Clifford algebra with Reduce

2004

CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C (B) -the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUMbased on Chevalley's recursive formula, and cmulRS -based on a non-recursive Rota-Stein sausage. Graßmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.

Science of Computer Programming, 2006

The role of computer algebra systems (CASs) is not limited to analyzing and solving mathematical and physical problems. They have also been used as tools in the development process of computer programs, starting from the specification and ending with the coding and testing phases. In this way one can exploit their powerful mathematical capacity during the development phases and, by the other way, take advantage of the speed performance of languages such as FORTRAN or C in the implementation. Among the mathematical features of CASs there are transformations allowing one to optimize the final code instructions. In this paper we show some kind of optimizations that can be done on new or existing algorithms, by extending some techniques that compilers apply currently to optimize the machine code. The results show that the CPU time taken by the optimized code is reduced by a factor that can reach 5. The optimizations are performed with a package built on a well known CAS: Mathematica.

The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.

1992

2000

Maple is a comprehensive general purpose computer algebra system. It is used primarily in education and scientific research in the sciences, in mathematics, and in engineering. Maple can do both symbolic and numerical calculations and has facilities for 2 and 3-dimensional graph- ical output. The newest version of Maple, Maple V Release 2, sports a new user interface that integrates

arXiv: Rings and Algebras, 2019

Clifford algebras are an active area of mathematical research. The main objective of the paper is to exhibit a construction of a matrix algebra isomorphic to a Clifford algebra of signature (p,q), which can be automatically implemented using general purpose linear algebra software. While this is not the most economical way of implementation for lower-dimensional algebras it offers a transparent mechanism of translation between a Clifford algebra and its isomorphic faithful real matrix representation. Examples of lower dimensional Clifford algebras are presented.

Journal of Symbolic Computation, 1988

This paper explains how computer algebra (Reduce) was used to analyse the expressions resulting from the complexity analysis of a family of algorithms for the isolation of real roots of polynomials. The expressions depend on sufficiently many parameters, and are sufficiently complex, that manual analysis would have been almost impossible. In addition, by analysing constant factors as well as the 0 form of the expressions, we obtain more information about the relative costs of algorithms with the same 0 complexity.

Recursion Theory, its Generalisations and Applications

Notice the instruction can be ]J specified by its characteristic numerical parameters <ao-1 ,]J,cr,>.. 1 , ... ,Ak> where we let ao-1 stand for a number which refers to the kind of the instruction and a stand for that 1 such that a is cr., a point taken up later on in this section when we l consider coding.

Proceedings of the 7th Wseas International Conference on Engineering Education, 2010

We investigate a possible application of computer algebra system as a tool in learning and teaching mathematics. There are analyzed the technical and didactic aspects of some mathematical expressions using different systems. The main purpose is to help pupils, students, teachers in better understanding of different computer algebra systems. Knowing the advantages and disadvantages of these systems, they became able for correct application of these systems. The examples are basically taken from secondary school, but also they can be part of other exercises in faculties. There are used the basic commands of the computer algebra with which the teachers and students are familiar..