New algorithms for the exact computation of the sign of algebraic expressions

摘要

The paper considers the problem of exact computation of the sign of algebraic expressions of real numbers represented in floating point arithmetic. We describe a new method for the exact computation and suggest some variations to improve the efficiency. The input data for the algorithm is represented by normalized floating point numbers with fixed mantissa length (machine numbers). The algorithm computes the exact value of the sign of the sum of machine numbers and it can be applied to exactly compute the sign of almost any algebraic expression. We suggest several variations of the original Exact Sign of a Sum Algorithm (ESSA) to improve the performance of the algorithm and we test the algorithms on different data sets. This includes the implementation of floating point filters based on interval analysis, a special algorithm for performing multiple bit-wise transformations on the numbers in lists and the application of different rules to reduce the number of iterations of the algorithm. The theoretical upper bound on the complexity of ESSA is O(l/sup 2/), where l is the number of elements of the input. The expected average, experimental complexity of the suggested algorithms is proportional to the length of the input lists and it is close to l/2 in most cases. We perform a comparison analysis among the algorithms. The comparisons are based on the computational efficiency of the algorithms on both well posed and ill posed data sets. The algorithms are verified by the computer implementations.