Robert M. Corless

On the LambertW function

The LambertW function is defined to be the multivalued inverse of the functionw →wew. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.

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.

A sequence of series for the Lambert W function

WC give a uniform treatment of several series expansions for the Lambert Iirfurrction, leading toaninfinite family of new series. Wealsodisrxrss standardization, cornplexbranches,a family of arbitrary-order iterative methods forcornputation of IT”,and give a theorem showing how to correctly solve another simpl~ and frequently occurring nonlinear equation.

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.

Compact finite difference method for integro-differential equations

In this paper, we give sixth order compact finite difference formula for second order integro-differential equations (IDE) with different boundary conditions, and both of error estimates and numerical experiments confirm our compact finite difference method can get fifth order of accuracy. We also adjust compact finite difference method for first order IDE and a system of IDE and give numerical experiments for them. Our algorithm even can solve nonlinear IDE and unsplit kernel of IDE. The most advantages of compact finite difference method for IDE are that it obtains high order of accuracy, while the time complexity to solve the matrix equations after we use compact finite difference method on IDE is O(N), and it can solve very general case of IDE.

A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots

The technique of solving systems of multivariate polynomial equations via rigenproblems has become a topic of active research (with applications in computer-aided design and {untrul theory, for example) at least since the papers [2, 6, 9]. one may approach the problem via various resultant formulations ,x by Grijbner bases. As more understanding is gained, it is becoming clearer that eigenvalue problems are the “weakly nonlinear nucleus to which the original, strongly nonlinear task may be reduced’ [13], Earl,v works mmcemt,rat,ed on the case of simple roots. An example of such was, the paper [5], which used a numerical adaptation of il resultant technique due to Lazard to attack the problem directly, without, reference to Grobner bases.

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.

Initial value problems for ODEs in problem solving environments

A program is presented for solving initial value problems for ODEs numerically in Maple. We draw upon our experience with a number of closely related solvers to illustrate the differences between solving such problems in general scientific computation and in the problem solving environments Maple and MATLAB.

A model of the combined effects of vortex-induced oscillation and galloping

In this paper, the interactions of two forms of wind-induced oscillation of a cylinder of square cross-section in cross flow are modelled by simply combining the mathematical models of each form taken separately. The Hartlen-Currie model for vortex-induced oscillation is added to Parkinson and Smiths quasi-steady model for galloping and the resulting nonlinear oscillator equations are solved by a perturbation technique, the method of multiple scales. The results are compared with recent experimental data.