An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations (original) (raw)

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

Foundations of Computational Mathematics, 2011

We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input. We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system, on the average.

Regeneration homotopies for solving systems of polynomials

Mathematics of Computation, 2010

We present a new technique, based on polynomial continuation, for solving systems of n polynomials in N complex variables. The method allows equations to be introduced one-by-one or in groups, obtaining at each stage a representation of the solution set that can be extended to the next stage until finally obtaining the solution set for the entire system. At any stage where positive dimensional solution components must be found, they are sliced down to isolated points by the introduction of hyperplanes. By moving these hyperplanes, one may build up the solution set to an intermediate system in which a union of hyperplanes "regenerates" the intersection of the component with the variety of the polynomial (or system of polynomials) brought in at the next stage. The theory underlying the approach guarantees that homotopy paths lead to all isolated solutions, and this capability can be used to generate witness supersets for solution components at any dimension, the first step in computing an irreducible decomposition of the solution set of a system of polynomial equations. The method is illustrated on several challenging problems, where it proves advantageous over both the polyhedral homotopy method and the diagonal equation-by-equation method, formerly the two leading approaches to solving sparse polynomial systems by numerical continuation.

An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

2011 Proceedings of the Thirteenth Workshop on Algorithm Engineering and Experiments (ALENEX), 2011

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, Lgp and Maple's Isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.

Computation of zeros of linear multivariable systems

Automatica, 1982

Several algorithms have been proposed in the literature for the computation of the zeros of a linear system described by a state-space model 1x1-A,B,C,D}. In this report we discuss the numerical properties of a new algorithm and compare it with some earlier techniques of computing zeros. The new approach is shown to handle both nonsquare and/or degenerate systems without difficulties whereas earlier methods would either fail or would require special treatment for these cases. The method is also shown to be backward stable in a rigorous sense. Several numerical examples are given in order to compare speed and accuracy of the algorithm with its nearest competitors.

On multiplicities in polynomial system solving

Transactions of the American Mathematical Society, 1996

This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations.

Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components

SIAM Journal on Numerical Analysis, 2001

In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of di erent dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by nding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity. The bound is sharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a nite number of polynomials.

Solving Polynomial Systems Equation by Equation

The IMA Volumes in Mathematics and its Applications, 2008

By a numerical continuation method called a diagonal homotopy, one can compute the intersection of two irreducible positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets that are not necessarily irreducible or even equidimensional. Of particular interest is the special case where one of the sets is defined by a single polynomial equation. This leads to an algorithm for finding a numerical representation of the solution set of a system of polynomial equations introducing the equations one by one. Preliminary computational experiments show this approach can exploit the special structure of a polynomial system, which improves the performance of the path following algorithms.

Accelerated Solution of Multivariate Polynomial Systems of Equations

SIAM Journal on Computing, 2003

We propose new Las Vegas randomized algorithms for the solution of a square nondegenerate system of equations, with well-separated roots. The algorithms use O(δ 3 n D 2 log(D) log(b)) arithmetic operations (in addition to the operations required to compute the normal form of the boundary monomials modulo the ideal) to approximate all real roots of the system as well as all roots lying in a fixed n-dimensional box or disc. Here D is an upper bound on the number of all complex roots of the system (e.g., Bezout or Bernshtein bound), δ is the number of real roots or the roots lying in the box or disc, and = 2 −b is the required upper bound on the output errors. For computing the normal form modulo the ideal, the efficient practical algorithms of [B. Mourrain and P. Trébuchet, in 61-88] can be applied. We also yield the bound O(3 n D 2 log(D)) on the complexity of counting the numbers of all roots in a fixed box (disc) and all real roots. For a large class of inputs and typically in practical computations, the factor δ is much smaller than D, δ = o(D). This improves by the order of magnitude the known complexity estimates of the order of at least 3 n D 4 + D 3 log(b) or D 4 , which so far are the record estimates even for the approximation of a single root of a system and for each of the cited counting problems, respectively. Our progress relies on proposing several novel techniques. In particular, we exploit the structure of matrices associated to a given polynomial system and relate it to the associated linear operators, dual space of linear forms, and normal forms of polynomials in the quotient algebra; furthermore, our techniques support the new nontrivial extension of the matrix sign and quadratic inverse power iterations to the case of multivariate polynomial systems, where we emulate the recursive splitting of a univariate polynomial into factors of smaller degree.

Hybrid Method for Solving Polynomial Equations

2000

We discuss how to decompose the zero set of a multivariate polynomial system with inexact coecien ts to a sequence of zero sets of reduced triangular sets in a numerically stable way.