F. Avnaim, J.-D. Boissonnat, O. Devillers, F. P. Preparata, and M. Yvinec, Evaluating signs of determinants using single-precision arithmetic, Algorithmica 17 (1997), no. 2, 111--132.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....have also released an efficient implementation of their algorithm, which works for integer matrices with 53 bit entries. Wiedemann [25] computes the determinant of a sparse matrix in a finite field (modulo a prime number, for example) by computing the characteristic polynomial. Avnaim et al. [2] give a specialized determinant sign algorithm for 2 2 and 3 3 matrices. The algorithm computes the determinant sign using arithmetic on b or b 1 bits, where b is a bound on the entry bitlength. Bronnimann and Yvinec [4] extend this algorithm, obtaining the lattice method for n n matrices ....

Francis Avnaim, Jean-Daniel Boissonnat, Olivier Devillers, Franco P. Preparata, and Mariette Yvinec. Evaluating signs of determinants using single-precision arithmetic. Research Report 2306, INRIA, BP93, 06902 Sophia-Antipolis, France, 1994.

....citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. SCG 01, June 3 5, 2001, Medford, Massachusetts, USA. Copyright 2001 ACM 1 58113 357 X 01 0006 . 5.00. ror in the expression evaluation [28, 43, 5]. Alternatively, others have developed improved algorithms for arbitrary precision arithmetic [39, 23, 8, 37, 6] Finally, it is possible to use filters that exploit features of the predicates or make assumptions on the maximum precision needed to evaluate a predicate [21, 39, 44] These ....

....deriving error bounds from the structure of the expression before its variables are specialized. They likewise have limited application to large problems, highprecision inputs, and iterative algorithms. Many more authors have focused on the problem of the sign of the determinant of a small matrix [14, 9, 21, 39, 5]. Little of that recent work applies to computing determinants of large matrices. Some of these approaches also limit the input precision, which is difficult in the context of dealing with arbitrary degree algebraic numbers. The modular arithmetic approach used in LiDIA [23] and extended by [8] ....

Francis Avnaim, Jean-Daniel Boissonnat, Olivier Devillers, Franco P. Preparata, and Mariette Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17(2):111--132, 1997.

....operations. From (1) this corresponds to matrices such that the condition number satis es log cond det A = O(n log kAk) and not n log kAk) as in the worst case. Along the same lines, the lattice algorithm of Brnnimann et Yvinec [12] generalizes to high dimensions the method of Avnaim et al. [4] for dimensions 2 and 3. Its complexity is analogous to (6) To have a better complexity for well conditioned matrices, arithmetic ltering has been much studied especially for algebraic geometry problems (see the introduction) The idea is to rapidly evaluate the sign of the determinant using ....

F. Avnaim, J.D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17:111132, 1997.

....to the error bound at run time. This error bound is used exactly as in the static case. Semi static filters can thus be designed to produce error bounds that are less conservative, but at the added cost of run time computation. Examples of semi static filters have been proposed by Avnaim et al. [1] and Burnikel et al. 3] Shewchuk [13] has also proposed an adaptive combination of semi static filters. In dynamic filters, the error bound is calculated in a syntax directed fashion, by considering how each primitive operation (such as add or muirply) propagates errors from input to output. ....

AVNAIM, F., BOISSONNAT, J.-D., DEVILLERS, O., PREPARATA, F., AND YVINEC, M. Evaluating Signs of Determinants using Single-Precision Arithmetic. Algorithmica 17 (1997), 111-132.

....incorrect path. This may lead to catastrophic errors. In the realm of geometric computations, the situation is particularly severe and known as the precision caused robustness problem [15, 18, 27, 36, 39] A popular approach to overcoming the robustness problem is the exact computation paradigm [1, 2, 4, 5, 9, 10, 7, 13, 25, 33, 40, 41]. The paradigm calls for the exact evaluations of all conditions and hence the exact computation of signs. In this paper, we consider the sign computation for the following class of real algebraic expressions. The value of a real algebraic expression is either a real number or undefined (in ....

F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17:111--132, 1997.

....the computation of E is carried along with the computation of E . Typically, for each arithmetic operation of E , a rule determines the error bound for the result of that operation based on the operands and error bounds on them. Static lters are implemented for instance in LN [11] semi static in [1, 5], and dynamic lters in Expr [19] and LEDA [6] Also Schewchuk, in [18] approximates E up to rst order error terms, then up to second order errors etc. until the sign can be safely determined, this procedure combines a dynamic lter according to our description with exact computation, since it can ....

....by their conditional probability of success. Remark : Many geometric predicates boil down to computing the sign of a determinant. Much eoeort has already been made towards the exact evaluation of signs of determinants, using various specic solutions such as Clarkson s or the lattice method [8, 1, 4], or using general solutions such as exact integer arithmetic [10] and modular arithmetic [2] For d Theta d determinants, the complexities range from O(d 3 log d) to O(d 4 log d) with a potentially large constant in the asymptotic bounds. Practically, all these methods are several orders of ....

[Article contains additional citation context not shown here]

F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17:111132, 1997.

....than integer arithmetic. The trend is reversing for software libraries as well, and there are several proposals to use floating point arithmetic to perform extended precision integer calculations. Fortune and Van Wyk [10, 9] Clarkson [4] and Avnaim, Boissonnat, Devillers, Preparata, and Yvinec [1] have described algorithms of this kind, designed to attack the same computational geometry robustness problems considered later in this report. These algorithms are surveyed in Section 4.1. Another differentiating feature of multiprecision libraries is whether they use multiple exponents. Most ....

....sufficient Implementation of Geometric Predicates 31 to operate on 10 Theta 10 matrices of 32 bit integers. Clarkson s algorithm is naturally adaptive; its running time is small for matrices whose determinants are not near zero 6 . Recently, Avnaim, Boissonnat, Devillers, Preparata, and Yvinec [1] proposed an algorithm to evaluate signs of determinants of 2 Theta 2 and 3 Theta 3 matrices of p bit integers using only p and (p 1) bit arithmetic, respectively. Surprisingly, this is sufficient even to implement the insphere test (which is normally written as a 4 Theta 4 or 5 Theta 5 ....

[Article contains additional citation context not shown here]

Francis Avnaim, Jean-Daniel Boissonnat, Olivier Devillers, Franco P. Preparata, and Mariette Yvinec. Evaluating Signs of Determinants Using Single-Precision Arithmetic. Manuscript available from http://www.inria.fr:/prisme/personnel/devillers/anglais/determinant, 1995.

....From (1) this corresponds to matrices such that the condition number satisfies log cond det A = O(n log #A#) and not #(n log #A#) as in the worst case. Along the same lines, the lattice algorithm of Bronnimann et Yvinec [12] generalizes to high dimensions the method of Avnaim et al. [4] for dimensions 2 and 3. Its complexity is analogous to (6) To have a better complexity for well conditioned matrices, arithmetic filtering has been much studied especially for algebraic geometry problems (see the introduction) The idea is to rapidly evaluate the sign of the determinant using ....

F. Avnaim, J.D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17:111--132, 1997.

....to the error bound at run time. This error bound is used exactly as in the static case. Semi static lters can thus be designed to produce error bounds that are less conservative, but at the added cost of run time computation. Examples of semi static lters have been proposed by Avnaim et al. [1] and Burnikel et al. 3] Shewchuk [13] has also proposed an adaptive combination of semi static lters. In dynamic lters, the error bound is calculated in a syntax directed fashion, by considering how each primitive operation (such as add or multiply) propagates errors from input to output. ....

Avnaim, F., Boissonnat, J.-D., Devillers, O., Preparata, F., and Yvinec, M. Evaluating Signs of Determinants using Single-Precision Arithmetic. Algorithmica 17 (1997), 111-132.

....number of intersection points may need to be computed to ascertain that there is no intersection between two polyhedra. A way to avoid these intersection computations is to reduce the test to computing the signs of some determinants [76] as in many other problems arising in Computational Geometry [2]. p p h t v i 1 i Fig. 6. Basic edge face intersection test (general faces) Consider a face from one polyhedron, defined by the ordered sequence of vertices around it, represented by their position vectors p 1 ; p l , expressed in homogeneous coordinates (that is, p i = p x i ; ....

Avnaim, F. Evaluating signs of determinants using single-precision arithmetic. Tech. Rep. 2306, INRIA, 1994.

....patterns. A variety of techniques have been designed to make geometric algorithms robust in the presence of high precision numerical computations (e.g. involving square roots) and degenerate geometric configurations (e.g. more than two collinear points or more than three cocircular points) [3, 14, 24, 28, 33, 34, 43, 47, 48, 49, 57, 76, 78, 85, 86, 87]. GeomLib adopts the paradigm of exact computation (see, e.g. Refs. 3, 14, 86] and uses the concept of degree [57] to characterize the arithmetic precision requirement of a geometric algorithm. Namely, a geometric algorithm of degree d requires in its computations a precision that is, in the ....

....computations (e.g. involving square roots) and degenerate geometric configurations (e.g. more than two collinear points or more than three cocircular points) 3, 14, 24, 28, 33, 34, 43, 47, 48, 49, 57, 76, 78, 85, 86, 87] GeomLib adopts the paradigm of exact computation (see, e.g. Refs. [3, 14, 86]) and uses the concept of degree [57] to characterize the arithmetic precision requirement of a geometric algorithm. Namely, a geometric algorithm of degree d requires in its computations a precision that is, in the worst case, about d times that of the input data. Since the arithmetic precision ....

[Article contains additional citation context not shown here]

F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17(2):111--132, 1997.

....1 Introduction One of the most fundamental invariants of a square matrix is the determinant. Applications for computing the determinant of a matrix are numerous. For integer matrices alone they include computational number theory [4] computational group theory [9] and computational geometry [2, 3]. In this paper we present a new algorithm for the determinant which is faster than any previously known. For a matrix A n n this algorithm requires O n 3 logn log A 2 log detA log 2 n Research was supported in part by Natural Sciences and ....

F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using singleprecision arithmetic. Algorithmica, 17:111--132, 1997.

....1 Introduction One of the most fundamental invariants of a square matrix is the determinant. Applications for computing the determinant of a matrix are numerous. For integer matrices alone they include computational number theory [4] computational group theory [9] and computational geometry [2, 3]. In this paper we present a new algorithm for the determinant which is faster than any previously known. For a matrix A 2 Z n n this algorithm requires O(n 3 (logn logkAk) 2 p log j detAj log 2 n) Research was supported in part by Natural Sciences and Engineering Research ....

F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using singleprecision arithmetic. Algorithmica, 17:111--132, 1997.

....assuming precise geometric primitives. But of course, should we have addressed the robustness issue, the core problem would have been the same since the computations we end up with consist in evaluating the sign of algebraic expressions (sections 2.2.2 and 3. 1) The reader is referred to [20, 1] for recent developments in this area. # 1 # 2 Figure 1: a)Intersection of two toleranced polygons (b)Worst case example of intersection Figure 1(a) shows an example of two toleranced polygons intersecting in a configuration that may produce zero, one, or two components in the intersection ....

....# ij might in general be non convex. However, from a practical point of view, the following boot strapping algorithm may give satisfactory results: 1. First, get a rough estimate p ij of p ij as the fraction of points satisfying condition I over a sample of reasonable size uniformly drawn in [0, 1] 6 , 2. Second, plug p ij into the estimator theorem (theorem 11.1) of [15] which in turn gives the sample size such that p ij is accurate within a factor # with a probability greater than 1 #. 9 For example, on the configuration of figure 3(b) an estimate on a sample of size 1000 gives ....

F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Research Report 2306, INRIA, BP93, 06902 SophiaAntipolis, France, 1994. http://www.inria.fr/prisme/personnel/devillers/publis.html.

.... arithmetic schemes designed speci cally for computational geometry; most of them are methods for exactly evaluating the sign of a determinant using IEEE double precision oating point arithmetic, and hence can be used to perform e.g. the orientation and incircle tests or even the insphere test [1]. Diculties arise, if the tests to be performed involve previously computed geometric objects which require extended precision to be exactly represented. A more general approach, which is not speci c to determinants or even predicates, are multiprecision packages [2,11,15] They allow arbitrary ....

F. Avnaim et al.: Evaluating signs of determinants using single-precision arithmetic, ( INRIA, 1995)

....one. We have presented a simple plane sweep algorithm of degree 7 to compute the L1 Voronoi diagram of segments in O(n log n) time. Although the degree is low compared to the Euclidean case, a robust implementation for arbitrary segments would require the use of arithmetic filters (see e.g. Refs. [4,7,13]) In the special case of segments in fixed orientations with slopes given by small constants the algorithm is of degree 1 and thus can be implemented robustly by ordinary means. This is typically the case for shapes in VLSI designs. Using the (2nd order) L1 Voronoi diagram of shapes in one layer ....

F.Avnaim, J.D. Boissonnat, O. Devillers, F. Preparata, M. Yvinec, "Evaluating signs of determinants using single precision arithmetic", Algorithmica, 17 1997, 11132.

....combining floating point filters and exact evaluation of predicates; exact computation is performed when the floating point filter fails to provide a certified answer, which is usually rare. New methods have been designed for the exact evaluation of signs of determinants and arithmetic expressions [13, 2, 7], and various exact, adaptive arithmetics [9, INRIA, BP 93, 06902 Sophia Antipolis, France. Jean Daniel.Boissonnat sophia.inria.fr Research partially supported by ESPRIT IV LTR Project No 28155 (Galia) University of British Columbia, Department of Computer Science, and University of North ....

Francis Avnaim, Jean-Daniel Boissonnat, Olivier Devillers, Franco P. Preparata, and Mariette Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17(2):111--132, 1997.

....with integer entries. The rst one is a method based on the Gram Schmidt orthogonalisation process which has been proposed by Clarkson [5] We review the analysis of Clarkson and propose a variant of his method. The second method is an extension to n Theta n determinants of the ABDPY method [1] which works only for 2 Theta 2 and 3 Theta 3 determinants. Both methods compute the signs of a n Theta n determinants whose entries are integers on b bits, by using an exact arithmetic on only b O(n) bits. Furthermore, both methods are adaptive, dealing quickly with easy cases and resorting ....

....for evaluating exactly the sign of a determinant can in many cases avoid to pay the price of a general purpose multi precision package. Clarkson [5] propose an eOEcient method to compute the sign of a determinant whose entries are integers. The so called ABDPY method, due to Avnaim and col. [1], is an alternative solution for 2 Theta 2 and 3 Theta 3 determinants. Using an uncommon multiple term arithmetic, Schewchuck [16] designs an adaptative implementation for low dimensional geometric predicates on AEoating point entries. In this paper, we propose a theoretical and experimental ....

F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Research Report 2306, INRIA, BP93, 06902 SophiaAntipolis, France, 1994.

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F. Avnaim, J.-D. Boissonnat, O. Devillers, F. P. Preparata, and M. Yvinec, Evaluating signs of determinants using single-precision arithmetic, Algorithmica 17 (1997), no. 2, 111--132.

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Francis Avnaim, Jean-Daniel Boissonnat, Olivier Devillers, Franco P. Preparata, and Mariette Yvinec. Evaluating Signs of Determinants Using Single-Precision Arithmetic. Manuscript available from http://www.inria.fr:/prisme/personnel/devillers/anglais/determinant, 1995.

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F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17(2):111--132, 1997.

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F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec, Evaluating signs of determinants using single-precision arithmetic, Algorithmica 17 (1997) 111--132.

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F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17:111--132, 1997.

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F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Algorithmica, 17:111--132, 1997.

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F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluating signs of determinants using single-precision arithmetic. Research Report 2306, INRIA, BP93, 06902 Sophia-Antipolis, France, 1994.

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