minimizing a TV penalized least squares functional is presented and compared with existing minimization schemes, and a variant of the cell-centered finite difference multigrid method of Ewing and Shen is implemented for solving the (large, sparse) linear subproblems.
Abstract: Total variation (TV) methods are very effective for recovering “blocky,” possibly discontinuous, images from noisy data. A fixed point algorithm for minimizing a TV penalized least squares functional is presented and compared with existing minimization schemes. A variant of the cell-centered finite difference multigrid method of Ewing and Shen is implemented for solving the (large, sparse) linear subproblems. Numerical results are presented for one- and two-dimensional examples; in particular, the algorithm is applied to actual data obtained from confocal microscopy.

1,113 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202327
202249
2021132
2020108
201999
2018117