Cancellation Errors in Multivariate Resultant Computation with Floating-point Numbers

摘要

We test five methods of computing resultant of multivariate polynomials with floating-point number coefficients, mostly from the viewpoint of numerical errors. The methods tested are a) Brown-Traub's subresultant PRS algorithm, b) Collins' algorithm based on polynomial interpolation, c) minor expansion method of Bezout's determinant, d) Gaussian elimination method (Bareiss' algorithm) of Bezout's determinant, and e) efficient Gaussian elimination method (Sasaki-Murao's algorithm) of Bezout's determinant. We find that method a) causes so large cancellation errors that it is almost useless practically. The errors caused by methods b) and d) are not so large as those by a) but still large. The methods c) and e) are quite safe against the cancellation error and they are efficient, too. Mechanisms of occurrence of large cancellation errors in methods a), b) and d) are clarified. By these, we show that approximate algebraic computation often causes large cancellation errors hence the error analysis and development of error-safe algorithms are very important.