Milestones

1. High Precision Numerics i) A first version of the zero finder for complex analytic functions has been delivered to Maple for installation. This extends the flexibility already attained in finding zeros of polynomials to more general functions. Given an analytic function and a rectangular region in the complex plane, it outputs all zeros on or inside the box. The application of the developing code to truncated zeta functions, for example, has already led to very interesting insights into their theory. ii) The reverse symbolic engineering project has led to the development of the IDENTIFY function, now installed in the newest release, Maple9, of Maple. This function is used to find closed forms for decimal approximations of numbers. Its scope is quite broad, identifying, for example, 3.146264369941972 as sqrt(2)+sqrt(3), and 1.098612288668110 as arcsinh(4/3). To experimentalists and computationalists, being able to identify numerical constants which appear in their work can often be the key to unravelling underlying theory. 2. Differential Equations i) Classification of Abel non-linear ODEs mapable into second order linear ODEs. ii) Development of a systematic algorithm for computing non-Liouvillian solutions of certain type, solving 92% of the linear examples of Kamke's book, with a small computational cost: only the factorization of a third degree polynomial. iii) Computation of the first solutions to Heun equations (non-degenerate cases) which can be expressed in terms of finite sums of hypergeometric equations. This result is also used to obtain families of special cases for the Heun functions. iv) Exact solutions of differential equations. We have recently (see ref: Burger, Labahn, van Hoeij) discovered new methods for finding exact solutions of linear odes having elliptic functions as their coefficients. In the case of second order equations we give a decision procedure for determining if an equation has a solution which is either an elliptic function or else a function which is doubly-periodic of the second kind. v) R. Burger, G. Labahn have recently (see ref: Burger, Labahn, van Hoeij) discovered new methods for finding exact solutions of linear odes having elliptic functions as their coefficients. In the case of second order equations we give a decision procedure for determining if an equation has a solution which is either an elliptic function or else a function which is doubly-periodic of the second kind. 3. Core Algebra i) A first modular GCD algorithm for computing multivariate GCDs over algebraic functions fields presented with one field extension. The algorithm generalizes to the case where the minimal polynomial factors - this has application to solving polynomial systems. ii) A new rational number reconstruction algorithm that is close to optimal in that it reduces the number of primes needed in a modular algorithm to within one or two of optimal. iii) Modular algorithms for factoring Ore (differential and differential) operators have been developed by Giesbrecht and Zhang. These are the first known to run in polynomial-time. iv) A new method for interpolating sparse multi-variate polynomials, based on a relatively stable version of Prony's algorithm, has been developed by M. Giesbrecht, G. Labahn and W.-s. Lee. Good stability is achieved by reduction to a generalized eigenvalue problem. We are completing a robust implementation and further investigating stability, especially on different bases and real evaluation points. 4. Symbolic Linear Algebra i) Systems of linear differential and recurrence equations [H. Cheng and G. Labahn] We have (see ref: Beckermann, Cheng and Labahn in Milestones) given an algorithm for transforming a matrix of Ore polynomials into one having a nonsingular leading or trailing matrix. This is central for finding the powers of zeros or poles of rational solutions of systems of recurrence or differential equations. ii) Systems of linear differential and recurrence equations [G. Labahn and Z. Li] We have investigated methods of transforming so called uncoupled Ore algebras into algebras where one establish criteria for determining when hyperexponential solutions exist for such a system. This covers cases where one is solving linear functional systems (systems having mixtures of recurrence and differential operators). These hyperexponential solutions include exponential functions (in case of differential operators) and hypergeometric functions (in case of recurrence operators).