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104
The Exact Computation Paradigm
, 1994
"... 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 fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 95 (10 self)
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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 fixedprecision. 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 massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
Just relax: Convex programming methods for subset selection and sparse approximation
, 2004
"... Abstract. Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electric ..."
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Cited by 91 (4 self)
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Abstract. Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical engineering, applied mathematics and statistics, but small theoretical progress has been made over the last fifty years. Subset selection and sparse approximation both admit natural convex relaxations, but the literature contains few results on the behavior of these relaxations for general input signals. This report demonstrates that the solution of the convex program frequently coincides with the solution of the original approximation problem. The proofs depend essentially on geometric properties of the ensemble of elementary signals. The results are powerful because sparse approximation problems are combinatorial, while convex programs can be solved in polynomial time with standard software. Comparable new results for a greedy algorithm, Orthogonal Matching Pursuit, are also stated. This report should have a major practical impact because the theory applies immediately to many realworld signal processing problems. 1.
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 90 (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...
Robust Geometric Computation
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
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Cited by 72 (11 self)
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Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...
A Separation Bound for Real Algebraic Expressions
 In Lecture Notes in Computer Science
, 2001
"... Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, kth root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the si ..."
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Cited by 38 (4 self)
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Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, kth 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
Approximate Euclidean Shortest Paths In 3Space
, 1994
"... Papadimitriou's approximation approach to the Euclidean shortest path (ESP) in 3space is revisited. As this problem is NPhard, his approach represents an important step towards practical algorithms. However, there are several gaps in the original description. Besides giving a complete treatment ..."
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Cited by 33 (4 self)
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Papadimitriou's approximation approach to the Euclidean shortest path (ESP) in 3space is revisited. As this problem is NPhard, his approach represents an important step towards practical algorithms. However, there are several gaps in the original description. Besides giving a complete treatment in the framework of bit complexity, we also improve on his subdivision method. Among the tools needed are rootseparation bounds and nontrivial applications of Brent's complexity bounds on evaluation of elementary functions using floating point numbers. Keywords: Approximate Euclidean Shortest Path, Exact Geometric Computation. 1. Introduction The Euclidean shortest path (ESP) problem can be formulated as follows: given a collection of polyhedral obstacles in physical space S, and source and target points s 0 ; t 0 2 S, find a shortest obstacleavoiding path between s 0 and t 0 . Here S is typically E 2 or E 3 . It is evident that this is a basic problem in applications such as ro...
A Descartes algorithm for polynomials with bitstream coefficients
 CASC, VOLUME 3718 OF LNCS
, 2005
"... The Descartes method is an algorithm for isolating the real roots of squarefree polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bitstreams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We s ..."
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Cited by 32 (3 self)
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The Descartes method is an algorithm for isolating the real roots of squarefree polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bitstreams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bitstream coefficients. To isolate the real roots of a squarefree real polynomial q(x) = qnx n +...+q0 with root separation ρ, coefficients qn  ≥ 1 and qi  ≤ 2 τ, it needs coefficient approximations to O(n(log(1/ρ)+τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ)+τ) 2) bit operations.
Towards an open curved kernel
 In Proc. Annual ACM Symp. on Computational Geometry
, 2004
"... Our work goes towards answering the growing need for the robust and efficient manipulation of curved objects in numerous applications. The kernel of the cgal library provides several functionalities which are, however, mostly restricted to linear objects. We focus here on the arrangement of conic ar ..."
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Cited by 31 (14 self)
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Our work goes towards answering the growing need for the robust and efficient manipulation of curved objects in numerous applications. The kernel of the cgal library provides several functionalities which are, however, mostly restricted to linear objects. We focus here on the arrangement of conic arcs in the plane. Our first contribution is the design, implementation and testing of a kernel for computing arrangements of circular arcs. A preliminary C++ implementation exists also for arbitrary conic curves. We discuss the representation and predicates of the geometric objects. Our implementation is targeted for inclusion in the cgal library. Our second contribution concerns exact and efficient algebraic algorithms for the case of conics. They treat all inputs, including degeneracies, and they are implemented as part of the library synaps 2.1. Our tools include Sturm sequences, resultants, Descartes ’ rule, and isolating points. Thirdly, our experiments on circular arcs show that our ∗ Work partially supported by the IST Programme of the EU as a
Almost tight recursion tree bounds for the Descartes method
 In Proc. Int. Symp. on Symbolic and Algebraic Computation
, 2006
"... We give a unified (“basis free”) framework for the Descartes method for real root isolation of squarefree real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound ..."
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Cited by 30 (3 self)
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We give a unified (“basis free”) framework for the Descartes method for real root isolation of squarefree real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiXi with integer coefficients ai  < 2 L, this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.
Real Algebraic Numbers: Complexity Analysis and Experimentation
 RELIABLE IMPLEMENTATIONS OF REAL NUMBER ALGORITHMS: THEORY AND PRACTICE, LNCS (TO APPEAR
, 2006
"... 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 metho ..."
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Cited by 30 (17 self)
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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 squarefree 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.