H. Brnnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174182, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for fast, exact determinant sign computation. The most powerful general technique computes the determinant of an integer matrix using modular arithmetic. The matrix determinant is computed modulo several primes, and the complete determinant is reconstructed from the residues. Bronnimann et al. [3] have recently improved the reconstruction step by avoiding the use of multiprecision arithmetic. The authors 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 ....

....of 1=p m 1 . Newton s algorithm is more efficient than Lagrange s when the determinant magnitude is significantly less than Hadamard s bound. See the cited work and Knuth [17] for the details of these reconstruction algorithms. A third reconstruction algorithm is given by Bronnimann et al. [3], based on Lagrange s algorithm. Theirs is different from all other known methods in that it requires only single precision operations (assuming single precision input) yet computes the sign correctly. Modular arithmetic has two main advantages. First, the modular computation can be carried out ....

[Article contains additional citation context not shown here]

Herve Bronnimann, Ioannis Emiris, Victor Pan, and Sylvain Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....symbolic computation. Among the methods used to increase the efficiency of exact computations are those based on interval arithmetic [28, 26] floating point filters [17, 18] lazy arithmetic [2] tuned computations [17, 18] precision driven computation [46] minimized intermediate computation [8, 5], fast hardware computation [42] and modular arithmetic [17, 5] Libraries supporting basic exact computation have been developed, with LEDA [38] and CORE [27] being notable examples. These libraries, however, support only linear computations and a limited set of algebraic computations, and are ....

.... efficiency of exact computations are those based on interval arithmetic [28, 26] floating point filters [17, 18] lazy arithmetic [2] tuned computations [17, 18] precision driven computation [46] minimized intermediate computation [8, 5] fast hardware computation [42] and modular arithmetic [17, 5]. Libraries supporting basic exact computation have been developed, with LEDA [38] and CORE [27] being notable examples. These libraries, however, support only linear computations and a limited set of algebraic computations, and are not sufficient for general boundary evaluation problems. While ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic w ith single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....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 algorithms have been successfully used for computing convex hulls and Voronoi diagrams of point sets and Boolean ....

....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] applies to matrices with integer entries only. Within this context, the iterative reconstruction algorithms are interesting in that their running times are determined by the magnitude of the determinant, which is not necessarily related to the precision required by a forward error based ....

Herve Bronnimann, Ioannis Emiris, Victor Pan, and Sylvain Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....symbolic computation. Among the methods used to increase the efficiency of exact computations are those based on interval arithmetic [29, 27] floating point filters [18, 19] lazy arithmetic [2] tuned computations [18, 19] precision driven computation [47] minimized intermediate computation [8, 5], fast hardware computation [43] and modular arithmetic [18, 5] Libraries supporting basic exact computation have been developed, with LEDA [39] and CORE [28] being notable examples. These libraries, however, support only linear computations and a limited set of algebraic computations, and are ....

.... efficiency of exact computations are those based on interval arithmetic [29, 27] floating point filters [18, 19] lazy arithmetic [2] tuned computations [18, 19] precision driven computation [47] minimized intermediate computation [8, 5] fast hardware computation [43] and modular arithmetic [18, 5]. Libraries supporting basic exact computation have been developed, with LEDA [39] and CORE [28] being notable examples. These libraries, however, support only linear computations and a limited set of algebraic computations, and are not sufficient for general boundary evaluation problems. While ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic w ith single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....an n n matrix A is a classical problem. Numerical methods are usually focused on computing the sign via an accurate approximation of the determinant. Among the applications are important problems of computational geometry that can be reduced to the determinant question; the reader may refer to [11, 12, 9, 10, 46, 43] and to the bibliographies therein. In symbolic computation, the problem of computing the exact value of the determinant is addressed for instance in relation with matrix normal forms problems [41, 28, 23, 51] or in computational number theory [16] In this paper we survey the known major results ....

H. Brnnimann, I.Z. Emiris, V.Y. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic. In Proc. 13th Annual ACM Symp. Comput. Geom., pages 174182, 1997.

....Nrs. DMS 9977392, CCR 9988177, and CCR 0113121 (Kaltofen) and by the Centre National de la Recherche Scientifique, Actions Incitatives No 5929 et Stic LinBox 2001 (Villard) Preprint submitted to ALA 2001 JCAM Special issue 3 December 2001 to the determinant question; the reader may refer to [11,12,9,10,46,43] and to the bibliographies therein. In symbolic computation, the problem of computing the exact value of the determinant is addressed for instance in relation with matrix normal forms problems [41,28,23,51] or in computational number theory [16] In this paper we survey the known major results ....

H. Bronnimann, I.Z. Emiris, V.Y. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic. In Proc. 13th Annual ACM Symp. Comput. Geom., pages 174--182, 1997.

....degeneracies must be taken into consideration. The numerical precision problem was solved in the theory of geometric algorithms by assuming infinite precision real arithmetic [28] For certain algorithms and geometric objects this assumption is realizable in practice by using exact arithmetic [1] [3], 4] 6] 15] 31] 37] Computing with exact arithmetic is in general more costly than using floating point arithmetic, and in certain cases not realizable because of the geometric primitives that need to manipulated. Here again there is a gap between what could in theory be handled by exact ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....direct use of these algorithms can be quite slow and many techniques have been proposed to improve their speed. These include algorithms that explicitly keep track of error in the expression evaluation [Yap97, Yap97, ABD 97] or improved algorithms for arbitrary precision arithmetic [She96, BEPP97, Pri91, KLPY99] or use of filters that exploit features of the predicates or make assumptions on the size of input or the size of the predicates [FV93, She96, YD95] These algorithms have been successfully used for computing convex hulls and Voronoi diagrams of point sets and Boolean combinations ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....use of these algorithms can be quite slow and many techniques have been proposed to improve their speed. These include algorithms that explicitly keep track of error in the expression evaluation [KLN91, Yap97, ABD 97] or improved algorithms for arbitrary precision arithmetic [She96, Gro97, BEPP97, Pri91, Bai93] or use of filters that exploit features of the predicates or make assumptions on the size of input or the size of the predicates [FV93, She96, YD95] These algorithms have been successfully used for computing convex hulls and Voronoi diagrams of point sets and Boolean combinations ....

....97] 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 [Gro97] and extended by [BEPP97] applies to matrices with integer entries only. Within this context, the iterative reconstruction algorithms are interesting in that their running times are determined by the magnitude of the determinant, which is not necessarily related to the precision required by a forward error based ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....of degree at most d. Algorithms of degree d can therefore be implemented exactly in the predicate arithmetic model of degree d. This model is motivated by recent results that show that evaluating the sign of a polynomial expression may be faster than computing its value (see [ABD 97, BY97, BEPP97, Cla92, She96] This model is however very conservative since the non availability of the arithmetics required by a predicate is assimilated to an entirely random choice of the value of the predicate. The second model, called the exact arithmetic of degree d, is more demanding. It assumes that ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174182, 1997.

....bound. Low precision, even floating point, suffices most of the time, since most instances of geometric predicates are easy. In addition, for some basic predicates like the sign of a determinant, there are alternative evaluation strategies that require arithmetic with relatively low precision [1, 2, 3]. Exact arithmetic would be more useful if high level geometric rounding algorithms were available. Virtually any geometric construction that produces new geometric data increases the bitlength of geometric coordinates. For example, suppose points are represented with homogeneous integer ....

H. Bronnimann, I. Emiris, V. Pan, S. Pion, Computing exact geometric predicates using modular arithmetic with single precision, Proc. Thirteenth Ann. Symp. Comp. Geom, pp. 174--182, 1997.

....R k R j is r.e. in k; j. In fact, this predicate is recursive, that is, there exists a program able to decide, for any pair of integers k and j, whether or not R k R j . From a more practical point of view, this implies that the Boolean operators on rational polyhedra are computable (see [6] for an efficient implementation) and that a subset is compact if and only if it is the intersection of a countable set of rational polyhedra. By the general notion of computability in domains (see the appendix) an element (A; B) 2 S[0; 1] n is computable if the set fkjR k (A; B)g is r.e. ....

....which exists for every subset since F is a bounded complete domain (see the appendix) P is the best continuous extension of p. It is possible to compute the image P (K; L) of any pair (K; L) of rational interval disks, as this reduces to the evaluation of the sign of a few polynomials over Q (see [6]) Then, from two increasing sequences of rational interval disks (increasing with respect to v) defining a pair of interval disks, one can compute an increasing sequence in F defining their relative position. The actual image is computed after a finite time. However, when this image is not a ....

H. Bronimann, J. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. ACM Annual Symposium on Computational Geometry, 1997.

....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 [10, 2, 6], and various exact, adaptive arithmetics [7, 16, 17, 19] and various floating point filters, both static and dynamic, have been experimentally tested [ 11, Second, researchers have investigated algorithms that give approximate results with provable properties and guarantees on efficiency ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

.... available publicly [42] The issues of exact computation versus floating point arithmetic are discussed in detail in a survey by Yap and Dub e [50] Recent work includes papers by Avnaim et al. 2] and Bronnimann and Yvinec [12] on the exact evaluation of integer determinants, Bronnimann et al. [11] on exact geometric predicates with single precision arithmetic, and Shewchuk s design and implementation [43] of four predicates based on adaptive floating point arithmetic. See the recent survey papers by Yap and Dub e [49, 50] Exact arithmetic is offered by the geometric software packages LEDA ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing Exact Geometric Predicates Using Modular Arithmetic with Single Precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, Nice, France, June 1997.

....or a determinant. Designing a specialized implementation for evaluating such predicates can in many cases avoid the use of a general purpose multi precision software. In the last few years, a number of algorithms have been proposed to reliably evaluate the sign of a determinant [ABD 97, BEPP97, BY97, Cla92] While these algorithms are general, their implementations have been restricted to determinants of small matrices of order (e.g. up to 6 Theta 6) or assume that the matrix entries are restricted based on the machine precision (e.g. 53 bits for IEEE double precision arithmetic) ....

....algorithm mistakenly terminates and produces a wrong answer. For a large matrix with a small determinant, Newton s algorithm lets us compute the determinant over as few as two finite fields, instead of several hundred. Recently, a new reconstruction method has been proposed by Bronnimann et al. BEPP97] where the reconstruction process gives an early exit when the determinant is fairly small. It is effective for matrices with short entries, but it seems to require that the determinant be computed over all of the fields prescribed by Hadamard s bound before reconstruction begins, and therefore ....

[Article contains additional citation context not shown here]

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....degeneracies must be taken into consideration. The numerical precision problem was solved in the theory of geometric algorithms by assuming infinite precision real arithmetic [28] For certain algorithms and geometric objects this assumption is realizable in practice by using exact arithmetic [1] [3], 4] 6] 15] 31] 37] Computing with exact arithmetic is in general more costly than using floating point arithmetic, and in certain cases not realizable because of the geometric primitives that need to be manipulated. Here again there is a gap between what could in theory be handled by exact ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....bound. Low precision, even floating point, suffices most of the time, since most instances of geometric predicates are easy. In addition, for some basic predicates like the sign of a determinant, there are alternative evaluation strategies that require arithmetic with relatively low precision [1, 2, 3]. Exact arithmetic would be more useful if high level geometric rounding algorithms were available. Virtually any geometric construction that produces new geometric data increases the bitlength of geometric coordinates. For example, suppose points are represented with homogeneous integer ....

H. Bronnimann, I. Emiris, V. Pan, S. Pion, Computing exact geometric predicates using modular arithmetic with single precision, Proc. Thirteenth Ann. Symp. Comp. Geom, pp. 174--182, 1997.

....sign of the determinant of a suitable matrix. The classic algorithm for computing the determinant of an integer matrix is based on the Chinese remainder theorem. A recent treatment may be found in [MC93] and an efficient implementation may be found in the LiDIA library [BBP95] Bronniman et al. BEPP97] improve this technique with a new Chinese remainder reconstruction algorithm. Much of the recent work on the exact determinant sign problem has focused on small matrices. Karasick et al. KLN91] present a variety of techniques based on exact interval arithmetic on matrices of order 2 4. ....

....were no larger than 65, 82, and 116 bits, respectively. Comparison of methods. In table 7, we show that the four stage filter improves on the speed of the general Chinese remainder algorithm as implemented in LiDIA. The routine we call Inria is Sylvain Pion s implementation of the algorithm in [BEPP97] The Inria code and LiDIA implement essentially the same algorithm, and both take advantage of IEEE double precision floating point to compute in modular arithmetic. The Inria code allows as input only matrices with entries up to 53 bits in length, the largest integers that will fit in a double. ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

....D = x Delta y Gamma x Delta y. From the mathematical viewpoint, the difference D is the determinant of the matrix x x y y : In computational geometry, there are known fast algorithms which check the sign of the determinant faster than known algorithms for computing it (see, e.g. [1, 2, 3, 4, 12]) These algorithms are not directly applicable in our case, because they cut down on the algebraic complexity (number of algebraic operations real number multiplications and additions) in the limit when the size of the matrix increases, while we are interested in cutting down the bit ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion, "Computing exact geometric predicates using modular arithmetic with single precision", Proc. 13th Annual ACM Symposium on Computational Geometry, 1997, pp. 174--182.

....geometry, and it is planned to incorporate the solver into the Computational Geometry Algorithms Library CGAL, a joint project of seven European sites (see http: www.cs.ruu.nl CGAL ) A more tuned implementation of exact arithmetic would be a natural next step. For instance, Bronnimann et al. [5] have shown that modular arithmetic can yield much faster computation of determinants; the technique might also apply to the matrix operations considered here. It is possible to handle nonlinear optimization problems with our approach as well; in computational geometry, typical examples are the ....

H. Bronnimann, I. Z. Emiris, V. Y. Pan, S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, 1997.

.... available publicly [41] The issues of exact computation versus floating point arithmetic are discussed in detail in a survey by Yap and Dub e [49] Recent work includes papers by Avnaim et al. 3] and Bronnimann and Yvinec [13] on the exact evaluation of integer determinants, Bronnimann et al. [12] on exact geometric predicates with single precision arithmetic, and Shewchuk s design and implementation [42] of four predicates based on adaptive floating point arithmetic. See the recent survey papers by Yap and Dub e [49, 48] Exact arithmetic is offered by the geometric software packages LEDA ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing Exact Geometric Predicates Using Modular Arithmetic with Single Precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174--182, Nice, France, June 1997.

....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 [6, 1, 3] or using general solutions such as exact integer arithmetic [9] 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. For all these methods we observe that they are, practically, several orders of magnitude slower than the straightforward, inexact AEoating point ....

....This is an important problem in computational geometry since many geometric predicates are expressible by determinants. The following methods are available to compute the exact sign of any determinant A 2 F d;d : ffl Exact integer or AEoating point arithmetic ffl Exact modular arithmetic [2] ffl Clarkson s reorthogonalization method [6, 3] ffl The lattice method [1, 3] In order to apply some of these methods, it is necessary to make the matrix entries integral by multiplying the matrix with a large enough power of 2. Note that this scaling does not change the determinant s sign, ....

H. Bronnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174182, 1997.

.... [BP] Recently, it turned out that some of the most fundamental problems of computational geometry (such as the computation of convex hulls and Voronoi diagrams) are reduced to the computation of det A or, more precisely, its sign, that is, testing whether det A = 0, det A 0, or det A 0 [BEPP97], BEPP98] BKMNSU] EC] Em] FVW] Y97] YD] In many areas of computational geometry, lower dimensional problems must be solved, and then n ranges between 2 and 10, usually staying below 5. In this class of applications, the matrix A is filled with long numbers, representing the real ....

....[C] though the correctness certification of the output of the proposed algorithm (based on the modified Gram Schmidt method) complicated and slowed down the computation. The algorithm of [ABDPY] competes with one of [C] for n 4 but does not work well for larger n. Recent progress reported in [BEPP97], BEPP98] relies on using symbolic algorithm that computes det A modulo several primes p 1 ; p k such that their product exceeds j det Aj. The papers propose effective algorithms for the recovery of the sign of det A from these data, based on some novel application and extensions of the ....

[Article contains additional citation context not shown here]

H. Bronnimann, I. Z. Emiris, V. Y. Pan, S. Pion, Computing Exact Geometric Predicates Using Modular Arithmetic with Single Precision, Proc. 13th Ann. ACM Symp. on Computational Geometry, 174-182, ACM Press, New York, 1997.

....recently achieved by combining numerical and algebraic techniques for the computation of the sign of algebraic and algebraic geometric predicates. This progress includes in particular the computation of the sign of matrix determinant, which is a major problem of practical geometric computations [BEPP97], BEPP99] PY99] We organize the paper as follows. In the next section, we will specify our models of computing. Then we cover polynomial rootfinding in section 3, solution of a polynomial system of equations in section 4, computation of approximate polynomial gcd in section 5 and computations ....

H. Bronnimann, E. Z. Emiris, V. Y. Pan, S. Pion, Computing Exact Geometric Predicates Using Modular Arithmetic with Single Precision, Proc. 13th Ann. ACM Symp. on Computational Geometry, 174-182, ACM Press, New York, 1997.

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H. Brnnimann, I. Emiris, V. Pan, and S. Pion. Computing exact geometric predicates using modular arithmetic with single precision. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 174182, 1997.

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