On the Design and Construction of Algebraic Manipulation Systems

We compare and contrast several techniques for the implementation of components of an algebraic manipulation system. On one hand is the mathematical-algebraic approach which characterizes (for example) IBM's Axiom. On the other hand is the more ad hoc approach which characterizes many other popular systems (for example, Macsyma, Reduce, Maple, and Mathematica). While the algebraic approach has generally positive results, careful examination suggests that there are significant remaining problems, especially in the representation and manipulation of analytical, as opposed to algebraic, mathematics. We describe some of these problems and some general approaches for solutions. INTRODUCTION Symbolic algebraic mathematics programs have been in use for several decades now. Some programs implement stand-alone algorithms which accomplish a particular task: say, factoring a univariate polynomial. Other programs appear as part of a monolithic system such as Macsyma, Mathlab '68, Axiom (formerly S..

Series Solutions of Algebraic and Differential Equations: A Comparison of Linear and Quadratic Algebraic Convergence

Speed of convergence of Newton-like iterations in an algebraic domain can be affected heavily by the increasing cost of each step, so much so that a quadratically convergent algorithm with complex steps may be comparable to a slower one with simple steps. This note gives two examples: solving algebraic and first-order ordinary differential equations using the MACSYMA algebraic manipulation system, demonstrating this phenomenon. The relevant programs are exhibited in the hope that they might give rise to more widespread application of these techniques. 1 Newton Iteration in a Power Series Domain Newton iteration is a powerful tool for developing approximations to solutions of various types of equations. Recently, the case of iteration in a power series domain has been studied in some detail as a notion relevant especially in the field of computer-aided algebraic computation. It has been shown by Kung and Traub [6] that there are fast procedures for finding Taylor-series type expansion..

Chapter 1 PROBLEM SOLVING ENVIRONMENTS AND SYMBOLIC COMPUTING

Abstract What role should be played by symbolic mathematical computation facilities in scientific and engineering “problem solving environments”? Drawing upon standard facilities such as numerical and graphical libraries, symbolic computation should be useful for: The creation and manipulation of mathematical models; The production of customoptimized numerical software; The solution of delicate classes of mathematical problems that require handling beyond that available in traditional machine-supported floating-point computation. Symbolic representation and manipulation can potentially play a central organizing role in PSEs since their more general object representation allows a program to deal with a wider range of computational issues. In particular Numerical, graphical, and other processing can be viewed as special cases of symbolic manipulation with interactive symbolic computing providing both an organizing backbone and the communication “glue ” among otherwise dissimilar components

A Review of Mathematica

this paper are my own, and do not necessarily represent the views of government sponsors or others mentioned below. I thank Paul Abbott, Robert Campbell, Steven Christensen, Sam Dooley, David Jacobson, Dan Grayson, Velvel Kahan, Roman Maeder, Kevin McIsaac, Michael Monagan, Steven Omohundro, Malcolm Slaney, Neil Soiffer, William N. Venables, Stephen Wolfram, and others, for enlightening discussions

A review of Macsyma

We review the successes and failures of the Macsyma algebraic manipulation system from the point of view of one of the original contributors. We provide a retrospective examination of some of the controversial ideas that worked, and some that did not. We consider input/output, language semantics, data types, pattern matching, knowledge-adjunction, mathematical semantics, the user community, and software engineering. We also comment on the porting of this system to a variety of computing systems, and possible future directions for algebraic manipulation system-building

Importing Pre-packaged Software into Lisp: Experience with Arbitrary-Precision Floating-Point Numbers

We advocate the use of Common Lisp as a powerful glue for building scientific computing environments. Naturally one then has to address mixing pre-existing (non Lisp) code into this system. We provide a specific example as an elaborate FORTRAN system written by David Bailey for arbitraryprecision floating-point numeric calculation. We discuss the advantages and disadvantages of wholesale importing into Lisp. A major advantage is being able to use state-of-the art packaged software sooner, while overcoming the disadvantages caused by FORTRAN's traditional batch orientation and weak storage model. In this paper we emphasize in particular how e#ective use of imported systems may require one to address the contrast between the functional (Lisplike) versus state-transition-based (Fortran-like) approaches to dealing with compound objects. While our example is high-precision floats, other highly useful packages including those for simulation, PDE solutions, signal processing, statistical comp..