We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixed-precision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.-Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original mass-produced computers were pocket calculators. Although one's first exposure to computers today is likely to be some non-numerical application, numeri...

real expressions with arbitrary precision. Figure 1 shows (part of) the LEDA manual page for reals. reals provide exact computation in a convenient way. In an implementation of a geometric algorithm in C++, reals can be used like doubles. The following example Max-Planck-Institut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. Supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II). y Fachbereich 14, Informatik, Universitat des Saarlandes, 66041 Saarbrucken, Germany. z Martin-Luther-Universitat Halle, Fachbereich Mathematik und Informatik, 06099 Halle, Germany. 0 arises in the computation of Voronoi diagrams of line segments [2]. For i, 1 i 3, let l i : a i x + b i

Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, k-th root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real. 1

We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (-Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non square-free polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.

We describe the implementation of the LEDA [MN95, Nah95] data type real. Every integer is a real and reals are closed under the operations addition, subtraction, multiplication, division and squareroot. The main features of the data type real are ffl The user--interface is similar to that of the built--in data type double.

This dissertation is about a novel technology for planar part manipulation. A device has been developed, called the Universal Planar Manipulator (UPM), based on a single actuator: a rigid, horizontal plate which vibrates in its own plane (3 degrees of freedom). Generic parts (bottles, tools, etc.) placed on the UPM's plate move as a result of friction. Our main contribution has been to develop control algorithms for the UPM which allow it to manipulate several parts in parallel. The UPM's actuation simplicity combined with its open workspace (no grippers) renders it appealing to existing industrial applications and novel interactive devices such as toys and active desks.

We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method’s good performance in practice. We improve the previously known bound by a factor of dτ, where d is the polynomial degree and τ bounds the coefficient bit size, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is � OB(d 4 τ 2) using a standard bound on the expected bit size of the integers in the continued fraction expansion. Moreover, using a homothetic transformation we improve the expected complexity bound to � OB(d 3 τ) under the assumption that d = O(τ). We show how to compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source C++ implementation in the algebraic library synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bit size up to 8000 bits and degree up to 1000. 1

Abstract. Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x)ofαover L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of α, which is based on p-adic techniques. Given mα(x) anda rational prime p which does not divide the discriminant disc(mα(x)) of mα(x), the algorithm computes the Frobenius automorphism of p in time polynomial in the size of p and in the size of mα(x). By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α. 1.

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We describe a new control method for vibrationsbased planar manipulation. We've developed a device -- the Universal Planar Manipulator (UPM) -- based on a single, horizontally-vibrating plate. Though minimalist in construction (one moving part), the UPM can manipulate several parts on its surface in parallel, simply using friction. Previously, we've shown that a sequence of rigid plate rotations can be computed which produces pre-specified part displacements. Here we present a new method based on a special motion primitive -- the "jet" -- which displaces a chosen part in a desired direction while keeping all others still. A jet allows one to say: "c'mon part, do a local motion ". Parallel manipulation then reduces to a round-robin application of jets. This technique is both faster and more robust than the old rotations-based method. Experiments on parallel trajectory following and part sorting are presented. With jets, the UPM becomes a practical technology for applications such as part singulation, feeding, sorting, food handling, product displays, and interactive devices such as active desks and toys.