This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called box-consistency, which approximates the notion of arc-consistency well-known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the Hansen-Segupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with state-of-the-art continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.

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. 1

Three extended real interval systems are defined and distinguished by their implementation complexity and result sharpness. The three systems are closed with respect to interval arithmetic and the enclosure of functions and relations, notwithstanding domain restrictions or the presence of singularities. 1 Overview Section 2 introduces the problem of defining closed interval systems. In Section 3, real and extended points and intervals are defined. In Section 4, the empty and entire intervals are used to close the extended interval system. Section ?? shows how incorrect conclusions have been reached about the result of certain interval arithmetic operator-operand combinations. The author is grateful to Professor Arnold Neumaier for originally raising this issue. Section 9 describes how to legitimately use IEEE floating-point arithmetic to obtain the sharp results described in Section ??. Section 6 generalizes extended interval arithmetic to define interval enclosures of functions, with...

This paper documents LIA InC++ library for local interval arithmetic in C++. The main innovation of the library is the idea of extending traditional interval arithmetic with "complement" intervals and discontinuous intervals. By these extensions it is possible to evaluate not only ranges of possible values (i.e., intervals) but ranges of impossible values as well. LIA InC++ contains classes for interval types and overloaded definitions for primitive interval arithmetic operators and functions. Open ended intervals can be used in addition to the traditional closed ones. Intervals of infinite width (e.g., (-,2], (-,),...) are accepted as inputs and are used for managing problems of overflowing values during function evaluation. The library uses double precision machine arithmetic with optional outward rounding, makes use of interval properties such as scalarity and symmetry, and uses some bit-level manipulations for efficient computation. LIA library is the most fundamental one in our In...

This paper reviews work on using interval arithmetic as the basis for next generation spreadsheet programs capable of dealing with rounding errors, imprecise data, and numerical constraints. A series of ever more versatile computational models for spreadsheets are presented beginning from classical interval arithmetic and ending up with interval constraint satisfaction. In order to demonstrate the ideas, an actual implementation of each model as a class library is presented and its integration with a commercial spreadsheet program is explained. 1 LIMITATIONS OF SPREADSHEET COMPUTING Spreadsheet programs, such as MS Excel, Quattro Pro, Lotus 1--2--3, etc., are among the most widely used applications of computer science. Since the pioneering days of VisiCalc and others, spreadsheet programs have been enhanced immensely with new features. However, the underlying computational paradigm of evaluating arithmetical functions by using ordinary machine arithmetic has remained the same. The wor...