Stephen M. Watt

The singular value decomposition for polynomial systems

This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coecients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD’s which gives exact results for interesting nearby problems, and give ecient algorithms for computing precisely how nearby. We generalize this to multivariate GCD computation. Next, we adapt Lazard’s u-resultant algorithm for the solution of overdetermined systems of polynomial equations to the inexact-coecient case. We also briefly discuss an application of the modified Lazard’s method to the location of singular points on approximately known projections of algebraic curves.

QR factoring to compute the GCD of univariate approximate polynomials

We present a stable and practical algorithm that uses QR factors of the Sylvester matrix to compute the greatest common divisor (GCD) of univariate approximate polynomials over /spl Ropf/[x] or /spl Copf/[x]. An approximate polynomial is a polynomial with coefficients that are not known with certainty. The algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle, by splitting and reversal if necessary. The algorithm has been tested on thousands of examples, including pairs of polynomials of up to degree 1000, and is now distributed as the program QRGCD in the SNAP package of Maple 9.

Numerical Implicitization of Parametric Hypersurfaces with Linear Algebra

We present a new method for implicitization of parametric curves, surfaces and hypersurfaces usingessen tially numerical linear algebra. The method is applicable for polynomial, rational as well as trigonometric parametric representations. The method can also handle monoparametric families of parametric curves, surfaces and hypersurfaces with a small additional amount of human interaction. We illustrate the method with a number of examples. The efficiency of the method compares well with the other available methods for implicitization.

Towards factoring bivariate approximate polynomials

A new algorithm is presented for factoring bivariate approximate polynomials over C[x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust.

Distance-based classification of handwritten symbols

We study online classification of isolated handwritten symbols using distance measures on spaces of curves. We compare three distance-based measures on a vector space representation of curves to elastic matching and ensembles of SVM. We consider the Euclidean and Manhattan distances and the distance to the convex hull of nearest neighbors. We show experimentally that of all these methods the distance to the convex hull of nearest neighbors yields the best classification accuracy of about 97.5%. Any of the above distance measures can be used to find the nearest neighbors and prune totally irrelevant classes, but the Manhattan distance is preferable for this because it admits a very efficient implementation. We use the first few Legendre-Sobolev coefficients of the coordinate functions to represent the symbol curves in a finite-dimensional vector space and choose the optimal dimension and number of bits per coefficient by cross-validation. We discuss an implementation of the proposed classification scheme that will allow classification of a sample among hundreds of classes in a setting with strict time and storage limitations.

A geometric-numeric algorithm for absolute factorization of multivariate polynomials

In this paper, we propose a new semi-numerical algorithmic method for factoring multivariate polynomials absolutely. It is based on algebraic and geometric properties after reduction to the bivariate case in a generic system of coordinates. The method combines 4 tools: zero-sum relations at triplets of points, partial information on monodromy action, Newton interpolation on a structured grid, and a homotopy method. The algorithm relies on a probabilistic approach and uses numerical computations to propose a candidate factorization (with probability almost one) which is later validated.

An OpenMath 1.0 implementation

The first official version of the OpenMath specification was released in December. This paper presents the first implementation of this standard, in the form of a C library. To ensure a faithful realization, a second, independent implementation with the same api was built using Aldor (A). We describe how the C library has been embedded in two main-stream computer algebra systems, Maple and Reduce, which can now communicate with each other and Aldor, and with specialized programs also linking the libraries. We discuss some of the problems encountered in developing the api, and the solutions we have chosen.

A numerical absolute primality test for bivariate polynomials

We give a new numerical absolute primality criterion for bivariate polynomials. This test is based on a simple property of the monomials appearing after a generic linear change of coordinates. Our method also provides a probabilistic algorithm for detecting absolute factors. We sketch an implementation and give timings comparing with two other algorithms implemented in Maple.

Scratchpad II: an abstract datatype system for mathematical computation

Scratchpad II is an abstract datatype language and system that is under development in the Computer Algebra Group, Mathematical Sciences Department, at the IBM Thomas J. Watson Research Center. Many different kinds of computational objects and data structures are provided. Facilities for computation include symbolic integration, differentiation, factorization, solution of equations and linear algebra. Code economy and modularity is achieved by having polymorphic packages of functions that may create datatypes. The use of categories makes these facilities as general as possible.

A first report on the A # compiler

Abstract Machine A major part of the A ] compiler is concerned with producingoptimized intermediate code, or Foam code. “Foam” is anacronym for “First Order Abstract Machine.” The abstractmachine is first order in the sense that it does not treat itstypes as values.Foam is designed to contain only those concepts whichcan have an efficient realization in both Lisp and C. Forexample it is not possible to take an address of a variablebecause that would be inefficient in Lisp (a closure wouldbe created). Nor are dynamic type tests allowed, as thatwould be inefficient in C. We have been asked how the lackof address arithmetic limits the potential performance ofcompiled A ] vs hand-coded C which uses pointers to traversearrays in inner loops. It is our experience that this is a minorconcern on current architectures with optimizing compilers.Foam is not restricted to the precise intersection of Cand Lisp. Some aspects are handled by support libraries.Big integer arithmetic is assumed as part of Foam, and thisis provided as a library for C. Also the memory model differsfrom both C and Lisp in some details: garbage collection isassumed (this is a run time support library in C) and it ispossible to make an explicit request to free storage (in Lispthis is ignored).A Foam program is comprised of a flat sequence of com-mands. Foam types have various sizes and uses. or example,“Char” is a text character whereas “Byte” is a charactersized integer, “DFlo” is a double precision floating point,“Ptr” can point to an array, record, arbitrary sized inte-ger, etc. Reference instructions contain the kind of refer-ence and the position, e.g., “Loc 3” refers to the third localvariable of the current function and “RElt 7 x 2” indicatesthe 2nd field of the record x, using the 7th layout format.Foam operations consist of instructions, such as “If b n,”which indicates that if b is true then proceed to label n, andbuiltin operations, e.g., “HIntLT a b” is a half-word-integerless-than comparison. The builtin operations are type spe-cific and conversion operations are generally provided. Adetailed description of Foam is given elsewhere [26].27