MARS: A Maple/Matlab/C Resultant-Based Solver Ioannis Z. Emiris Aaron Wallack INRIA Cognex Corp oration B.P.93 1 Vision Drive Sophia-Antip olis 06902, France Natick MA 01760, USA [email protected] [email protected] http://www.inria.fr/safir/emiris Dinesh Mano cha Department of Computer Science, UNC Chap el Hill NC 27599-3175, USA [email protected] http://www.cs.unc.edu/~dm Abstract economics and optimization [MM94 ], and molecular biol- ogy [EM96 ]. The main op erations in these applications can The problem of computing zeros of a system of p olynomial b e classi ed into twotyp es. First, the simultaneous elimina- equations has b een well studied in the computational litera- tion of one or more variables from a given set of p olynomial ture. Anumb er of algorithms have b een prop osed and many equations to obtain a \symb olically smaller" system. This computer algebra and public domain packages provide the problem arises, for instance, in graphics and mo deling ap- capability of computing the ro ots of p olynomial equations. plications where the implicit expression of a curve or surface Most of these implementations are based on Grobner bases is precisely the resultant p olynomial. Second, the compu- which can b e slow for even small problems. In this pap er, tation of all numeric solutions of a system of p olynomial we present a new system, MARS, to compute the ro ots of equations. Our practical motivation is fast computation of a zero dimensional p olynomial system. It is based on com- solutions of a zero-dimensional system with 10 variables or puting the resultant of a system of p olynomial equations less. followed by eigendecomp osition of a generalized companion Elimination theory, a branch of classical algebraic geom- matrix. MARS includes a robust library of Maple func- etry,investigates the condition under which sets of p olyno- tions for constructing resultant matrices, an ecient library mials have common ro ots. Its results were known at least of Matlab routines for numerically solving the eigenprob- a century ago and still app ear in mo dern treatments of al- lem, and C co de generation routines and a C library for gebraic geometry, although often in non-constructive form. incorp orating the numerical solver into applications. We il- The main result is the construction of a single resultantpo- lustrate the usage of MARS on various examples and utilize lynomial of n homogeneous p olynomial equations in n un- di erent resultant formulations. knowns, such that the vanishing of the resultant is a neces- sary but not always sucient condition for the given system to have a nontrivial solution. This resultant is known as the 1 Intro duction multipolynomial resultant the given system of p olynomial equations [AS88 , Man94 , MC94 ]. The multip olynomial re- Finding the solution of a system of nonlinear p olynomial sultant of the system of p olynomial equations can b e used equations over a given eld is a classical and fundamental for eliminating the variables and computing the numeric so- problem in the computational literature which has b een ex- lutions of a given system of p olynomial equations. The same tensively studied. approach is also valid in the non-homogeneous context, as Recently, a great deal of interest in solving nonlinear p o- illustrated later. lynomial systems has come from di erent applications. It Given a zero-dimensional system, the computation of all includes computer algebra [Ren92], rob otics [MC94 , RR95, common solutions can be reduced to an eigenvalue prob- WC97 ], computer graphics [Man94 ], geometric and solid lem. In this resultant-eigendecomp osition technique, each mo deling [Hof89 , MD95 ], computer vision [Emi97 , WM98 ], eigenvalue corresp onds to one variable of a ro ot, and the Permission to make digital or hard copies of all or part of this work for asso ciated eigenvector characterizes the other variables of p ersonal or classro om use is granted without fee provided that copies the ro ot. This approachisvery useful for rep eatedly solv- are not made or distributed for pro t or commercial advantage and ing similar systems b ecause the symb olic pro cessing is only that copies b ear this notice and the full citation on the rst page. To copy otherwise, to republish, to p ost on servers or to redistribute to p erformed once and numerically solving the system reduces lists, requires prior sp eci c p ermission and/or a fee. ISSAC'98, to instantiating co ecients of the resultant matrix followed c Rosto ck, Germany. 1998 ACM 1-58113-002-3/ 98/ 0008 $5.00 244 by eigendecomp osition. rent homotopy implementations and algorithms su er from Main Contributions: The main contribution of our work many problems. The di erent paths b eing followed may not is a software package consisting of Maple, Matlab and C b e geometrically isolated. As a result, each path has to b e libraries for solving zero-dimensional systems a more thor- at times followed with impractically tight tolerances, which ough review can b e found in [WEM98 ]. Given a system, slows down the overall algorithm. MARS computes the resultant as a matrix p olynomial and Multip olynomial resultant algorithms provide the most numerically solving the resultant matrices. MARS simpli- ecient metho ds as far as asymptotic complexityis con- es the task of incorp orating a numerical multip olynomial cerned for solving a system of p olynomial equations by solver into a user's application. We present a number of eliminating variables. One of their main advantages is the issues in the design and implementation of this library and fact that the resultant can always b e expressed in terms of highlight its p erformance on a numb er of examples. matrices and determinants. We will later describ e di erent The rest of the pap er is organized as follows. The next techniques for construction of resultant matrices b elow. Sys- section discusses alternate approaches, existing implemen- tems suchas Axiom, Maple, Mathematica and Reduce tations and their limitations. Section 3 outlines the main only o er matrix expressions for the resultantoftwo univari- approach in using resultant matrices for reducing system- ate p olynomials, either as Sylvester's matrix or as B ezout's solving to a problem in linear algebra. In particular, sub- matrix. Some use of resultants can also b e found in other section 3.1 mentions the di erent matrix formulations and systems, suchasCASA, develop ed at RISC-Linz. how they are constructed, whereas the following subsection Di erent sp ecialized mo dules based on resultant ma- shows the matrix op erations, typically p erformed numeri- trices exist for solving systems of p olynomial equations, cally, applied to approximate all common ro ots. Section 4 e.g. [CGT97 , CP93 , Emi97, KM95 , KS96, MP97 , Reg95 ]. describ es the main architects of our library and the pack- Typically, these programs would rely on Linpack, Eispack, age's organization, and section 5 discusses implementation Lapack,orMatlab for their numerical calculations. All of details and features of the MARS package. We illustrate the these programs implement one or, exceptionally,two kinds power and adaptability of our library, including the available of matrices, and are not designed for wide distribution, so interfaces and the automatic generation of C co de, in sec- they lack in user-friendliness. There is currently a very in- tion 6 by studying concrete examples. Section 7 reviews the teresting e ort in the context of FRISCO for developing a p erformance and practical complexityof MARS.We sum- general library of resultant functions in C++, to which the marize and conclude with further work in section 8. second author is participating. 2 Related work 3 Resultant-based system solving There is a long history of using resultant-based approaches There is more than one way to solve arbitrary p olynomial to study and solve systems of p olynomial equations. Re- systems by using resultants, yet here we fo cus on the one cently, certain practical results that have established re- metho d presenting the strongest practical interest. Namely, sultants, along with Grobner bases and continuation tech- we are interested in constructing resultant matrices, whose niques, as a metho d of choice in solving zero-dimensional determinants express nontrivial multiples of the resultant p olynomial systems. For systems of medium size, the appli- p olynomial and which, furthermore, reduce the computation cations highlighted earlier illustrate the comparative advan- of all common zeros to a problem in linear algebra. The tages of resultant-based metho ds: resultants can strongly symb olic part of matrix construction can strongly exploit exploit p olynomial structure, they reduce the nonlinear p olynomial structure, whereas the manipulation of the ma- problem to one in linear algebra, and combine a symb olic trices b ene ts from the current state-of-the-art in numerical with a numeric approach. linear algebra. Belowweoverview b oth stages and explain Grobner bases have b een studied for a longer time and of- how to reduce the nonlinear problem to the computation of fer an array of general implementations to eciently handle eigenvalues and eigenvectors of a square matrix. zero-dimensional systems. For the purp oses of illustration, we mention only very few representatives, namely GB [Fau95 ] 3.1 Symb olic computation and the PoSSo/FRISCO library [FRI97 ]. Most computer al- The computation of resultants typically relies on construct- gebra systems, like Axiom, Mathematica, Maple and Re- ing matrices whose determinant is either the exact resultant duce have a package for computing the Grobner bases of an p olynomial or, more generally, a nontrivial multiple of it.