Computing exact geometric predicates using mo dular arithmetic with single precision z Herv Brnnimann Ioannis Z Emiris Victor Y Pan Sylvain Pion Intro duction Abstract We prop ose an ecient metho d that deter mines the sign of a multivariate p olynomial expression Most of geometric predicates can b e expressed as com with integer co ecients This is a central op eration on puting the sign of an algebraic expression In principle which the robustness of many geometric algorithms de one may compute such expressions by using oating p ends The metho d relies on mo dular computations point arithmetic with a xed nite precision fp arith for which comparisons are usually thought to require metic but then the roundo errors may easily lead to multiprecision Our novel technique of recursive relax the wrong sign This problem is often referred to as the ation of the moduli enables us to carry out sign deter robustness problem in computational geometry mination and comparisons by using only oating p oint One solution to the robustness problem is to explic computations in single precision The metho d is highly itly handle numerical inaccuracies so as to design an parallelizable and is the fastest of all known multipreci algorithm that do es not fail even if the numerical part sion metho ds from a complexitypoint of view Weshow of the computation is done approximately or how to compute a few geometric predicates that reduce to analyze the error due to the fp imprecision to matrix determinants We discuss implementation Such designs are extremely involved and haveonlybeen eciency which can be enhanced by go o d arithmetic done for a few algorithms The general solution it has lters We substantiate these claims by exp erimental been widely argued is to compute the predicates ex results and comparisons to other existing approaches actly This can b e achieved in many This metho d can be used to generate robust and e ways computing the algebraic expressions with in cient implementations of geometric algorithms includ nite precision with a nite but much higher pre ing solid mo deling manufacturing and tolerancing and cision that can be shown to suce or by using numerical computer algebra algebraic representation an algorithm that p erforms a sp ecic test exactly In of curves and p oints symb olic p erturbation Sturm se the last category much work has fo cused on comput quences and multivariate resultants ing the sign of the determinant of a matrix with inte ger entries which applies to many geometric Keywords computational geometry exact arithmetic tests suchasorientation tests incircle tests compar robustness mo dular computations single precision RNS ing segmentintersections as well as to algebraic prim Residue Numb er Systems itives such as resultants and algebraic representations of curves and surfaces Recently some techniques have INRIA SophiaAntip olis BP Route des Lucioles SophiaAntip olis Cedex FRANCE This researchwas par b een devised for handling arbitrary expressions and fp tially supp orted by the ESPRIT IV LTR Pro ject No representation CGAL In our presentpaperwe prop ose a metho d that de z Department of Mathematics and Computer Science Lehman College City University of NewYork Bronx NY USA termines the sign of a multivariate p olynomial expres Work on this pap er was p erformed while visiting the second sion with integer co ecients using no op erations other author and was supp orted by NSF Grants CCR and than mo dular arithmetic and fp computations with and PSCUNY Awards and a xed nite single precision The latter op erations can be p erformed very fast on usual computers The Chinese remainder algorithms enable us to p erform ra tional algebraic computations mo dulo several primes that is with a lower precision and then to combine them together in order to recover the desired output ently and The algorithm of Hung and value The latter stage of combining the values mo d Parhami corresp onds to single application of the ulo smaller primes however was always considered a second stage of our recursive relaxation of the mo duli b ottleneck of this approach b ecause higher precision Such a single application suces in the context of the computations were required at this stage Our pap er goal of that is application to divisions in RNS prop oses a new technique whichwecallrecursive relax but in terms of the sign determination of an integer ation of the moduli and which enables us to resolvethe this only works for an absolutely larger input The pa latter problem Due to this technique we correctly re per gives probabilistic estimates for early termina coverthesignofaninteger from its value reduced mo d tion of Newtons interp olation pro cess whichwe apply ulo several smaller primes and we only use some sim in our probabilistic analysis of our algorithm Its main ple lower precision computations at the recovery stage sub ject is an implementation of an algorithm computing This should make our algorithms of some indep endent multidimensional convex hulls The pap er do es not interest also for the theory and practice of algebraic use our techniques of recursive relaxation of the mo duli computing Our deterministic algorithms and of and it do es not contain the basic equations of sections and resp ectively sp ecify our approachand our section our technique based on Lagranges and Newtons inter p olation formulae resp ectively Our algorithm of sec tion gives a probabilistic simplication of algorithm Exact sign computation using mo dulararithmetic Preliminary exp erimental results and running times are discussed in section In general our metho ds are com Mo dular computations Our mo del of a computer is that parable in sp eed to other exact metho ds and even faster of a fp pro cessor that p erforms op erations at unit cost for particular inputs by using bbit precision eg in the IEEE double precision standard we have b It is a realistic mo del as it covers the case of most workstations used Related work Performing exact arithmetic is usually in research and industry We will use mainly one basic exp ensive Thus it is customary to resort to arithmetic prop erty of fp arithmetic on such a computer for all lters those lters safely evaluate a predicate in four arithmetic op erations and for computing a square most cases in order to avoid p erforming a more ex ro ot to o but we will not need it the computed result pensive exact implementation The dicult cases arise is always the fp representation that b est approximates when the expression whose sign we wish to compute is the exact result This means that the relative very small For typical lters the smaller this quantity error incurred by an op eration returning x is at most the slower the lter this is referred to as adap b blog jxjbc and that the absolute error is at most tivity Mo dular arithmetic displays an opp osite kind of All logarithms in this pap er are base In particular adaptivity with a smaller quantity fewer mo duli have op erations p erformed on pairs of integers smaller than to be computed hence the test is faster Typically b are p erformed exactly as long as the result is also when lters fail they also provide an upp er b ound on b smaller than the absolute value of the expression whose sign wewish Let m m be k pairwise relatively prime inte to compute see many details and estimates in k Q gers and let m m For anynumber x not nec This b ound can then be used to determine how many i i essarily an integer welet x x mo d m b e the only mo duli should be taken Mo dular arithmetic is there i i m m i i number in the range such that x x is a fore complementary to the ltering approach We also i multiple of m This op eration is always among the observe this in section i standard op erations b ecause it is needed for reducing Residue Number Systems RNS express and ma the arguments of p erio dic functions nipulate numb ers of arbitrary precision by their mo duli To b e able to p erform arithmetic mo dulo m on inte with resp ect to a given set of numb ers They have b een i gers by using fp arithmetic with bbit precision we will p opular b ecause they provide a cheap and highly paral b assume that m Performing mo dular multi lelizable version of multiprecision arithmetic It is im i m m i i plications of two integers from the interval p ossible here to give a fair and full accountonRNSbut can b e done bymultiplying these numb ers and return Knuth and Aho Hop croft and Ullman provide ing their pro duct mo dulo m The pro duct is smaller a go o d intro duction to the topic From a complexity i b than in magnitude and hence is computed exactly point of view RNS allows to add and multiply numb ers Performing additions can be done very much in the in linear time Its weak p oint is that sign computation m m i i same way but since the result is in the range and comparisons are not easily p erformed and seem to taking the sum mo dulo m is more easily achieved by require full reconstruction in multiple precision which i adding or subtracting m if necessary Integral divi defeats its purp ose This is precisely the issue that our i sions mo dulo m can b e computed using Euclids algo pap er handles i rithm we will need them in this pap er only in section The closest predecessors of our work are appar k rather than k mo duli Recursivelywe will reduce Therefore arithmetic mo dulo m can b e p erformed us i the solution to the case of a single mo dulus m where ing fp arithmetic with bbit precision provided that b x x We