scholarly journals Bounded Fixed Point Iteration

1991 ◽  
Vol 20 (359) ◽  
Author(s):  
Hanne Riis Nielson ◽  
Flemming Nielson
Keyword(s):  
Fixed Point ◽  
Abstract Interpretation ◽  
Higher Order ◽  
Monotone Functions ◽  
Additive Functions ◽  
Fixed Point Iteration ◽  
Exponential Bound ◽  
Strictness Analysis ◽  
Long Chains

In the context of abstract interpretation for languages without higher-order features we study the number of times a functional need to be unfolded in order to give the least fixed point. For the cases of total or monotone functions we obtain an exponential bound and in the case of strict and additive (or distributive) functions we obtain a quadratic bound. These bounds are shown to be tight in that sufficiently long chains of functions can be shown to exist. Specializing the case of strict and additive functions to functionals of a form that would correspond to iterative programs we show that a linear bound is tight. This is related to several analyses studied in the literature (including strictness analysis).

A syntactic method for finding least fixed points of higher-order functions over finite domains

1997 ◽  
Vol 7 (4) ◽  
pp. 357-394
Author(s):  
TYNG-RUEY CHUANG ◽  
BENJAMIN GOLDBERG
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Normal Forms ◽  
Abstract Interpretation ◽  
Higher Order ◽  
Semantic Domain ◽  
Recent Developments ◽  
Strictness Analysis ◽  
Finite Domains ◽  
Higher Order Functions

This paper describes a method for finding the least fixed points of higher-order functions over finite domains using symbolic manipulation. Fixed point finding is an essential component in the calculation of abstract semantics of functional programs, providing the foundation for program analyses based on abstract interpretation. Previous methods for fixed point finding have primarily used semantic approaches, which often must traverse large portions of the semantic domain even for simple programs. This paper provides the theoretical framework for a syntax-based analysis that is potentially very fast. The proposed syntactic method is based on an augmented simply typed lambda calculus where the symbolic representation of each function produced in the fixed point iteration is transformed to a syntactic normal form. Normal forms resulting from successive iterations are then compared syntactically to determine their ordering in the semantic domain, and to decide whether a fixed point has been reached. We show the method to be sound, complete and compositional. Examples are presented to show how this method can be used to perform strictness analysis for higher-order functions over non-flat domains. Our method is compositional in the sense that the strictness property of an expression can be easily calculated from those of its sub-expressions. This is contrary to most strictness analysers, where the strictness property of an expression has to be computed anew whenever one of its subexpressions changes. We also compare our approach with recent developments in strictness analysis.

scholarly journals Fitness Conditions for fixed Point Iteration

1992 ◽  
Vol 21 (384) ◽  
Author(s):  
Flemming Nielson ◽  
Hanne Riis Nielson
Keyword(s):  
Fixed Point ◽  
Abstract Interpretation ◽  
Flow Analysis ◽  
Worst Case ◽  
Fixed Point Iteration ◽  
Average Case ◽  
Data Flow Analysis ◽  
Tail Recursion ◽  
Carry Over ◽  
Primitive Recursive

This paper provides a link between the formulation of static program analyses using the framework of abstract interpretation (popular for functional languages) and using the more classical framework of data flow analysis (popular for imperative languages). In particular we show how the classical notions of fastness, rapidity and k-boundedness carry over to the abstract interpretation framework and how this may be used to bound the number of times a functional should be unfolded in order to yield the fixed point. This is supplemented with a number of results on how to calculate the bounds for iterative forms (as for tail recursion), for linear forms (as for one nested recursive call), and for primitive recursive forms. In some cases this improves the ''worst case'' results of H.R. Nielson and F. Nielson: Bounded Fixed Point Iteration, but more importantly it gives much better ''average case'' results.

A Fixed-Point Iteration Method With Quadratic Convergence

2012 ◽  
Vol 79 (3) ◽  
Author(s):  
K. P. Walker ◽  
T.-L. Sham
Keyword(s):  
Fixed Point ◽  
Nonlinear Equations ◽  
Unit Root ◽  
Iteration Method ◽  
Equation System ◽  
Higher Order ◽  
Iteration Algorithm ◽  
System Of Nonlinear Equations ◽  
Fixed Point Iteration ◽  
Root Functions

The fixed-point iteration algorithm is turned into a quadratically convergent scheme for a system of nonlinear equations. Most of the usual methods for obtaining the roots of a system of nonlinear equations rely on expanding the equation system about the roots in a Taylor series, and neglecting the higher order terms. Rearrangement of the resulting truncated system then results in the usual Newton-Raphson and Halley type approximations. In this paper the introduction of unit root functions avoids the direct expansion of the nonlinear system about the root, and relies, instead, on approximations which enable the unit root functions to considerably widen the radius of convergence of the iteration method. Methods for obtaining higher order rates of convergence and larger radii of convergence are discussed.

scholarly journals Differentiable Forward and Backward Fixed-Point Iteration Layers

IEEE Access ◽  
2021 ◽  
Vol 9 ◽  
pp. 18383-18392
Author(s):  
Younghan Jeon ◽  
Minsik Lee ◽  
Jin Young Choi
Keyword(s):  
Fixed Point ◽  
Fixed Point Iteration

scholarly journals A Study of Nonlinear Boundary Value Problem

2021 ◽  
Author(s):  
Noureddine Bouteraa ◽  
Habib Djourdem
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Higher Order ◽  
Nonlinear Boundary Value Problem ◽  
Point Boundary ◽  
Error Estimations ◽  
Fractional Differential

In this chapter, firstly we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-order fractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions and the error estimations are also given. Secondly, we cover the multi-valued case of our problem. We investigate it for nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler. Two illustrative examples are presented at the end to illustrate the validity of our results.

scholarly journals On the Equivalence of Implicit Kirk-Type Fixed Point Iteration Schemes for a General Class of Maps

2019 ◽  
Vol 07 (01) ◽  
pp. 123-137
Author(s):  
Alfred Olufemi Bosede ◽  
Hudson Akewe ◽  
Omolara Fatimah Bakre ◽  
Ashiribo Senapon Wusu
Keyword(s):  
Fixed Point ◽  
General Class ◽  
Fixed Point Iteration
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