Solving Systems of Polynomial Equations

Appeared in IEEE Computer Graphics and Applications, March 1994

Dinesh Manocha

Department of Computer Science

University of North Carolina,

Chap el Hill, NC 27599-3175

Abstract:

Current geometric and solid mo deling systems use semi-algebraic sets for de ning the

b oundaries of solid ob jects, curves and surfaces, geometric constraints with mating rela-

tionship in a mechanical assembly,physical contacts b etween ob jects, collision detection.

It turns out that p erforming many of the geometric op erations on the solid b oundaries or

interacting with geometric constraints is reduced to nding common solutions of the p oly-

nomial equations. Current algorithms in the literature based on symb olic, numeric and

geometric metho ds su er from robustness, accuracy or eciency problems or are limited

to a class of problems only.

In this pap er we present algorithms based on multip olynomial resultants and matrix

computations for solving p olynomial systems. These algorithms are based on the linear

algebra formulation of resultants of equations and in many cases there is an elegant re-

lationship b etween the matrix structures and the geometric formulation. The resulting

algorithm involves singular value decomp ositions, eigendecomp ositions, Gauss elimination

etc. In the context of oating p oint computation their numerical accuracy is well under-

stood. We also present techniques to make use of the structure of the matrices to improve

the p erformance of the resulting algorithm and highlight the p erformance of the algorithms

on di erent examples.

Supp orted in part by a Junior FacultyAward and DARPA Contract DAEA 18-90-C-0044

1 Intro duction

Geometric and solid mo deling deal with issues of representation and manipulation of phys-

ical ob jects. These elds have received much attention throughout the industrial and

academic communities for more than three decades. Most of the current formulations

of geometric ob jects curves and surfaces are in terms of p olynomial equations and in

many application like b oundary computations, the problems are reduced to manipulating

a p olynomial systems. Polynomial equations are used for representing semi-algebraic sets.

The imp ortance of semi-algebraic sets has b een established in many other applications

of geometric computation b esides b oundary representations. For example, the geometric

constraints asso ciated with kinematic relationships in a mechanical assembly comp osed

of prismatic and revolute joints de ne semi algebraic sets. Other applications include the

representations of voronoi diagrams of set-theoretic mo dels [1], formulation of con gura-

tion space of a rob ot in motion planning applications [2] etc. A fundamental problem in

these geometric computations is that solving systems of p olynomial equations. In particu-

lar, the problems of curveintersection, ray tracing curves and surfaces, inverse kinematics

of serial or parallel mechanisms, collision detection, computing the distance from a p oint

to a curve, nding a p oint on the bisector b etween two curves or a p oint equidistant from

three curves and solving geometric constraint systems are reduced to nding ro ots of non-

linear p olynomial equations [3, 4, 1]. The need for solving algebraic equations also arises in

surface intersection algorithms, for nding starting p oints on each comp onent and lo cat-

ing singularities [5, 6 ], manipulating o set curves and surfaces [7] and geometric theorem

proving [5]. Most of these problems have b een extensively studied in the literature.

The currently known techniques for solving non-linear p olynomial systems can b e classi-

ed into symb olic, numeric and geometric metho ds. Symb olic metho ds based on resultants

and Gr}obner bases algorithm can b e used for eliminating variables and thereby reducing

the problem to nding ro ots of univariate p olynomials. These metho ds have their ro ots

in algebraic geometry.However, the current algorithms and implementations are ecient

for sets of low degree p olynomial systems consisting of up to three to four p olynomials

only. The ma jor problem arises from the fact that computing ro ots of a univariate p oly-

nomial can b e ill{conditioned for p olynomials of degree greater than 14 or 15, as shown by

Wilkinson [8]. As a result, it is dicult to implement these algebraic metho ds using nite

precision arithmetic and that slows down the resulting algorithm. As far as the use of

algebraic metho ds in geometric and solid mo deling is concerned, the current viewp ointis

that they have led to b etter theoretical understanding of the problems, but their practical

impact is unclear [7, 9].

The numeric metho ds for solving p olynomial equations can b e classi ed into iterative

metho ds and homotopy metho ds. Iterative techniques, like the Newton's metho d, are

go o d for lo cal analysis only and work well if we are given go o d initial guess to each so- 1

lution. This is rather dicult for applications likeintersections or geometric constraint

systems. Homotopy metho ds based on continuation techniques have a go o d theoretical

background and pro ceed by following paths in the complex space. In theory, each path

converges to a geometrically isolated solution. They have b een implemented and applied

toavariety of applications [10]. In practice the current implementations have many prob-

lems. The di erent paths b eing followed may not b e geometrically isolated and thereby

causing problems with the robustness of the approach. Moreover, continuation metho ds

are considered to b e computationally very demanding and at the moment restricted to

solving dense p olynomial systems only. Recently, metho ds based on interval arithmetic

have received a great deal of attention in computer graphics and geometric mo deling. The

resulting algorithms are robust, though their convergence can b e relatively slow.

For some particular applications, algorithms have b een develop ed using the geometric

formulation of the problem. This includes sub division based algorithms for curves and

surface intersection, ray tracing. In the general case sub division algorithms have limited

applications and their convergence is slow. Their convergence has b een improved byB ezier

clipping [11]. However, for low degree curveintersections algebraic metho ds have b een

found to b e the fastest in practice. Similarly algorithms based on the geometric prop erties

of mechanismshave b een develop ed for a class of kinematics problems, constraint systems

and motion planning problems.

In this pap er we present an algorithm for nding ro ots of p olynomial equations and

demonstrate its p erformance on a numb er of applications. The algorithm uses resultants

to eliminate variables from a system of p olynomial equations. The resultant formulations

linearize a non-linear p olynomial system and the resultant, therefore, can b e expressed in

terms of matrices and determinants. In particular, we show that the resultant of a system

of p olynomial equations corresp onds to the determinantofa matrix polynomial and the

problem of nding its ro ots can b e reduced to an eigenvalue problem or a generalized

eigenvalue of a matrix p encil. There is an elegant relationship b etween the kernels of

these matrix p olynomials and the variables b eing eliminated, which is b eing used for

computing the rest of the variables and birational maps. We consider algorithms for

sparse as well as dense p olynomial systems and illustrate them on the problems of curve

and surface intersections, nding distance from a p oint to the curve, birational maps and

geometric constraint systems. Later on we make use of the sparsity of the matrices to

improve the p erformance of the algorithm. The resulting algorithms are robust in nature

and their numerical accuracy is well understo o d. For most of the linear algebra problems,

backward stable algorithms are known. This is in contrast with nding ro ots of high

degree univariate p olynomials. Furthermore, ecient implementations of backward stable

routines are available as part of LAPACK [12] and we used them in our applications.

The rest of the pap er is organized in the following manner. In Section 2 we review the 2

results from elimination theory.We consider sparse as well as dense p olynomial systems.

In Section 3, we brie y review some of the techniques from linear algebra and matrix

computations used in the pap er. In Section 4 we showhow the resultants of p olynomial

systems can b e expressed in terms of matrix p olynomials. Furthermore, there is a direct

relationship b etween the variables b eing eliminated and the kernels of the matrix p oly-

nomials. In Section 5 we illustrate the algorithm on intersection of curves and surfaces,

nding distance of a p oint to a curve or a surface, representation of birational maps and

geometric constraint systems. In Section 6 we discuss the p erformance and limitations of

the current algorithm and highlight future directions for p erformance improvement and

nally in Section 7 we conclude the pap er.

2 Background

A semi-algebraic set is obtained by a nite numb er of Bo olean set op erations union, inter-

section, di erence applied to half spaces de ned by algebraic inequalities. These sets are

used to de ne solids and their b oundaries consist of zero, one and two dimensional algebraic

sets in 3-space. Algebraic sets are de ned as common zeros of a system of p olynomial

equations. Common examples of b oundary sets include parametric curves and surfaces de-

ned byB ezier and B-spline formulations. For many op erations likeintersections, o sets

and blends on these b oundary sets, the resulting algorithms involve computing common

ro ots of p olynomial equations. Elimination theory, a branch of classical algebraic geome-

try, deals with conditions for common solutions of a system of p olynomial equations. Its

main result is the construction of a single resultant p olynomial of n homogeneous p oly-

nomial equations in n unknowns, such that the vanishing of the resultant is a necessary

and sucient condition for the given system to have a non-trivial solution. We refer to

this resultant as the multipolynomial resultant and use it in the algorithm presented in the

pap er.

2.1 Resultants

Resultants are used for eliminating a set of variables from given equations and the geomet-

ric, symb olic and numeric problems can b e cast in that manner. Many formulations for

computing the resultant of a system of p olynomial equations are known in the literature.

The classical literature in algebraic geometry deals with resultant of dense p olynomial

systems [13 , 14]. The resultant of sparse p olynomial systems has received a considerable

amount of attention in the recent literature and many formulations have app eared [15 , 16]

based on the BKK b ound relating the numb er of solutions of a sparse system to mixed

volumes of Newton p olytop es. All these formulations of resultants express them in terms 3

of matrices and determinants. For many sp ecial cases, corresp onding to n =2;3;4;5;6,

where n is the numb er of equations, ecient formulations of resultants expressed as the

determinant of a matrix, are given in [14 ]. The most general formulation of resultant

for dense p olynomial systems expresses it as a ratio of two determinants [13]. For sparse

p olynomials a single determinant formulation is known for multigraded systems [15] and

a formulation equivalent to Macaulay's formulation has b een highlighted in [16]. The

ongoing activity in sparse elimination metho ds is imp ortant as many p olynomial systems

resulting from applications in geometric constraint systems, intersection, o sets and blends

are sparse [7, 17 ].

Resultants have gained a lot of imp ortance in geometric and solid mo deling literature.

Following Sederb erg's thesis [3] on implicitization of curves and surfaces, resultants have

b een applied to motion planning [2] and many other geometric problems, as surveyed in

[18]. Most of the earlier practical applications of resultants were limited to low degree curve

and surface intersections [7]. The idea of combining resultant formulations with matrix

computations has b een prop osed with Macaulay's formulation in [19]. In particular, [19]

describ e a general formulation of the resultant of dense p olynomial system in terms of

matrices and show that the solutions can b e computed by reducing it to a linear eigenvalue

problem. It turns out their resultant algorithm corresp onds to the Macaulay's formulation

[13] and [19] do not takeinto account that the resultant is expressed as a ratio of two

determinants. As a result, their approachmay not work whenever the lower determinant

vanishes. Furthermore, for many cases of p olynomial systems consisting of up to 5 6

equation, ecient formulations of resultant given by Bezout, Dixon, Morley and Coble etc.

result in non-linear matrix p olynomials. The algorithm presented in this pap er is not based

on any particular formulation of the resultant. Rather it uses the fact that resultants can

b e expressed in terms of determinants of non{linear matrix p olynomials. This approach

has b een sp ecialized to particular problems of curve and surface intersections. In [20] the

problem of curveintersection is analyzed using resultants and algorithms are presented

based on Bezout resultantoftwo p olynomials and matrix computations. Higher order

intersections corresp onding to tangential intersections or singular p oints are considered

as well. For intersection of triangular or tensor pro duct surfaces, [6] use the fact that

the implicit representation of such surfaces corresp onds to the determinant of a matrix

and prop osed a representation and evaluation of the intersection curve in terms of matrix

computations.

3 Matrix Computations

In this section we review techniques from linear algebra and numerical analysis used in

our algorithm. We also discuss the numerical accuracy of the problems in terms of their 4

condition numb ers and the algorithms b eing used.

3.1 QR Factorization

The QR factorization of an m n matrix A is given by

A = QR;

where Q is an m m orthogonal matrix and R is an m n upp er triangular matrix. More

details on its computations are given in [22].

3.2 Singular Value Decomp osition

The singular value decomp osition SVD is a p owerful to ol which gives us accurate in-

formation ab out matrix rank in the presence of round o errors. Given A,amnreal

matrix then there exist orthogonal matrices U and V such that

A = UV

where U is a m n orthogonal matrix, V is n n orthogonal matrix and is a n n

diagonal matrix of the form

= diag ; ;...; :

1 2 n

Moreover, ... 0. The 's are called the singular values,

1 2 n i

3.3 Eigenvalues and Eigenvectors

Given a n n matrix A, its eigenvalues and eigenvectors are the solutions to the equation

Ax = sx;

where s is the eigenvalue and x 6= 0 is the eigenvector. The eigenvalues of a matrix are

the ro ots of its characteristic p olynomial, corresp onding to determinantA sI. As a

result, the eigenvalues of a diagonal matrix, upp er triangular matrix or a lower triangular

matrix corresp ond to the elements on its diagonal. Ecient algorithms for computing

eigenvalues and eigenvectors are well known, [22 ], and their implementations are available

as part LAPACK [12 ].

3.4 Power Iterations

The largest or the smallest eigenvalue of a matrix and the corresp onding eigenvector

can b e computed using the Power method [22]. Power metho d involves multiplication of a 5

matrix bya vector and after a few steps it converges to the largest eigenvalue of a metho d.

Given a matrix, A, the technique starts with a vector q and p erforms computation of the

form

z = Aq ;

i i1

q = z = k z k;

i i i

Aq : = q

i i

After a few iterations, corresp onds to the eigenvalue of maximum magnitude and q is

k k

the corresp onding eigenvector.

3.5 Sparse Matrix Computations

The general formulation of resultants corresp onding to Macaulay formulation results in

sparse matrices [13 ]. In such cases wewant to make use of the sparsity of the matrix

in computing its eigendecomp osition. The order of Macaulay matrix is a function of the

numb er of p olynomials and the degrees of the p olynomial. The sparsity of the matrix

increases with the degrees of the p olynomials or the numb er of equations.

Algorithms for sparse matrix computations are based on matrix vector multiplications

as highlighted in the Power iterations. For our applications, we use the algorithm high-

lighted in [23] for computing the invariant subspaces and thereby the eigendecomp osition

of a sparse matrix.

3.6 Generalized Eigenvalue Problem

Given n n matrices, A and B, the generalized eigenvalue problem corresp onds to solving

Ax = sBx:

We represent this problem as eigenvalues of A sB. The vectors x 6= 0 corresp ond to the

eigenvectors of this equati on. If B is non{singular and its condition numb er de ned in the

next section is low, the problem can b e reduced to an eigenvalue problem bymultiplying

b oth sides of the equation by B and thereby obtaining:

B Ax = sx:

However, B mayhave a high condition numb er and such a reduction can cause numerical

problems. 6

4 Algorithm for Solving Equations

In this section, we describ e the algorithm for solving a system of non-linear equations

assuming that they have nite numb er of common solutions. We initially showhow the

resultants are b eing used to linearize the problem in terms of matrices and determinants.

In particular, we obtain matrix p olynomials and the problem of ro ots computation is b eing

reduced to nding eigenvalues of a matrix p olynomial and the corresp onding vectors in its

kernel. This algorithm is illustrated on many applications in the next section.

Given a system of n equations in n unknowns,

F x ;x ;... ;x = 0

1 1 2 n

F x ;x ;... ;x = 0

2 1 2 n

F x ;x ;... ;x = 0:

n 1 2 n

Let the degrees of these equations b e d ;d ;...;d , resp ectively. The resultant, Rx , is

1 2 n 1

obtained by eliminating the variables x ;x ;...;x from these equations. The resultantisa

2 3 n

p olynomial in x and its ro ots corresp ond to the x co ordinate of each solution of the given

1 1

multivariate system. The degree of the resultant is equal the total numb er of non-trivial

solutions of the given system of equations. The degree is is b ounded by the BKK b ound

for sparse system and Bezout b ound for dense p olynomial systems. Di erent formulations

of resultant express it as determinant of a matrix or as ratio of two determinants. In either

case, the entries of the resulting matrices are p olynomial functions of x .

In case, a single matrix formulation is not p ossible for the given system, we use the

u-resultant formulation to solve the given system of equations [24]. In particular, we add

a p olynomial

F x ;x ;... ;x =u +u x + ... + u x

n+1 1 2 n 0 1 1 n n

to the given system of equations. The u 's are symb olic variables. The resultant is obtained

by eliminating the variables x ; ...;x from the n + 1 equations and is a p olynomial in

1 n

u ;u ;...;u . It is known as the u-resultant of original system of p olynomial equations [24 ].

0 1 n

Moreover, the u-resultant is expressed as a ratio of two determinants, DetM=D etD.

However the entries of D are indep endent of the u 's. This is a prop erty of Macaulay's

formulation and the u-resultants [24]. As a result, if the matrix D is non-singular, the

resultant of the F ;F ;...;F corresp onds exactly to the determinantof M. In case, D

1 2 n+1

is singular, we replace M by its largest non-singular minor as shown in [17].

Given M, whose entries are p olynomials in the u 's, the u-resultant corresp onding to

its determinant can b e factored into linear factors of the form [24]:

DetM= u + u + ... + u

i0 0 i1 1 in n

i=1 7

where k is the total numb er of non-trivial solution and ; ; ;...; are the

i0 i1 i2 in

pro jective co ordinates of a solution of the given system of equations. Let us cho ose a

sp ecialization of the variables:

u = x ; u = 1; u =0; u =0; ...; u =0:

0 1 1 2 3 n

The determinantof M obtained after sp ecialization is a p olynomial in x and its ro ots

corresp ond exactly to the x co ordinate of each solution of the given multivariate sys-

tem. Thus, the determinant corresp onds exactly to the resultantofF ;F ;...;F , Rx ,

1 2 n 1

obtained after eliminating x ;x ;...;x . As a result, given any system of n p olynomial

2 3 n

equations whose co ecients are numeric constants, we can eliminate n 1variables and

express the resultant as determinant of a matrix Mx .

Multip olynomial resultants linearize a non-linear p olynomial system. In other words,

they take a system of non-linear p olynomial equations, say F ;F ;...;F , and reduce it

1 2 n

to a linear system of the form

d d d T T

Mx 1 x ... x ... x x ...... x = 00 ... 0 : 1

1 2 n 2 3 n

Mx is a square matrix and its entries are p olynomials in x . The entries of the vector

1 1

consist of p ower pro ducts of x ;x ;...;x the actual arrangement of the p ower pro ducts

1 2 n

of these variables is a function of the degrees of the p olynomial and the formulation of

resultant b eing used. This linearization has the prop erty that for any given solution

; ;...; , of the given system, M is a singular matrix and the vector in its

1 2 n 1

kernel is obtained by substituting x = ;x = ;...;x = in the vector consisting

1 1 2 2 n n

of p ower pro ducts highlighted in 1. We use this prop erty along with those of matrix

p olynomials to compute the ro ots of the p olynomial equations.

4.1 Matrix Polynomials

The matrix Mx highlighted in the previous section can b e expressed as a matrix p oly-

nomial:

Mx =M +M x +M x + ... + M x ; 2

1 0 1 1 2 2 l

where M are numeric matrices and l is the maximum degree of x in any term of Mx .

i 1 1

All M have the same order, say m m. The determinantof Mx corresp onds exactly to

i 1

the resultant of the given equations and we are interested in its ro ots. Furthermore, for a

given ro ot , the kernel of M is used to compute rest of the co ordinates, ; ;...; .

1 1 2 3 n

Let us assume that the leading matrix, M is non-singular and well-conditioned. As a

do es not intro duce severe numerical errors. Let result computation of M

1 1

Mx ; and M ; 0 i

1 i 1 i

l l 8

Mx is a monic matrix p olynomial. Its determinant has the same ro ots as do es the

determinantof Mx . Let x = b e a ro ot of the equation, DeterminantMx =0:

1 1 1 1

As a result M is a singular matrix and there is at least one non trivial vector in its

kernel. Let us denote that m 1vector as v . That is

M v = 0; 3

where 0 is a m 1null vector. The ro ots of the determinantof Mx corresp ond to the

eigenvalues of C highlighted in the following theorem [17 ]:

Theorem 4.1 Given the matrix polynomial, M x the roots of the polynomial corre-

sponding to its determinant are the eigenvalues of the matrix

0 1

0 I 0 ... 0

B C

0 0 I ... 0

B C

. . . .

C = ; 4

B C

......

B C

0 0 0 ... I

@ A

M M M ... M

0 1 2 l1

where 0 and I are m m nul l and identity matrices, respectively. Furthermore, the

eigenvector of C corresponding to the eigenvalue x = has the form:

1 1

2 l1 T

[v v v ... v] ;

where v is the vector in the kernel of M as highlighted in 3.

Many a times the leading matrix M is singular or close to b eing singular due to high

condition numb er. Some techniques based on linear transformations are highlighted in

[17], such that the problem of nding ro ots of determinant of matrix p olynomial can b e

reduced to an eigenvalue problem. However, there are cases where they do not work. For

example, when the matrices have singular p encils. In such cases, we reduce the intersection

problem to a generalized eigenvalue problem using the following theorem [17]:

Theorem 4.2 Given the matrix polynomial, Mx the roots of the polynomial corre-

sponding to its determinant are the eigenvalues of the generalized system C x C , where

1 1 2

0 1

I 0 0 ... 0

B C

0 I 0 ... 0

B C

. . . .

C =

B C

......

B C

0 0 ... I 0

@ A

0 0 ... 0 M

l 9

0 1

0 I 0 ... 0

B C

0 0 I ... 0

B C

. . . .

C = ; 5

B C

......

B C

0 0 0 ... I

@ A

M M M ... M

0 1 2 l1

where 0 and I are m m nul l and identity matrices, respectively.

The ro ots of the determinantof Mx corresp ond to the eigenvalues of C or C x C .

1 1 1 2

In many applications we are only interested in the real solutions or solutions lying in a

particular domain. The QR or QZ algorithm for eigenvalue computation returns all the

eigenvalues of a given matrix and it is dicult to restrict them to nding eigenvalues in a

particular domain [22 ].

Let us assume that is a simple eigenvalue of C. In the rest of the pap er, we carry

out the analysis on the eigenvalues of C and the resulting algorithm is similar for the

eigenvalues of the p encil C x C . Since is a simple eigenvalue, the kernel of C I

1 1 2 1 1

has dimension one represented as

2 l1 T

V =[v v v ... v] :

Furthermore, we know that v =[v v ... v ] corresp onds to the vector in the kernel

1 2 m

of M . Given v ,we use the relationship highlighted in 1 to compute the x ; ...;x

1 2 n

co ordinates:

d d d T

1 x ... x ... x x ...... x = v v ...v

2 n n 1 2 m

2 3

. For example, =

In many cases may corresp ond to an eigenvalue of multiplicity greater than one of

C. There are two p ossibilities:

; ;...; is a solution of multiplicity greater than one of the given system of

1 2 n

equations. In this case, the algebraic of multiplicityof is greater than one but

the geometric multiplicity corresp onding to the dimension of the kernel of C I

is still one.

There maybetwo solutions of the given equations of the form ; ;...; and

1 2 n

0 0

. As a result the kernel of C I has dimension greater than one. ;...; ;

1 1

n 2

The problem of computing higher multiplicity ro ots can b e numerically ill-conditioned.

However, in many cases it is p ossible to identify higher multiplicity eigenvalues of a matrix

by identifying clusters of eigenvalues and using the knowledge of the condition number of

the clusters. More details of its application to nding solutions of p olynomial equations

are given in [17]. 10

Given a higher multiplicity eigenvalue, ,we compute its geometric multiplicityby

computing the SVD of C I. The geometric multiplicity corresp onds to the number of

singular values equal to zero. In case, the geometric multiplicity is one, the relationship

highlighted in 1 is used to compute ; ;...; for each . Otherwise there are twoor

2 3 n 1

more vectors in the kernel of M . The vectors computed using linear algebra routines

may corresp ond to anytwovectors in the vector space corresp onding to the kernel. As

a result, it is dicult to compute , , ...; from them. To solve the problem we

2 3 n

substitute x = and solve the n 1 equations:

1 1

F ;x ;...;x = 0

1 1 2 n

F ;x ;...;x = 0

2 1 2 n

F ;x ;...;x = 0;

n1 1 2 n

for x , x ..., x . The solutions obtained are veri ed by substituting into F ;x ;...;x =

2 3 n n 1 2 n

Wehave implemented the algorithm using linear algebra libraries. The three ma jor

parts of the algorithm are:

1. Use a suitable resultant formulation to linearize the problem in terms of matrix

p olynomials. The entries of the Macaulay matrix are actually the co ecients of the

p olynomial equations. The resultant formulations of Bezout, Dixon corresp onding

to two or three equations result in matrix entries b eing p olynomial functions of the

co ecients.

2. Reduce the problem to an eigenvalue problem. This involves estimating the condition

numb er of the leading matrix of the matrix p olynomial. In some cases a linear

rational transformation is involved on the matrices to improve the conditioning of

the leading matrix. Finally,we reduce the problem to an eigenvalue or a generalized

eigenvalue problem.

3. Compute the eigendecomp osition of the given matrix and recover the common ro ots

from the eigenvalues and the eigenvectors. In a few instances, this mayinvolve

identifying higher order eigenvalues using knowledge of clusters, using the SVD to

know the geometric multiplicity of the eigenvalue and p ossibly solving a system of

n 1 equations in n 1 unknowns.

All the three parts mentioned ab ove are relatively simple to implement, given the linear

algebra routines. A ma jor feature of the algorithm is that at each stage the numerical

accuracy of the op erations involved is well understo o d. As a result, we are able to come up 11

with tight b ounds on the accuracy of the resulting solution. In fact, the higher multiplicity

eigenvalues are determined from their condition numb ers. For most cases, we are able to

accurately compute the ro ots using the 64 bit IEEE oating p oint arithmetic available on

most workstations.

5 Applications

In this section we highlight the application of the equation solving algorithm to some prob-

lems in b oundary computations and geometric constraint systems. Initially,we consider an

example of intersection of three surfaces from [10]. In particular, Morgan uses continuation

metho ds to solve this problems and argues that approaches based on symb olic reduction

lead to severe numeric problems. Secondly,we consider the problem of nding a distance

from a p oint to a curve or a surface. It is well known that these problems can b e reduced

to solving non-linear algebraic equations and they are frequently encountered in the com-

putation of o set curves and surfaces and voronoi surfaces [7, 1]. The matrix relationship

expressed in 1 is also used for computing the birational maps. These birational maps

are expressed in terms of ratio of determinants as opp osed to algebraic expression. They

are useful for the inversion problem and expressing the rational relationships b etween the

variables b eing eliminated and the co ecients of the given p olynomial system. Finally,we

discuss the application of this algorithm to geometric constraint systems.

5.1 Intersection of Three Surfaces

Given three algebraic surfaces, F x; y ; z = 0; Fx; y ; z = 0; Fx; y ; z = 0, we

1 2 3

are interested in their common p oints of intersection. This problem arises frequently in

the b oundary computations [5]. It turns out that this problem corresp onds to solving

3 equations in 3 unknowns. In case, we are given parametric surfaces, we can either

implicitize them and reduce it to solving for 3 equations or deal with the parametric

variables and reduce the problem to solving 6 equations in 6 unknowns.

The total numb er of solutions in the complex space is b ounded by the pro duct of the

three degrees. Resultants and Gr}obner bases have b een used to eliminate twovariables

from these three equation. The resulting problem corresp ond to nding ro ots of a univari-

ate p olynomial, which can b e ill-conditioned on low degree p olynomials. We illustrate it

on an example from [10] and also analyze the accuracy of the algorithm presented in this

pap er.

Consider the intersection of a sphere, a cylinder and a plane describ ed by the following

equations:

2 2

1:6e 3 x +1:6e3 x 1 = 0

1 2

2 2 2

5:3e4 x +5:3e4 x +5:3e4 x +2:7e2 x 1 = 0

1 2 3 12

1:4e 4 x +1:0e4 x +x 3:4e3 = 0:

1 2 3

According to [10], these equations havetwo real solutions of norm ab out 25 and a com-

plex conjugate pair of order 10 . The two real solutions havea physical meaning. After

eliminating x and x from these equations, the eliminant is [10 ]:

2 3

4 3

6:38281970398352 x 7:12554854545301e9 x +

1 1

1:89062308408416e19 x +9:36558635415069e20 x

1:15985845720186e22:

where the co ecients have b een rounded to 15 digits. In the original p olynomial system

there is a range of 4 orders of magnitude in the co ecients and in the eliminant the range

is 22. As a result, it is very dicult to accurately compute the ro ots using the IEEE 64

bit arithmetic, which allows 16 digits of precision. These kind of problems are commonly

faced in algebraic algorithms.

We analyze the accuracy and robustness of our algorithm on this problem. We use the

Macaulay formulation of the resultant and it linearizes the problem into

Mx v = 0: 6

The matrix Mx corresp onding to the resultant formulation is:

0 1

1:6e 3 0 0 0 0 0 0 1+1:6e3 x ; 0 0

0 1:6e3 0 0 0 0 0 0 1+ 1:6e3 x 0

B C

0 0 1:6e3 0 0 0 0 0 0 1+1:6e3 x

B C

5:3e4 0 0 5:3e4 0 0 0 5:3e4 x +0:027 x 1 0 0

B 1 C

B C

0 5:3e 4 0 0 5:3e 4 0 0 0 5:3e 4 x +0:027 x 1 0

B C

0 0 5:3e 4 0 0 5:3e 4 0 0 0 5:3e 4 x +0:027 x 1

B C

B 0 1:0e 4 0 1 0 0 1:4e 4 x 3:4e 3 0 0 0 C

@ A

0 0 1:0e 4 0 0 0 1 1:4e 4 x 3:4e 3 0 0

0 0 0 0 0 1 1:0e 4 0 1:4e 4 x 3:4e 3 0

0 0 0 0 0 0 0 1:0e 4 1 1:4e 4 x 3:4e 3

and the right hand side vector is

3 2 2 2 3 2 T

v =x x x x x x x x x x x x 1 :

3 2 2 3 2 3

2 2 2 3 3 3

Although the Macaulay resultant expresses the resultant as a ratio of two determinants,

the lower matrix is non-singular and consist of numerical entries only.

It follows from Bezout's theorem that the system has 4 solutions in the complex space.

In fact, the determinant of the matrix, say Mx , highlighted ab ove corresp onds to the

eliminant up to a constant. We treat it as a matrix p olynomial and reduce the problem

to an eigenvalue problem.

In this case, Mx isa1010 matrix p olynomial of degree 2 in x . Let us express it

1 1

as:

Mx = M x +M x +M :

1 2 1 1 0

1 13

The fact that the determinantof Mx has degree 4 implies that M is singular. As

1 2

a result, we cannot reduce the problem to an eigenvalue problem using Theorem 4.1. It

turns out that the condition number of M is 4:0911e03. As a result, we p erform a linear

transformation y =1=x and we reduce the problem to nding eigenvalues of the matrix

0 I

C = :

1 1

M M M M

2 1

0 0

It follows that y = 0 is an eigenvalue of multiplicity6 of C.We use this information in

cho osing the appropriate shifts in the double shift QR algorithm for eigendecomp osition,

as describ ed in [22]. This knowledge of some of the eigenvalues sp eeds up its convergence.

The 4 nonzero eigenvalues are

0:04037383; 0:04037383; 1:35035361e 10 + 5:773741e 10 i; 1:35035361e 10 5:773741e 10 i;

1 . Substituting x =1=y and computing x and x from the eigenvectors of where i =

1 2 3

C results in two real solutions to the original equations:

x ;x ;x =4:76851749893; 3:39419223; 0:00720701519;

1 2 3

x ;x ;x = 24:76851768; 3:3941908959; 0:006528173:

1 2 3

The solutions computed are accurate up to 8 digits of accuracy.Further accuracy can

be achieved by a few iterations of Newton's metho d on the original equations and using the

solutions from eigendecomp osition pro cedure as the start p oints for Newton's iteration.

Wehave used this algorithm on the intersection of three surfaces with more than 100

intersections and are able to compute the solutions with go o d accuracy.

5.2 Distance from a Point to a Curve or Surface

The problem of computing the distance of a p oint to a curve or a surface comes up

rep eatedly in solid mo deling computations. Some examples include computation of the

medial axis transform,voronoi surfaces, o set curve and surfaces etc. [7, 1]. It is well

known in literature that this problem can either b e p osed as an optimization problem

minimizing the distance function or reduce it to solving algebraic equations. Let X; Y

beapoint and F x; y = 0 b e the curve. Let x ;y b e the p oint on the curve closest to

p p

X; Y . Then there are two equations:

F x ;y = 0

p p

Y y F x ;y X x F x ;y = 0:

p x p p p y p p 14

It turns out that these equations have more than one real solution and therefore, any

algorithm based on lo cal metho ds may converge to a wrong solution [1]. Algorithms based

on constrained optimization may converge to a lo cal minima of the function. Thus, none

of the previous technique is able to solve this problem in a reasonable manner.

Using the algebraic formulation the problem reduces to solving for x ;y in these two

p p

equations. For a curve of degree n there are n solutions to these equations. We used the

Bezout formulation of the resultantoftwo equations and solved the problem using the

eigenvalue formulation. In particular, we eliminate x from the given equations and the

Bezout formulation results in a n n matrix, whose entries are a p olynomial of degree n

in y . This is eventually reduced to an eigenvalue problem of a matrix of order n . The

resulting algorithm works well in practice. We know from the equations that the ro ots of

the given system contain a p oint on the curve closest to X; Y . This p oint is identi ed by

computing the distances b etween the ro ots and X; Y and nding the ro ot corresp onding

to the minimum distance. Similarly the problem of nding the distance from a p ointtoa

surface is reduced to solving three p olynomials in three unknowns.

Many other problems likeray-tracing parametric curves and surfaces, nding singular

p oints on algebraic curves and surface can b e reduced to solving two p olynomial equations

in two or three unknowns. This algorithm has b een successfully applied to these problems

[17].

5.3 Birational Maps

Birational maps play a fundamental role in algebraic geometry. They have also gained

imp ortance in solid mo deling for their use in many applications. Many problems related

to b oundary computation are easily solved using birational maps. The most common

example is that of inversion problem for rational curves and surfaces. Given a rational

parametric surface, expressed in pro jective co ordinates as:

Fs; t; u= x; y; z; w= Xs; t; u;Y s; t; u;Zs; t; u;Ws; t;u;

where X s; t; u;Y s; t; u;Zs; t; u and W s; t; u are homogeneous p olynomials of degree

n. Common examples of this formulation are the triangular and tensor pro duct B ezier

patches used in geometric and solid mo deling. A rational surface de nes a map from the

s; t; u pro jective plane to the pro jective space de ned by x; y; z; w. In applications involving

parametric surfaces an imp ortant problem is that of computing the s; t; u co ordinates, given

a p oint on the surface, x ;y ;z ;w . This is the inversion problem.

0 0 0 0

It turns out that the exact relationship b etween the p oints on the surface, Fs; t; u,

and the parameters s; t; u can b e expressed as a rational map as well:

F x; y; z; w= s; t; u= Sx; y; z; w;Tx; y ; z ; w ;Ux; y ; z ; w ; 15

where eachof Sx; y; z; w, T x; y; z; w and U x; y; z; w is a homogeneous p olynomial.

As a result, we see that F and F de ne rational maps b etween the s; t; u and x; y; z; w

space.

Algorithms to compute the birational maps are known in the literature. They are based

on Gr}obner bases [5] or multip olynomial resultants [18]. However, direct applications of

these algorithms su er from accuracy and eciency problems as highlighted in [7]. We

make use of the fact that multip olynomial resultants linearize a non-linear problem by

eliminating some variables as shown in 1. In particular, we express the birational maps

in terms of ratio of determinants and use matrix computations for their computation. At

the moment, the algorithm is limited to systems of p olynomial equations, such that their

resultant can b e expressed as a single determinant. Such formulations are known for 2; 3

or 4 equations and most applications of curve and surface mo deling involving birational

maps fall in this category.

Lets consider the resultantofthen equations, F ;F ;...;F expressed as a single

1 2 n

matrix in 1. The entries of the matrix are functions of the co ecients of the given

p olynomial equations and x .To compute, x ; ...;x we make use of the kernel of Mx .

1 2 n 1

In particular, we represent Mx as

1 0

M x M x ... M x

1;1 1 1;2 1 1;m 1

C B

M x M x ... M x

2;1 1 2;2 1 2;m 1

C B

C : Mx =B

. . .

C B

. . .

......

A @

M x M x ... M x

m;1 1 m;2 1 m;m 1

Let us consider the p oints which make Mx singular. In this case, x corresp ond to the

1 1

ro ots of the determinantof Mx . For a generic choice of the p oint the kernel of Mx

1 1

has dimension one this has to b e the case for birational maps to b e de ned. Lets treat

eachpower pro duct in the vector highlighted in 1 as a separate variable and denote the

resulting vector as

0 0 0

v =[1 v v ... v ] :

2 3 m

It follows from this notation that x = v , for i n.Taking the n equations denoted by

Mx v and applying the Cramer's rule it follows that

1 16