Let x =(x 1�x 2 � �xn) beavector of real numbers. x is said to possess an integer relation if there exist integers ai such that a 1x 1 + a 2x 2 + + anxn = 0. Recently Ferguson and Forcade discovered practical algorithms [7, 8, 9] which, for any n, either nd a relation if one exists or else establish bounds within which no relation can exist. One obvious application of these algorithms is to determine whether or not a given computed real number satis es any algebraic polynomial with integer coe cients (where the sizes of the coe cients are within some bound). The recursive form of the Ferguson-Forcade algorithm has been implemented with multiprecision arithmetic on the Cray-2 supercomputer at NASA Ames Research Center. The resulting computer program has been used to probe the question of whether or not certain constants involving � e, and satisfy any simple polynomials. These computations established that the following constants cannot satisfy any algebraic equation of degree eight or less with integer coe cients whose Euclidean norm is 10 9 or less: e = � e + � log e � � e � =e � = , and log e. Stronger results were obtained in several cases. These computations thus lend credence to the conjecture that each of the above mathematical constants is transcendental.

The advent of inexpensive, high-performance computers and new efficient algorithms have made possible the automatic recognition of numerically computed constants. In other words, techniques now exist for determining, within certain limits, whether a computed real or complex number can be written as a simple expression involving the classical constants of mathematics. These techniques will be illustrated by discussing the recognition of Euler sum constants, and also the discovery of new formulas for π and other constants, formulas that permit individual digits to be extracted from their expansions.

For each positive integer n ≥ 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n = 2 case is equivalent to the standard continued fraction algorithm. For n = 3, it reduces to a new iteration of the triangle. Cubic irrationals that are roots of x 3 +kx 2 +x−1 are shown to be precisely those numbers with purely periodic expansions. For general positive integers n, it reduces to a new iteration of an n dimensional simplex. Algebraic numbers that are roots of x n + kx n−1 + x n−2 − x − 1 are precisely those with purely periodic expansions. 1

In this paper, we present a novel method for designing polynomial FIR predictors for fixed-point environments. Our method yields filters that perform exact prediction of polynomial signals even with short coefficient word lengths. Under ordinary coefficient truncation or rounding, prediction capability degrades, or may be totally lost. With the proposed method, the filters are designed so that the predictive properties are exactly preserved in fixed-point implementations. The proposed filter design method is based on integer programming (IP) and can be directly applied to any fixed-point FIR design specifications which can be formulated in a form of linear constraints on the filter coefficients. 1. Introduction By their nature, digital devices handle numbers using a finite number of bits per digit [1]. In many embedded applications using highly optimized, small and less power consuming application specific integrated circuits (ASICs) it would be desirable to get by with low precision...

In this paper we present an algorithm for finding successive best approximations of a line by lattice points in three dimensions. Our algorithm is primarily an extension of the work of Furtwangler [13], which has been generalised for arbitrary radius functions, lattices and initial bases. We show that, after a finite number of initialisation iterations, the algorithm will produce all best approximations to the line above a certain height. Conversely, we show that, after initialisation, all convergents of the algorithm are best approximations (with one possible exception). We also provide a numerical example to illustrate the algorithm. 1 Introduction The best approximation of a line by lattice points is a problem of fundamental interest in the design of algorithms. The problem is one of finding those lattice points which lie successively closer to the given line as we move along the line away from the origin. That is, given an M-dimensional lattice\Omega\Gamma defined (though not un...

We study the following problem: given x 2 IR n either find a short integer relation m 2 ZZ n ; so that ! x; m ?= 0 holds for the inner product ! : ; : ? ; or prove that no short integer relation exists for x: Hastad, Just, Lagarias and Schnorr (1989) give a polynomial time algorithm for this problem. We present a stable variation of the HJLS--algorithm that preserves lower bounds on (x) for infinitesimal changes of x: Given x 2 IR n and ff 2 IN this algorithm finds a nearby point x 0 and a short integer relation m for x 0 : The nearby point x 0 is 'good' in the sense that no very short relation exists for points x within half the x 0 --distance from x: On the other hand if x 0 = x then m is, up to a factor 2 n=2 ; a shortest integer relation for x: Our algorithm uses, for arbitrary real input x; at most O(n 4 (n + log ff)) many arithmetical operations on real numbers. If x is rational the algorithm operates on integers having at most O(n 5 + n 3 (log ff) 2 +...

Abstract: The aim of this survey is to discuss multidimensional continued fraction and Euclidean algorithms from the viewpoint of numeration systems, substitutions, and the symbolic dynamical systems they generate. We will mainly focus on two types of multidimensional algorithms, namely, unimodular Markovian ones which include the most classical ones like e.g. Jacobi-Perron algorithm, and algorithms issued from lattice reduction. We will discuss these algorithms motivated by the study of substitutive dynamical systems, symbolic dynamical systems with low complexity function, and discrete geometry.

A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in [9]) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the triangle sequence. We give a new criterion for when a triangle sequence uniquely describes a pair of numbers and give the first explicit examples of triangle sequences that do not uniquely describe a pair of reals. Finally, this dual approach yields that the triangle sequence is topologically strongly mixing, meaning in particular that it is topologically ergodic.

Let x =(x 1�x 2 � �xn) beavector of real numbers. x is said to possess an integer relation if there exist integers ai not all zero such thata 1x 1 + a 2x 2 + + anxn =0. Beginning ten years ago, algorithms were discovered by one of us which, for any n, are guaranteed to either nd a relation if one exists or else establish bounds within which no relation can exist. One of those algorithms has been employed to study whether or not certain fundamental mathematical constants satisfy simple algebraic polynomials. Recently one of us discovered a new relation- nding algorithm that is much more efcient, both in terms of run time and numeric precision. This algorithm has now been implemented on high-speed computers using multiprecision arithmetic. Using these programs, several of the previous numerical results on mathematical constants have been extended, and other possible relationships between certain constants have been studied. This paper describes this new algorithm, summarizes the numerical results, and discusses other possible applications. In particular, it is established that none of the following constants satis es a simple, low-degree polynomial: (Euler's constant), log � log � 1 (the imaginary part of the rst zero of Riemann's zeta function), log 1, (3) (Riemann's zeta function evaluated at 3), and log (3). Several classes of possible additive andmultiplicative relationships between these and related constants are ruled out. Results are also cited for Feigenbaum's constant, derived from the theory of chaos, and two constants of fundamental physics, derived from experiment.

A survey of