Appeared in Proceedings of ISSAC'95 USA. bounds. of a matrix polynomial (or a ratio of matrix polynomials).

Transcription

1 Appeared in Proceedings of ISSA'95 Nmeric-Symbolic Algorithms for Ealating One-Dimensional Algebraic Sets Shankar Krishnan Dinesh Manocha Department of ompter Science Uniersity of North arolina hapel Hill, N USA fkrishnas,manochag@csnced Abstract: We present ecient algorithms based on a combination of nmeric and symbolic techniqes for ealating one-dimensional algebraic sets in a sbset of the real domain Gien a description of a one-dimensional algebraic set, we compte its projection sing resltants We represent the reslting plane cre as a singlar set of a matrix polynomial as opposed to roots of a biariate polynomial Gien the matrix formlation, we make se of algorithms from nmerical linear algebra to compte start points on all the components, partition the domain sch that each reslting region contains only one component and ealate it accrately sing marching methods We also present techniqes to handle singlarities for well-conditioned inpts The reslting algorithm is iteratie and its complexity is otpt sensitie It has been implemented in oating-point arithmetic and we highlight its performance in the context of compting intersection of high-degree algebraic srfaces Introdction The problem of ealating one-dimensional algebraic sets is fndamental in nmeric and symbolic compting It can be simply stated as nding roots of n ane algebraic eqations in (n + ) nknowns Algebraic sets are widely sed for representing objects and constraints in compter graphics, geometric modeling, robotics, compter ision and moleclar modeling Many of the fndamental problems like srfacesrface intersection, osets of cres and srfaces, Voronoi sets generated by cres and srfaces in geometric modeling [Hof89], kinematic analysis of a redndant robot [ra89], robot motion planning [an88], object recognition in compter ision [PK92] and conformation space of moleclar chains [H] correspond to ealating one-dimensional algebraic sets In most cases we are interested in ealating all the components in the sbset of the real domain All these problems hae been extensiely stdied in the literatre It is well known in algebraic geometry that cres of gens zero hae a rational parametrization [Abh9] How- Spported in part by a Sloan Fondation Fellowship, Uniersity Research Award, NSF Grant R , ONR ontract N , and ARPA ontract DABT eer ery few cres hae gens zero and in most applications the problem of ealating algebraic sets corresponds to classifying all components and singlarities of the cre and compting a nmerical approximation sing piecewise straight lines or splines with garanteed nmerical error bonds Main Reslt: We present nmeric algorithms for ealation of one-dimensional algebraic sets in a sbset of the real space They isolate each component of the cre and ealate them with garanteed error bonds Gien a cre, we compte a birationally eqialent algebraic plane cre sing resltants The latter is represented as the singlar set of a matrix polynomial (or a ratio of matrix polynomials) Gien the matrix representation, we make se of algorithms based on eigenales and complex tracing to compte a start point on each component of the cre in the gien domain Frthermore we partition the domain sch that each reslting region has a niqe component We present algorithms for ealating each component, preenting component jmping and handling singlarities The oerall algorithm has been implemented in nite precision arithmetic and works well for well-conditioned problems Its main adantages are eciency and accracy The reslting algorithm is iteratie and its performance is a fnction of the degree of the algebraic cre, the nmber of components in the gien domain and the accracy desired In practice, we hae been able to ealate cres of degree as high as 324 in a few seconds on a SGI Onyx Prior Work: There is a considerable amont ofwork in classic and modern literatre related to ealation of algebraic cres Eery algebraic space cre is birationally eqialent to an algebraic plane cre and the latter can be compted sing Gr}obner bases [Bc89] and resltants Gien an algebraic plane cre, techniqes for desinglarization based on qadratic transformations are gien in [Wal5, Abh9, AB88] Howeer, the reslting algorithm can be exponential in the degree of the cre Algorithms based on ollins' cylindrical algebraic decomposition (AD), [ol75], hae been sed for ealating all components of algebraic cres [Arn83, SS83] Howeer, its worst case complexity is dobly exponential in the nmber of ariables For plane cres, improed polynomial time algorithms based on AD hae been presented in [AF88, AM88] Howeer, the exponent in terms of N (the degree of the cre) is rather high Frthermore, these algorithms are implemented sing exact arithmetic, which makes them slow in practice Other algorithms inclde those based on Whitney's stratied sets and gap theorems [an88] In practice all these algorithms are

2 ecient for low degree cres only Nmerical and nite precision algorithms based on interal arithmetic [Moo79] and homotopy methods [GZ79, Mor92] hae been sed for ealating algebraic sets While the former are slow in practice, the latter hae been restricted to zero-dimensional algebraic sets and ser from problems like component jmping There is a considerable amont of emphasis in the modeling literatre to ealate srface intersections and oset cres [Hof89, Hof9, SN9, M9, Hoh9] and in ision literatre to compte aspect graphs [PK92] This incldes algorithms for compting all components inclding the closed loops Howeer, these algorithms are somewhat restrictie and cannot be sed for ealating general algebraic cres The combination of resltant formlations and matrix comptations hae been sed for ealating zero-dimensional algebraic sets in [Laz83, AS86, Man92, Man94] In this paper, we se them for ealating one-dimensional algebraic sets Organization : The rest of the paper is organized in the following manner In section 2, we reiew some literatre on resltants and matrix comptations and formlate the problem in terms of matrix polynomials Section 3 describes algorithms for compting a start point on each component of the algebraic cre and in section 4 we highlight the tracing algorithm taking care of component jmping and singlarities We present the implementation and performance of the algorithm for srface intersections in section 5 Section 6 addresses isses of robstness and some measres we take to recoer from ill-conditioned inpts We conclde in section 7 2 Backgrond A set of polynomial eqations whose soltion corresponds to a one-dimensional algebraic set is gien by: F(; ; w; w2;:::;wn,) = F2(; ; w; w2;:::;wn,) = Fn(; ; w ; w2;:::;wn,) = : We assme that this algebraic set consists of proper components only Moreoer, we are only interested in ealating the all the components of the cre inside the region D = [U ;U 2][V ;V 2][W (;) ;W (;2) ] [W (2;) ;W (2;2) ] ::: [W (n,;);w (n,;2)] 2 < n+ Formally, the fnctions F i, i = ;2;:::;n, are the components of a ector fnction F : D! < n, D < n+ The soltion to the problem are elements of D that map to the zero ector nder F Since or algorithm works in doble precision oating point arithmetic, the reslts of or algorithm will not always match the otpt specication gien aboe Therefore, we modify it to t or framework Along with the set of eqations, or algorithm reqires the denition of a ector norm k : k on < n and a small positie constant ( machine precision) as additional inpts Then the otpt set consists of all those elements x 2 D sch that k F (x) k < In anmber of modeling and isalization applications, ales of can be proided It is not clear if it is so in the general case We eliminate n, ariables from these eqations sing mltipolynomial resltant algorithms Almost all the projections are one-to-one and reslt in a birationally eqialent cre In or case, we perform a generic linear transformation and eliminate w ;:::;w n, from the reslting set The resltant can be expressed in terms of matrices and determinants In particlar, single determinant formlations are known for ales of n = 2;3;4;5;6 [Dix8, Jo9, M27, SZ94] We se M(; ) to represent the reslting matrix polynomial The most general formlation of the resltant expresses it as a ratio of two determinants [Mac2] Let s denote the top and bottom matrices as P(; ) and Q(; ) respectiely A similar formlation for P(; ) for sparse polynomial systems has been highlighted in [E93] In or case, we will make se of the matrix formlation and represent itasannealated determinant It is possible that P(; ) and Q(; ) are singlar, while the resltant is nonzero In sch cases we se the leading non-anishing minor of P(; ) and represent itasm(; ) It contains the resltant and an extraneos factor The algorithm ealates the reslting algebraic set and sbstittes the ales back into the original eqations to discard the soltions corresponding to the extraneos factor The degree of the algebraic cre, N, is gien by the Bezot or Bernstein bond of the gien system of eqations The plane cre birationally eqialent to the algebraic cre corresponds to the singlar set of M(; ) For the general Macalay's formlation the plane cre corresponds to the dierence of singlar set of P(; ) and singlar set of Q(; ) taking into accont the mltiplicities of indiidal factors For the rest of the paper, we perform a nmber of nmerical comptations like determinants, eigenales, singlar ales of M( i; i) and similar analysis is applicable in the general case based on the ratio of P(; ) and Q(; ) Gien a point on the plane cre, ( ; ), the corresponding point on the space cre, (w ;w2 ;:::;wn, ), is compted sing the kernel of M( ; ) [Man92] The algorithm for ealation of algebraic cres comptes a start point on each component and traces the reslting component The tracing algorithm ses the local geometry of the cre (like deriatie information) to determine sccessie points We shall now describe or method to ealate partial deriaties based on the matrix representation 2 Deriatie omptation We denote the determinant of the matrix M(; )asd(; ) D (; ) and D (; ) represent the rst order partial deriaties with respect to and To be able to trace throgh the cre we need to ealate D( ; ), D ( ; ) and D ( ; ) for a gien point ( ; ) accrately and eciently To compte the rst and higher order partials, we se a simple ariation of Gassian elimination [M9] The basic idea is to compte the partial deriatie of each matrix entry at the beginning of comptation and pdate the deriatie information along with each step of Gassian elimination In this case, we modify the matrix strctre sch that entry consists of a tple G ij (; )=(gij(; );gij ( ;);gij ( ;)); where gij( ; ) and gij( ; ) represent the partial deriaties of g ij(; ) with respect to and at ( ; ) The reslting matrix strctre is then of the form 2 3 M(; )= 4 G (; ) ::: G n (;) ::: G n (;) ::: G nn(; ) To compte D( ; );D ( ; ) and D ( ; ), we perform Gassian elimination We consider the matrix formed by rst entry of each tple (eqialent to M( ; )) and proceed to compte its determinant sing Gassian elimination As a side eect we change the entry in the other 5 :

3 tples Assme we are operating on the ith and kth rows of the matrix Atypical step of Gassian elimination is of the form g kj = g kj, g ki g g ii ij, where g kj represents the element in the k th row and j th colmn of the matrix In the new formlation this step is replaced by three steps: g kj = g kj, g ki g ii g ij, g kj = g kj, (g ki g ij +g ki g ij )g ii,(g ki g ij )g ii (g ii ) 2, and gkj = gkj, (g g ki ij +g ki g )g ij ii,(g ki g ij )g ii (g ii ) 2 We make a choice for the piot element based on the rst tple (ie g ij entry) After Gassian elimination is complete, we compte D( ; );D ( ; ) and D ( ; ) as D( ; ) = Q n gii, P i= D n g ( ; )=D( ; ) ii i= g ii, and D ( ; )= P n g D( ; ) ii i= g ii This procedre can be easily extended to compte the higher order partial deriaties as well The accracy of the reslting algorithm is improed by partial and complete pioting as well For the general Macalay's formlation, we indiidally compte the partials of the determinants of P(; ) and Q(; ) and make se of qotient rle to compte the partials of the resltant 3 omptation of Start Points and Loop haracterization In this section, we shall describe algorithm for compting the start points on eery component of the cre This is ery critical for the tracing algorithm Inside the domain [U ;U 2][V ;V 2], the plane cre consists of two types of components: open and closed Open components hae at least one point lying on the bondary of the domain losed components, on the other hand, lie completely within the rectanglar domain We refer to them as loops In practice, nding start points on loops is signicantly harder than those on open components 3 Start Points on Open omponents From the denition of open components, it is clear that its endpoints mst lie on the bondary of the domain Starting points on all these components can be obtained by soling the system after sbstitting one of = U, = U 2, = V and = V 2 into M(; ) This reslts in a niariate matrix ^M() or ^M() depending on the ariable sbstittion Withot loss of generality, we can assme that the ariable was sbstitted The reslting matrix ^M() can then be written as ^M() = d^md+ d,^md,+:::+ ^M + ^M, where the ^Mi's are nmeric matrices We need to nd all the soltions of this matrix polynomial The roots of ^M() hae a one-to-one correspondence with the eigenales of the companion matrix [Man92] 2 In ::: 3 6 = () ::: In,M,M,M2 :::,Md,, where M i = ^M d ^M i [GLR82]In case ^M d is singlar or illconditioned, the problem is redced to a generalized eigenale problem [Man92] Algorithms to compte all the eigenales are based on QR orthogonal transformations [GL89] They compte all the real as well as complex eigenales To compte a real or complex sbset of the roots, iteratie algorithms are gien in [Man94] If there are few real soltions (two or three in the domain), the latter algorithm is signicantly faster than the QR algorithm For general Macalay's formlation, eigenales comptations are performed on P(; i) and Q(; i) and we take a dierence of 2 L L 3 2 A B L 2 Figre : haracterization of loops the two sets acconting for the mltiplicities of the indiidal roots We label this method as bondary-ale comptation and it gies both the endpoints of eery open component The diclty in identifying start points on closed components lies in the fact that loops hae no sch simple characterization as the one for open components Howeer, we show that we can se a simple algebraic property that wold gide s to at least one point oneery loop 32 Identifying Start Points on Loops The cre D(; ) is an algebraic plane cre in the complex projectie plane dened by and We are, howeer, interested only in nding the part that lies in the portion of the real plane dened by (; ) 2 [U ;U 2][V ;V 2] If we relax this restriction so that one of the ariables, say, can take complex ales, the intersection cre is dened as a continos set consisting of real and complex components Before we gie or loop characterization, we introdce some basic denitions Denition Trning points are points on the plane cre where the tangent ector, as projected in the (; ) space, is parallel to the or parameter axis Moreoer, one of the partial deriaties (with respect to or ) of the cre is For eg, -trning points are points where the tangent is parallel to the axis We classify -trning points into left -trning points and right -trning points Apoint ( ; ) is a left -trning point if the cre goes into the complex domain in the left neighborhoodof (=,, where is a small positie ale) A point ( ; ) is a right -trning point if the cre goes into the complex domain in the right neighborhood of (= +) Lemma If the cre in the real domain [U ;U 2][V ;V 2] consists of a closed component, then two arbitrary complex conjgate paths meet at one of the real points (corresponding to a trning point) on the loop Proof: The proof is obios from the continity properties in the complex space Since the loop is isolated in the real space, it is connected to other components in the complex space onsider the loop shown in Fig At line L, the roots of occr at points A and B The roots come closer together as is changed from L to L 2 At L 2, the roots coincide to form a doble root This is precisely where the trning point occrs When is changed from L 2 to L 3, the doble root corresponding to becomes complex And since all the coecients of the cre are real, the complex roots mst occr in conjgate pairs

4 2 2 A B Q2=(2,2) Q3=(2,3) Q=(,) s D (a) (d) 2 Figre 2: Possible path in complex space 2 Since we need only one point on eery loop to trace it completely, we restrict orseles to left -trning points only The domain has changed from the real plane to a three dimensional space formed by, r and i, where r and i are the real and imaginary ales of We se bondaryale comptation followed by complex tracing (tracing in complex space) to determine trning points on loops Since all complex roots occr in conjgate pairs, it sces to trace only one of them (with positie imaginary ales) Howeer it is not enogh to only trace the complex paths obtained after performing bondary-ale comptation on M(U ;) For example, consider Fig2 A complex path arises from point D (a right-trning point) and ends in (left-trning point of the loop) The complex path is shown in dotted lines This path will neer be detected by the preios method When a complex path toches the real plane the imaginary part (of ) mst pass throgh some small constant ale before redcing to zero These are precisely the common points of the cre with the plane i = In other words, we are trying to nd all the real soltions to the eqation DET(M(; r + i)) = Expanding ot the expression and collecting the real and imaginary terms we can write DET(Mr(; r)+im i (; r)) = (2) It is easy to show that the soltions (; r) satisfying eqation (2) also satisfy DET(R(; r)) =, where Mr(; r),m R(; r)= i (; r) M (3) i (; r) Mr(; r) As before, the soltions to (3) can be posed as the singlar set of matrix R(; r) This singlar set is a discrete point set and the order of the matrix is twice that of M(; ) Initially we form the companion matrix of R(; r), r, similar to the one in Eq() We compte all the eigenales of r at = U (we expect all of them to be complex) We se the ones with positie imaginary points as starting points and trace all the paths in increasing direction ntil it either crosses the = U 2 plane or become real All the real ales of r are points lying ery close to the trning points of the intersection cre They are denoted by ( r; r) Then ( r; r) is sed as an initial gess to conerge to the trning point sing inerse power iterations When complex tracing is done, all the trning points which are (,) R R7 R2 (2,) 2 (,) R6 (b) R3 R5 (,2) R4 (2,2) R4 R46 (2,) R42 R45 (c) R43 (,2) Figre 3: (a)omponent jmping (b)first-leel decomposition (c)second-leel decomposition (d)tracing step potentially part of loops are obtained The details of tracing sing inerse power iterations are presented in the next section 4 Tracing Gien the start points, we ealate the cre sing or tracing algorithm A nmber of algorithms for tracing based on local iteratie methods hae been sed in homotopy methods, srface interrogations and soltions of dierential eqations [Hof89, Mor92] Gien a point on the cre, an approximate ale of the next point is obtained by taking a small step size in a direction determined by the local geometry of the cre Based on the approximate ale, these algorithms se local iteratie methods like Newton's method to trace back on to the cre Gien a start point, these algorithms are applicable to trace algebraic cres as well Howeer, the three main isses concerning tracing algorithms are: onerging back on to the cre 2 Preenting component jmping 3 Ability to handle singlarities and trace throgh mltiple branches The conergence problems arising from the behaior of Newton's method are well known It is rather diclt to predict the conergence of Newton's method on high degree eqations omponent jmping can occr when two components of the cre are relatiely close to each other as shown in g3(a) In this case, the tracing algorithm can jmp from point A on component topoint B on component 2 Most implementations circment this problem R44

5 by choosing ery small and conseratie step sizes Bt this still cannot garantee correctness and moreoer, slows down the algorithm Singlarities are points where the cre selfintersects or the tangent ector anishes They typically lead to mltiple branches arond the singlar point The tracing algorithm has to determine these singlar points ef- ciently and trace all the branches We present an ecient tracing algorithm that can resole all these isses most of the time In particlar weintrodce a techniqe called component splitting and tracing based on inerse power iterations Singlarities are also handled eciently for well-conditioned inpts 4 omponent Splitting After performing bondary-ale comptation and loop detection, a seqence of points are obtained on the cre((; ) 2 [U ;U 2] [V ;V 2]) which either correspond to starting points on open components or some point on loops Using these points the plane cre is traced completely withot missing any important cre featres The idea behind component splitting is that if there are only two bondary points inside a region with no loops, these points belong to the same component of the cre Frther there exists exactly one component of the cre inside this region Therefore, the prpose of the algorithm is to sbdiide the original domain into smaller regions sch that each region contains exactly one cre component We now describe the working of component splitting The inpt into this rotine is a rectanglar domain, specied as [L ;H ] [L ;H ], and a set of points, S, onthe cre inside this domain S coers all the components of the intersection cre inside the domain If the cardinality of S is two, then we are assred of a single cre component and the decomposition terminates If the cardinality ofsis greater than two the algorithm sbdiides the domain along isolines (lines of constant or in the domain) determined by the ales of points of S The isolines chosen at eery point cold either be a -isoline ( = )ora-isoline ( = ) The algorithm arbitrarily chooses the -isoline to sbdiide the domain If sbdiision is not possible (all the points in S hae coordinates as L or H ), then -isoline is chosen for sbdiision In the process, new points corresponding to the intersections of the isolines with the cre are generated and inserted into the appropriate regions The component splitting algorithm is then applied recrsiely to each newly created region Sbdiisions of domains are not carried ot indenitely If the dimensions of a domain become smaller than a speci- ed tolerance, the sbdiision is stopped and checked for singlarities Informally, singlarities are points on the cre where the cre self-intersects or has mltiple branches In the presence of singlarities (exclding csps), no leel of decomposition can prodce sbdomains with one simple cre component nless the singlar point is determined accrately If the algorithm is nable to isolate single cres in a domain after repeated leels of sbdiision, then one of two cases can occr The cre has a singlarity, or Two components of the cre are ery close At this point, minimization of an energy fnction E(; ) distingishes the two cases E(; ) =(D(; ) 2 + D (; ) 2 + D (; ) 2 ) (4) where D(; ) is the determinant ofm(; ), and D (; ) and D (; ) are the partial deriaties of D(; ) with respect to and respectiely The minimization is applied with the midpoint of the region as the initial point A minimm ale of zero (with tolerance) corresponds to a singlarity A non-zero minimm ale means that the cre has two ery close components If there is a singlarity, then sbdiision is done at the singlar point and component splitting is performed at each sbdomain Singlarities of a high degree cre can be ery sensitie to small inpt pertrbations and oating point errors Therefore, the algorithm reports a singlarity if the minimm ale obtained is smaller than a ser-specied ale This method is ssceptible to nmerical errors especially if the singlarities lie ery close to each other We, howeer, beliee that sch pathological cases are rare in practice The psedocode for the aboe algorithm is gien below: omponentsplitting(domain, Xsection points, tolerance) If (there are only two Xsection points) trace the cre inside the region and retrn 2 If (region size is smaller than tolerance) { Apply singlarity criterion { If there is a singlarity Sbdiide the domain at the singlar point along both axes Find all intersection points along the sbdiided cres for each sbregion, do omponentsplitting(sbregion, new points, tolerance) Retrn 3 If (domain conergence is slow) { Diide the domain at midpoint of one of the interals { ompte the intersections of the cre with the diiding line { For each of the two sbregions, do omponentsplitting( sbregion, new points, tolerance) { Retrn else { Diide the domain along isolines from eery Xsection points For the point (L ; ), the corresponding line is = { ompte the intersection of the cre with each sch line { For each sbregion, do omponentsplitting( sbregion, new points, tolerance) { Retrn After performing this algorithm, a set of cres traced inside each region is obtained Some of these are parts of the same cre component By matching their endpoints, they are connected appropriately to obtain the original cre in the [U ;U 2][V ;V 2] domain This algorithm garantees No component jmping - tracing is performed only inside a region that is garanteed to contain jst one cre Singlarity detection - Dring all stages of the algorithm, geometrically isolated singlar points are always bracketed

6 Domain Decomposition Bisection Figre 4: omparing or algorithm s Bisection Figre 5: ase of slow conergence of or algorithm Two leels of the decomposition algorithm hae been highlighted on a particlar cre in Figs3(b) and 3(c) The nmber of leels of sbdiision performed by the algorithm depends directly on the presence of singlarities or close components in the cre The decomposition step is similar in natre to that of interal arithmetic based algorithms Howeer, for most cases or algorithm performs fewer leels of decomposition Interal arithmetic based algorithms [Sny92] perform a nmber of decompositions as a fnction of the accracy parameter These algorithms are sally robst, bt are slow in practice Or algorithm performs abot an order of magnitde faster than interal arithmetic based methods (as applied to srface intersections) 4 Performance and onergence Fig4 proides a comparison between ordinary bisection (corresponding to diiding the domain into eqal hales at each step) and or component splitting algorithm It can be seen that in the case depicted by the gre, or method performs mch better than bisection In fact, on an aerage, or method achiees the desired leel of sbdiision mch faster than bisection This is becase or method is a form of gided sbdiision as opposed to blind partitioning adopted by bisection Howeer, there are instances when the algorithm does not redce the region size appreciably This sally happens when the set of intersections are ery close to the corners of the region One sch example is illstrated in g5 These cases can be detected easily thogh When sch instances are encontered, bisection is performed once on the domain to break the symmetry (of points in set S) omponent splitting is then performed on each half The step size of tracing is determined by the size of each region The algorithm gien aboe is sed to partition the domain of the cre into regions with a single cre component Its complexity is a fnction of the nmber of components and the separation of the components into arios regions For most practical cases, there are a few and wellseparated components in the real domain and the algorithm performs well for sch cases In many ways the nderlying philosophy is rather similar to AD based algorithms for partitioning the domain into regions Or algorithm ses an ecient and accrate zero-dimensional soler [Man94] and works well sing nite precision algorithm On the other hand, the AD based algorithm [Arn83] compte all the extremal and trning points sing prely symbolic methods and exact arithmetic Een thogh this method garantees that the soltion is always topologically reliable, they are impractical becase of their large memory reqirements and poor eciency 42 Tracing in lower dimension After component splitting, the entire domain ([U ;U 2] [V ;V 2]) is sbdiided into smaller regions each with exactly one cre segment It also retrns the two endpoints of the cre inside the region Starting from one of the endpoints, the tracing algorithm comptes sccessie cre points sing the local geometry of the cre ntil the other endpoint is reached Let the component be Gien a point Q =( ; ) the skeleton of the tracing algorithm is gien below Frthermore, it is reqired that any point on the piecewise representation of the cre is not more than apart from the cre ompte D ( ; ) and D ( ; ), the partial deriaties of the cre with respect to and, respectiely This is the ector (on the plane) normal to the plane cre Gien the normal ector, nd the nit ector corresponding to the tangent Let this ector be (t ;t ) Find an approximate point Q 2 =( 2; 2), where 2 = + t S, and 2 = + t S, where S is the step size and S Using ( 2; 2), conerge back to the cre atq 3= ( 2; 3), if j t j > j t j, ortoq 3=( 3; 2), if j t j > j t j sing inerse power iterations A single tracing step is shown in g3(d) The two main components of the tracing algorithm are the choice of step size and tracing back to the cre component sing inerse power iterations We explain each of them in detail For the rest of the analysis we will assme that Q 3 =( 2; 3) Inerse Power Iterations: In a single step of the tracing algorithm, we need to compte the eigenale of M( 2;) which is closest to 2 (g3(d)) As a reslt, we compte the companion matrix from M( 2;) (see eq () ) and set s = 2 Therefore, we need to compte the smallest eigenale of the matrix, si The smallest eigenale of, si corresponds to the largest eigenale of (, si), Instead of compting the inerse explicitly (which isnmer- ically nstable), we se inerse power iterations [GL89] To sole the matrix system eciently, we se LU decomposition of the matrix (, si) sing Gassian elimination We also make se of the strctre of the matrix to redce its complexity Gien s, let B =, si B is of the form: I n I n ::: ::: B = ::: I n I n P P 2 P 3 ::: P m A

7 where is a fnction of s The LU decomposition of B has the form: I n ::: I n I n ::: I B n ::: I B AB n ::: A ::: R R 2 ::: L m ::: U m where L m and U m correspond to the LU decomposition of R m R i's can be easily compted from the P i's LU decomposition can sometimes face nmerical problems if the matrix B is ill-conditioned In sch cases, QR factorization can be sed, where Q and R are orthogonal and pper trianglar matrices respectiely Akey property ofinerse power iteration is that it conerges to the eigenale closest to s If the closest eigenale to s is a complex conjgate pair, the power method does not conerge to any real ale In sch cases, the tracing algorithm chooses a smaller step size for comptation Step Size omptation: The step size S is chosen to preent component jmping Toaoid component jmping the following constraints are imposed on Q 2 Let the closest distance of Q 2 to the domain bondary be as shown in g3(d) As a reslt, any point onany other component of the cre is at least away Frthermore, the distance from Q 2 to is at most If<, the inerse power iteration garantees that after the conergence of power iteration the reslting point is still on Therefore, a bond on the choice of stepsize is gien by the condition <S< We initially choose a ale of S and check whether S < If this constraint is not satised we rene the ale of S sing a binary search oer the range [;S] Ths making se of component splitting and inerse power iterations, we aoid component jmping dring tracing 5 Implementation and Performance The algorithm has been implemented and its performance was measred on a nmber of models The algorithm ses existing EISPAK [GBDM77] and LAPAK [ABB + 92] rotines for some of the matrix comptations At each stage of the algorithm, we can compte bonds on the accracy of the reslts obtained based on the accracy and conergence of nmerical methods adopted like eigenale comptation, power iteration and Gassian elimination The algorithm was implemented on an SGI Onyx workstation This workstation has 28MB of main memory and a specfp rating of 97 In most applications, the nmber of components in the real domain is small and well separated and the algorithm performs ery well in sch cases 5 Application to Srface Intersection In this section, we shall discss the application of or algorithm to a particlar case, srface-srface intersection In most applications, the problem appears as intersection of two parametric srfaces A special case of parametric srface is the tensor-prodct Bezier srface (simply called Bezier patch) A Bezier patch, F(s; t), of degree m n is represented as: F(s; t) = m i= n j=v ijbi m (s)bj n (t), where = hw ijx ij;w ijy ij;w ijz ij;w iji are the control points of V ij the patch in homogeneos coordinates and B m i (s) = s i (,s) m,i is the Bernstein polynomial The domain of the m i srface is dened on the nit sqare s; t in the (s; t) plane Gien two Bezier srfaces, F(s; t) and G(; ) F(s; t) = (X(s; t); Y(s; t); Z(s; t); W(s; t)) G(; ) = (X(; ); Y (; ); Z(; ); W (; )) represented in homogeneos coordinates, their intersection cre is dened as the set of common points in 3-space and is gien by the ector eqation F(s; t) =G(; ) This reslts in the following set of three eqations in for nknowns: F(s; t; ; ) =X(s; t)w(; ), X(; )W (s; t) = F2(s; t; ; ) =Y(s; t)w (; ), Y (; )W (s; t) = (5) F3(s; t; ; ) =Z(s; t)w (; ), Z(; )W (s; t) = ; and the domain of the intersection cre is(s; t; ; ) 2 [; ] [; ] [; ] [; ] The ariables s and t are eliminated sing Dixon's resltant [Dix8] If the patch F(s; t) is of degree m n, then the Dixon's resltant obtained by eliminating s and t from F ;F 2 and F 3 is a biariate matrix (M(; )) of order 2mn The singlar set of this matrix corresponds to the intersection cre The intersection cre was ealated sing or algorithm for a nmber of cases Some of them are shown in gs6 and 7 It takes abot, 3 seconds to ealate algebraic cres of degree or more with abot three or for components in the domain Fig8 shows intersection of two goblets Each of them is composed of 72 bicbic Bezier patches The intersection cre is a piecewise algebraic space cre of degree 324 The oerall algorithm initially prnes ot non-intersecting pairs of srfaces sing bonding boxes Eentally, itealates 57 pairs of intersecting srfaces and the reslting algorithm takes abot 2:8 seconds In all the examples, the Eclidean norm was sed and the ale of was,5 We beliee that a nmber of optimization techniqes can be incorporated in or implementation to gie better reslts Unfortnately, there are no existing benchmarks aailable to test or algorithm Frther, there are ery few pblished performance reslts on srface intersection algorithms Interal arithmetic has been sed to compte srface intersections In compting the intersection cre between a bmpy sphere and a cylinder, [Sny92] reports a rnning time of seeral mintes on a HP9 series 835 workstation [BHHL88, BR9] se qadratic transformations to remoe singlarities from the intersection cre and trace it in higher dimensions [MD94a] proide ecient algorithms to ealate zero-dimensional algebraic sets Methods of [AF88, AM88] ealate the topological types of plane algebraic cres sing exact arithmetic These algorithms proide ery accrate reslts, bt are inecient for high degree cres 6 Problem onditioning The three most important considerations in the ealation of algebraic cres are robstness, accracy and eciency There is a clear trade-o between these three, since the larger the robstness enhancement and accracy comptations the slower the exection time of the algorithm While it is almost impossible to proide an algorithm that can satisfy all of them completely together, one mst at least target an algorithm that can proide a good fraction of both in most cases and can be ne-tned according to the reqirements of the application In order to garantee robstness, a general algorithm mst be able to determine the conditioning of the problem

8 intersection cre Figre 6: omponent Splitting applied to this intersection Figre 7: Patches haing (i) close intersection cres (ii) intersecting in a singlarity The conditioning becomes more signicant becase of errors introdced by nmerical comptations If the inpt data changes by, the otpt reslts will change by a fnction () For ery small ales of, there may exist a constant sch that () [Hof89] If is small the problem is said to be well-conditioned A large ale of signies an ill-conditioned problem The ale is called the condition nmber Howeer, it is nontriial to calclate for general algebraic problems, as shown by [Ren94] In or case, the algorithm works well for well-conditioned inpts Or algorithm (which redces starting point comptation and tracing to soling eigensystems) can enconter illconditioned inpts if the system has higher mltiplicity eigenales Since the redction process inoles ronding errors, the reslting system will hae a clster of eigenales near the original mltiple eigenale Howeer, it has been shown that the arithmetic mean of the clster is sally mch less sensitie to small pertrbations than indiidal eigenales [Kat8] In sch cases, or algorithm comptes the arithmetic mean of the clster by following a simple heristic which monitors the condition nmber of indiidal eigenales [MD94b] 7 onclsion We hae presented a general algorithm based on nmeric and symbolic methods for ealation of algebraic cres in arbitrary dimensions It comptes a birationally eqialent plane cre sing resltants and represents it as a matrix polynomial The reslting algorithm is based on nmerical matrix comptations and geometric characterization of Figre 8: Intersecting goblets the cre It has been sed to accrately compte intersection of high degree srfaces In terms of eciency it otperforms earlier methods based on AD, desinglarization etc and it is more robst and accrate (in terms of identifying all components and singlarities) as compared to specic algorithms for srface intersection presented in [Hof89, Hoh9, SN9] References [AB88] SS Abhyankar and Bajaj omptations with algebraic cres In Lectre Notes in ompter Science, olme 358, pages 279{284 Springer Verlag, 988 [ABB + 92] E Anderson, Z Bai, Bischof, J Demmel, J Dongarra, J D roz, A Greenbam, S Hammarling,

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