Paul S. Wang

P-adic reconstruction of rational numbers

In a recent paper, Wang [1981] introduces a p-adic algorithm for the construction of partial fraction decompositions. This differs from the usual p-adic algorithms for factorisation or the computation of greatest common divisors ([Wang, 1978], [Wang, 1980], [Moore & Norman, 1981]) in that the p-adic image is used to reconstruct rational numbers, rather than integers.

An improved multivariate polynomial factoring algorithm

A new algorithm for factoring multivariate polynomials over the integers based on an algorithm by Wang and Rothschild is described. The new algorithm has improved strategies for dealing with the known problems of the original algorithm, namely, the leading coefficient problem, the bad-zero problem and the occurrence of extraneous factors. It has an algorithm for correctly predetermining leading coefficients of the factors. A new and efficient p-adic algorithm named EEZ is described. Bascially it is a linearly convergent variable-by-variable parallel construction. The improved algorithm is generally faster and requires less store then the original algorithm. Machine examples with comparative timing are included.

FINGER: A Symbolic System for Automatic Generation of Numerical Programs in Finite Element Analysis

FINGER is a LISP-based system to derive formulas needed in finite element analysis, and to generate FORTRAN code from these formulas. The generated programs can be used with existing, FORTRAN-based finite element analysis packages. This approach aims to replace tedious hand computations that are time consuming and error prone. The design and implementation of FINGER are presented. Techniques for generating efficient code are discussed. These include automatic intermediate expression labelling, interleaving formula derivation with code generation, exploiting symmetry through generated functions and subroutines. Current capabilities include generation of material matrices, strain-displacement matrices and stiffness matrices. FINGER contains a package, called GENTRAN, that translates symbolic formulas into FORTRAN. GENTRAN can generate functions, subroutines and entire programs. Thus, it is also of interest as a general-purpose FORTRAN code generator, Aside from the finite element application, the techniques developed and employed are useful for automatic code generation in general.

MACSYMA from F to G

A descriptive, non-technical tutorial on MACSYMA, a well-known and widely used Computer Algebra system, is presented. Several examples of its capabilities are exhibited using actual MACSYMA input and output. A discussion of computer-based symbolic mathematical computation is motivated by pointing out inherent difficulties with familiar numeric computations. The inner workings of Computer Algebra systems are briefly discussed in addition to some on-going work on MACSYMA and future directions.

Polynomial Factorization

A new coefficient bound is established for factoring univariate polynomials over the integers. Unlike an overall bound, the new bound limits the size of the coefficients of at least one irreducible factor of the given polynomial. The single-factor bound is derived from the weighted norm introduced in Beauzamy et al. (1990) and is almost optimal. Effective use of this bound in p-adic lifting results in a more efficient factorization algorithm. A full example and comparisons with known coefficient bounds are included.

A p-adic algorithm for univariate partial fractions

Partial fractions is an important algebraic operation with many applications in applied mathematics, physics and engineering. It is also an important operation in any computer symbolic and algebraic system. Among other things, it is used in the integration algorithm.

Factoring multivariate polynomials over algebraic number fields

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented in the algebraic manipulation system MACSYMA.** Some machine examples with timing are included.

Design and protocol for Internet accessible mathematical computation

Mathematical computing can become easily accessible and conveniently usable on the Internet. The distributed Internet Accessible Mathematical Computation (IAMC) system can supply mathematical computing powers widely through TCP/IP, the Web, or email. The overall IAMC architecture, a feasibility study, elements of the Mathematical Computation Protocol (MCP), the design of IAMC client and IAMC server, the server interface to compute engine, and the API design for a Java implementation of MCP are presented.

Code Generation and Optimization for Finite Element Analysis

The design and implementation of a software system for automatically generating code for finite element analysis are described. Exact symbolic computational techniques are employed to derive strain-displacement matrices and element stiffness matrices. Methods for dealing with the excessive growth of symbolic expressions in practical computations are discussed. Automatic FORTRAN code generation and optimization are described with emphasis on improving the efficiency of the resultant code. The generated code can be used, without modification, with a FORTRAN-based finite element analysis package.

WME: towards a web for mathematics education

Reported is an approach for Web-based Mathematics Education (WME). The WME framework is a distributed system that aims to create a Web for mathematics education. Components of the WME framework include the Mathematics Education Markup Language (MeML) for page markup, regular Web servers to deliver pages, WME Page Processors to enable common Web browsers to receive MeML pages, and a variety of WME services (mathematical and educational) to supply power and interactivity to MeML pages. The MeML language supports the creation of efficient, effective, and dynamic mathematics education content. The WME architecture is designed so that mathematics education contents (static and/or dynamic) and remote WME services can be built independently, maintained independently, deployed easily, and still interoperate and reinforce one another on a global scale.