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53
Towards Exact Geometric Computation
, 1994
"... Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To rec ..."
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Cited by 88 (6 self)
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Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown 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 fixedprecision arithmetic. We suggest that in many cases, implementors should make robustness a nonissue 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...
Greatest Factorial Factorization and Symbolic Summation
 J. Symbolic Comput
, 1995
"... This paper is selfcontained, 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 GFFform of a polynomial. In S ..."
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Cited by 58 (7 self)
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This paper is selfcontained, 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 GFFform of a polynomial. In Section 3 we investigate the relation to the dispersion function (Abramov, 1971) and discuss "shiftsaturated" polynomials which are polynomials with sufficiently nice GFFform. Due to lattice properties of K[x] with respect to gcd, a minimal shiftsaturated polynomial sat(p) can be assigned to each p 2 K[x]. The canonical Sform 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 Sform 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 "HermiteOstrogradsky Formula" for rational function integration. In addition, discussing minimaldegree answers to...
Using Program Slicing to Simplify Testing
 EUROSTAR'94
, 1994
"... 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 tha ..."
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Cited by 55 (33 self)
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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 ...
NearOptimal Parameterization of the Intersection of Quadrics: II. A Classification of Pencils
, 2005
"... ..."
Distributed control of spatially invariant systems
 IEEE Transactions on Automatic Control
, 2002
"... 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 system ..."
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Cited by 34 (0 self)
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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 finitedimensional 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, infinitedimensional systems, optimal control, robust control, spatially invariant systems.
MAPC: A library for Efficient and Exact Manipulation of Algebraic Points and Curves
"... 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, ..."
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Cited by 28 (8 self)
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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.
Intersecting Quadrics: An Efficient and Exact Implementation
 IN ACM SYMPOSIUM ON COMPUTATIONAL GEOMETRY  SOCG’2004
, 2004
"... 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]. Unlike ..."
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Cited by 23 (2 self)
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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]. Unlike
Efficient and accurate brep generation of low degree sculptured solids using exact arithmetic
 In ACM/SIGGRAPH Symposium on Solid Modeling
, 1997
"... 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 bou ..."
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Cited by 20 (8 self)
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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
An abstract interpretation approach for automatic generation of polynomial invariants
 In 11th Static Analysis Symposium
, 2004
"... www.cs.unm.edu/~kapur Abstract. A method for generating polynomial invariants of imperative programs is presented using the abstract interpretation framework. It is shown that for programs with polynomial assignments, an invariant consisting of a conjunction of polynomial equalities can be automatic ..."
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Cited by 16 (4 self)
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www.cs.unm.edu/~kapur Abstract. A method for generating polynomial invariants of imperative programs is presented using the abstract interpretation framework. It is shown that for programs with polynomial assignments, an invariant consisting of a conjunction of polynomial equalities can be automatically generated for each program point. The proposed approach takes into account tests in conditional statements as well as in loops, insofar as they can be abstracted to be polynomial equalities and disequalities. The semantics of each statement is given as a transformation on polynomial ideals. Merging of paths in a program is defined as the intersection of the polynomial ideals associated with each path. For a loop junction, a widening operator based on selecting polynomials up to a certain degree is proposed. The algorithm for finding invariants using this widening operator is shown to terminate in finitely many steps. The proposed approach has been implemented and successfully tried on many programs. A table providing details about the programs is given. 1