An improved algorithm for the isolation of polynomial real zeros

THe Collins-Loos algorithm for computing isolating intervals for the zeros of an integer polynomial requires the evaluation of polynomials at rational points. This implies the use of arbitrary precision integer arithmetic. It is shown how careful use of single precision, floating point arithmetic within the context of a slightly modified algorithm can make the calculation considerably faster and no less exact. Typically, 95% or more of the evaluations can be done without exact arithmetic. The precise speedup depends on the relative costs of the arithmetic in a given implementation. The implementation on DEC KL-10 computer is some 5 to 10 times faster than the original Univac 1110 implementation in SAC-I.