Interval Arithmetic, Affine Arithmetic, Taylor Series Methods: Why, What Next? (original) (raw)
Related papers
Interval approximation of higher order to the ranges of functions
Computers & Mathematics with Applications, 1996
Bernstein and B-spline forms are generalized to multivariate polynomials. These forms are combined with a type of Taylor form for multivariate functions to generate realizable forms for multivariate functions.
arXiv (Cornell University), 2020
The interval numbers is the set of compact intervals of R with addition and multiplication operation, which are very useful for solving calculations where there are intervals of error or uncertainty, however, it lacks an algebraic structure with an inverse element, both additive and multiplicative This fundamental disadvantage results in overestimation of solutions in an interval equation or also overestimation of the image of a function over square regions. In this article we will present an original solution, through a morphism that preserves both the addiction and the multiplication between the space of the interval numbers to the space of square diagonal matrices.
Interval Root Finding and Interval Polynomials: Methods and Applications in Science and Engineering
Polynomial Paradigms: Trends and Applications in Science and Engineering, 2022
Polynomial systems are at the heart of a wide range of scientific fields. Multiple forms of polynomials are often involved in attempts to reduce currently open or hard problems to problems already solved. Because of their ever-increasing importance in many practical applications, polynomials span the entire spectrum of various disciplines of mathematics, science and engineering. In practice, a recurring problem is to determine the roots of polynomials under parametric uncertainty or imprecision. Interval mathematics is a subtle body of principles and methods for manipulating systems involving quantifiable uncertainties. Interval methods have proven successful and reliable for computing guaranteed enclosures of roots of polynomials under parametric uncertainty. This "self-validating" feature of interval analysis makes it competitive and preferable to the ordinary approximation methods in many practical applications. In this connexion, by the pursuit of the quest of reliable knowledge amidst uncertainty, the raison d'être of the present work is a systematic investigation of uncertain polynomials. We begin by laying out a rigorous algebraic foundation for ordinary polynomials and some of their generalizations, and then, we go further to define two new generalizations of ordinary and trigonometric polynomials, namely, generalized n-adic polynomials and n-adic S-polynomials. These two generalizations are so framed as to provide a rigorous mathematical foundation for interval and set-valued polynomials. Next, we formalize the theories of interval algebra and interval polynomials and show how to obtain guaranteed interval enclosures of families of generalized real polynomials. Afterwards, we describe how to use infinite polynomials with Taylor models to compute finer interval enclosures. Moreover, we examine some of the interval methods for root finding, with special attention is paid to the interval branch and prune method. Finally, we put forward a more refined interval branch and prune algorithm. In order to illustrate the significance of our algorithm, we compute a finer interval enclosure for an example introduced by Eldon Hansen and G. William Walster in 2003. We show that our interval enclosure is sharper than the optimal result given by Hansen and Walster. The algorithms of this work are coded using the software package, InCLosure. Many numerical examples are given, showing that all the results are guaranteed interval enclosures.
Interval Computations: Introduction, Uses, and Resources
1996
Interval analysis is a broad Þeld in which rigorous mathematics is associated with with scientiÞc computing. A number,of researchers worldwide have produced a voluminous literature on the subject. This article introduces interval arithmetic and its interaction with established mathematical theory. The article provides pointers to traditional literature collections, as well as electronic resources. Some successful scientiÞc and engineering applications
Formal Aspects of Correctness and Optimality of Interval Computations
Formal Aspects of Computing, 2006
An interval is a continuum of real numbers, defined by its end-points. Interval analysis, proposed by R. Moore in the 50's, concerns the discovery of interval functions to produce bounds on the accuracy of numerical results that are guaranteed to be sharp and correct. The last criterion, correctness, is the main one since it establishes that the result of an interval computation must always contains the value of the related real function.
A new class of interval methods with higher order of convergence
Computing, 1989
A New Class of Interval Methods with Higher Order of Convergence. In this paper we introduce a new class of interval methods for enclosing a simple root of a nonlinear equation. For each nonnegative integer p we describe an iterative procedure belonging to this class which requires p + 1 function values and an interval evaluation of the second derivative per step. The order of convergence of the iterative procedure grows exponentially with p. For p_>4 this order is strictly greater than Key words." Nonlinear equations, order of convergence.
Polynomial Precise Interval Analysis Revisited
Lecture Notes in Computer Science, 2009
We consider a class of arithmetic equations over the complete lattice of integers (extended with −∞ and ∞) and provide a polynomial time algorithm for computing least solutions. For systems of equations with addition and least upper bounds, this algorithm is a smooth generalization of the Bellman-Ford algorithm for computing the single source shortest path in presence of positive and negative edge weights. The method then is extended to deal with more general forms of operations as well as minima with constants. For the latter, a controlled widening is applied at loops where unbounded increase occurs. We apply this algorithm to construct a cubic time algorithm for the class of interval equations using least upper bounds, addition, intersection with constant intervals as well as multiplication.
Scientific Research and Essays, 2011
The problem of tightening computable overestimation error bounds arising from the solution of linear interval system of equations by some interval methods is considered. This study describes the quality of enclosure of these methods with a view of diminishing the overestimation bounds due to excessive use of interval operations. In particular, two different interval multiplication operations are examined and applied on the Krawczyk's method. It also compare note with results obtained from the well known Hansen-Bliek-Rohn method. It is shown that our results are never worse than the bounds afforded by the Bauer-Skeel and Hansen-Bliek-Rohn bounds. MSC(2000): 65G20, 65G30.
Interval arithmetic: From principles to implementation
Journal of the ACM, 2001
We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication, and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard's specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems, we derive data that are sufficiently detailed to convert directly to a program for efficiently implementing the interval operations. Finally, we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily closed.