John P. Fitch

Course notes

Algebraic manipulation covers branches of software, particularly list processing, mathematics, notably logic and number theory, and applications largely in physics. The lectures will deal with all of these to a varying extent. The mathematical content will be kept to a minimum.Algebraic manipulation covers branches of software, particularly list processing, mathematics, notably logic and number theory, and applications largely in physics. The lectures will deal with all of these to a varying extent. The mathematical content will be kept to a minimum.

Applications of algebraic manipulation programs in physics

This paper describes the application to three areas of physics of computer programs that carry out formal algebraic manipulation. The application areas discussed are celestial mechanics, general relativity and quantum electrodynamics. The paper describes typical problems from each of these disciplines which can be solved using algebraic manipulative systems and presents sample programs for the solution of these problems using several algebra systems. For each discipline a review of published work acknowledging the use of algebra programs is presented and the most advanced applications are discussed in detail. In particular the Lie transform, Petrov classification and Kahanes simplification procedure are reviewed from the standpoint of algebra programs. A number of simple examples are used to introduce the reader to the capabilities of an algebra program and a brief review of the technical problems of algebraic manipulation is given. Further applications of such systems to mathematics, chemistry and engineering are briefly mentioned in the text and relevant work is referenced in the bibliography but the main emphasis is placed on applications in theoretical physics. However, the simple examples indicate, and the applications in the physical sciences confirm, that algebra systems are capable of exploitation over a much wider area than is covered in the present review. This review was completed in December 1971.

Implementing LISP in a high-level language

The problems of producing an efficient but robust implementation of a high‐level language by building on existing compilers are discussed with reference to a LISP system consisting of an interpreter, a compiler and a large collection of support routines.

A note on compacting garbage collection

A variation of the Haddon and Waite compacting garbage collector is presented that needs only bounded workspace, but which has typical runtime proportional to the size of the heap rather than n log n for a heap of size n. The algorithm has been measured in the context of a LISP system where it has been seen to behave close to its optimum. The relationship of this algorithm to one due to Lang and Weigbreit is also explained.

A review of algebraic manipulative programs and their application

This paper describes the applications area of computer programs that carry out formal algebraic manipulation. The first part of the paper is tutorial and severed typical problems are introduced which can be solved using algebraic manipulative systems. Sample programs for the solution of these problems using several algebra systems are then presented. Next, two more difficult examples are used to introduce the reader to the true capabilities of an algebra program and these are proposed as a means of comparison between rival algebra systems. A brief review of the technical problems of algebraic manipulation is given in the final section

Problem #8: random walk

This problem was brought to my attention by Professor Violet Cane of Manchester University, and derives from correlated random walks, in statistics. It turned out to be more difficult than I thought, causing difficulties to a number of algebra systems.This problem was brought to my attention by Professor Violet Cane of Manchester University, and derives from correlated random walks, in statistics. It turned out to be more difficult than I thought, causing difficulties to a number of algebra systems.

General relativity and the application of algebraic manipulative systems

The paper describes some applications of symbolic algebra systems to problems of general relativity including the derivation of the field equations, the Petrov classification of a metric, and the solution of the field equations in the presence of matter in a simple case. Attention is drawn to the strictly algebraic difficulties encountered in this work.

A simple method of taking nth roots of integers

When performing algebraic calculations by computer one is occasionally faced with expressions like 4(1/2). A particular example of this is in the statistics calculation described in [1], where the expressions involve[EQUATION]When all we require is the principal (positive) root the expression collapses. Dealing with (q2)(1/2) for an explicit q is not difficult; this note is concerned with the treatment of the 4(1/2).When performing algebraic calculations by computer one is occasionally faced with expressions like 4(1/2). A particular example of this is in the statistics calculation described in [1], where the expressions involve[EQUATION]When all we require is the principal (positive) root the expression collapses. Dealing with (q2)(1/2) for an explicit q is not difficult; this note is concerned with the treatment of the 4(1/2).

Problems #3 and #4 in REDUCE and MACSYMA

The algebra systems REDUCE and MACSYMA are used to solve SIGSAM Problem #3, the Reversion of a Double Series, and SIGSAM Problem #4, the Lie Transform Solution of the Harmonic Oscillator. All the algorithms used have been previously published, so the times may be used to compare both systems and algorithms independently.

Double by single length division

Arbitrary precision arithmetic packages are built around methods for multiplying single length numbers to get a double length result and dividing double length numbers by single length ones to get a quotient and remainder. Existing systems use one of three strategies to do the double by single division:1) descend to machine code,2) exploit the fact that if the number base is small enough there is no difficulty,3) Use a variant on what we term the recursive algorithm as displayed below.