Recurrent relations and speed-up of computations using computer algebra systems

Abstract

Implementation of various numerical methods often needs organisation of computation using complex iterative formulae, i.e. cycles. The time for performing a program on a computer and the necessary storage capacity are largely dependent on form of these formulae. It is desirable to construct computations so that each iterative step might use the results obtained at the previous steps as completely as possible. This implies the association of the given formulae with the recurrent relations, that bring about the same result but economise the arithmetic operations. A special algebraic method has been created that provides for the automatic construction of such recurrent relations. This method and main ways of its use in the algorithms of cycle optimisation, in the computer algebra systems and in algorithms of automatic parallel programs construction are explained. Many examples and programs are given.