Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floating-point arithmetic. Such implementations have many well-known problems, here informally called "robustness issues". To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixed-precision arithmetic. We suggest that in many cases, implementors should make robustness a non-issue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encomp...

This paper is self-contained, no difference field knowledge but only basic facts from algebra are required. In the following we briefly review its sections. Section 2 presents the basic GFF notions, in particular the Fundamental Lemma and an algorithm for computing the GFF-form of a polynomial. In Section 3 we investigate the relation to the dispersion function (Abramov, 1971) and discuss "shift-saturated" polynomials which are polynomials with sufficiently nice GFF-form. Due to lattice properties of K[x] with respect to gcd, a minimal shift-saturated polynomial sat(p) can be assigned to each p 2 K[x]. The canonical S-form of a rational function is introduced as the quotient of two polynomials with denominator of type sat(p). In Section 4 rational telescoping is treated; based on S-form representation, Theorem 4.1 explains why factorials rather than powers play the essential role in summation. Section 5 presents a new and algebraically motivated approach to Gosper's algorithm; together with the basic notions of GFF and Symbolic Summation 3 Section 2 this section can be read independently from the rest of the paper. In Section 6 we consider the general rational summation problem from GFF point of view. Two new algorithms are given. The first one works iteratively similar to the approach sketched by Moenck (1977). His approach is implemented in the computer algebra system Maple to sum rational functions, but due to several gaps in Moenck's original description the Maple algorithm fails on certain rational function inputs as observed by the author of this paper; see Example 6.6. The second algorithm provides an analogue to what is called "Horowitz' Method" or "Hermite-Ostrogradsky Formula" for rational function integration. In addition, discussing minimal-degree answers to...

Program slicing is a technique for automatically identifying all the lines in a program which affect a selected subset of variables. A large program can be divided into a number of smaller programs (its slices), each constructed for different variable subsets. The slices are typically simpler than the original program, thereby simplifying the process of testing a property of the program which only concerns a subset of its variables. Some aspects of a program's computation are not captured by a set of variables, rendering slicing inapplicable. To overcome this difficulty we make a program introspective, adding assignments to denote these `implicit' computations. Initially this makes the program longer. However, slicing can now be applied to the introspective program, forming a slice concerned solely with the implicit computation. We improve the simplification power of slicing using program transformation. To illustrate our approach we consider the implicit computation which ...

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We present MAPC, a library for exact representation of geometric objects -- specifically points and algebraic curves in the plane. Our library makes use of several new algorithms, which we present here, including methods for nding the sign of a determinant, finding intersections between two curves, and breaking a curve into monotonic segments. These algorithms are used to speed up the underlying computations. The library provides C++ classes that can be used to easily instantiate, manipulate, and perform queries on points and curves in the plane. The point classes can be used to represent points known in a variety of ways (e.g. as exact rational coordinates or algebraic numbers) in a unified manner. The curve class can be used to represent a portion of an algebraic curve. We have used MAPC for applications dealing with algebraic points and curves, including sorting points along a curve, computing arrangement of curves, medial axis computations, and boundary evaluation on curved primitives. As compared to earlier algorithms and implementations utilizing exact arithmetic, our library is able to achieve more than an order of magnitude improvement in performance.

Abstract. Constant propagation aims at identifying expressions that always yield a unique constant value at run-time. It is well-known that constant propagation is undecidable for programs working on integers even if guards are ignored as in non-deterministic flow graphs. We show that polynomial constants are decidable in non-deterministic flow graphs. In polynomial constant propagation, assignment statements that use the operators +, −, ∗ are interpreted exactly but all assignments that use other operators are conservatively interpreted as non-deterministic assignments. We present a generic algorithm for constant propagation via a symbolic weakest precondition computation and show how this generic algorithm can be instantiated for polynomial constant propagation by exploiting techniques from computable ring theory. 1

We present the first complete, exact and efficient C++ implementation of a method for parameterizing the intersection of two implicit quadrics with integer coefficients of arbitrary size. It is based on the nearoptimal algorithm recently introduced by Dupont et al. [4].

Abstract—We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the control of vehicular platoons, flow control, microelectromechanical systems (MEMS), smart structures, and systems described by partial differential equations with constant coefficients and distributed controls and measurements. For fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator (LQR), P and, optimal controllers can be obtained by solving a parameterized family of standard finite-dimensional problems. We show that optimal controllers have an inherent degree of decentralization, and this provides a practical distributed controller architecture. We also prove a general result that applies to partially distributed control and a variety of performance criteria, stating that optimal controllers inherit the spatial invariance structure of the plant. Connections of this work to that on systems over rings, and systems with dynamical symmetries are discussed. Index Terms—Distributed control, infinite-dimensional systems, optimal control, robust control, spatially invariant systems.

We present efficient representations and algorithms for exact boundary computation on low degree sculptured CSG solids using exact arithmetic. Most of the previous work using exact arithmetic has been restricted to polyhedral models. In this paper, we generalize it to higher order objects, whose boundaries are composed of rational parametric surfaces. The use of exact arithmetic and representation guarantees that a geometric algorithm is numerically accurate and is likely to be required for perturbation techniques which handle degeneracies. We present efficient algorithms for computing the intersection curves of trimmed parametric surfaces, decomposing them into multiple components for e cient point location queries inside the trimmed regions, and computing the boundary of the resulting solid using topological information and component classification

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