A Practical Method for Floating-Point Gröbner Basis Computation

Abstract

Computing Gröbner bases with inexact coefficients is eagerly desired in industrial applications, but the computation with floating-point numbers is quite unstable if performed naively. In previous papers, the present author clarified that large term-cancellations occur frequently making the computation unstable, and he proposed a method of removing the harm of exact term-cancellations and a method of estimating the amounts of inexact term-cancellations. However, the estimation is very rough and never satisfactory. The inexact cancellations cause the accuracy loss of the output system, hence it is important to estimate their amounts precisely. In this chapter, we propose a practical method of estimating the amounts of inexact cancellations fairly well and very simply. We have tested our method by several examples, and obtained reasonable results. We also propose a simple device to reduce the amounts of exact term-cancellations.

Work supported in part by Japan Society for the Promotion of Science under Grants 19300001.

Appendix: Effective Floating-Point Numbers

The efloat number was proposed by Kako and Sasaki in 1997 [6] so as to detect the cancellation errors automatically. (Similar ideas would be proposed by other authors). We represent an efloat as $\#\mathrm{E}[f,\, e]$ (big-efloat as $\#\mathrm{BE}[f,\, e]$), where $f$ is a floating-point number representing the value of this efloat, either of fixed precision or of arbitrary precision, and $e$ is a fixed-precision floating-point number representing an approximate error of $f$. The efloat arithmetic is as follows: (we put $a = \#\mathrm{E}[f_a, e_a]$ and $b = \#\mathrm{E}[f_b, e_b]$).

$$\begin{aligned}&a + b \Longrightarrow \#\mathrm{E}[f_a + f_b,\ \max \{e_a,e_b\}], \\&a - b \Longrightarrow \#\mathrm{E}[f_a - f_b,\ \max \{e_a,e_b\}], \\&a \times b \Longrightarrow \#\mathrm{E}[f_a \times f_b,\ \max \{|f_b e_a|,|f_a e_b|\}], \\&a \div b \Longrightarrow \#\mathrm{E}[f_a \div f_b,\ \max \{|e_a/f_b|,|f_a e_b/f_b^2|\}]. \end{aligned}$$

Thus, the value-part of efloat is nothing but the conventional floating-point value. On the other hand, the error-part of efloat represents the cancellation error approximately; the rounding errors are neglected in determining the error-parts. Suppose the coefficients of the input polynomials contain relative errors of $\varepsilon _\mathrm{init}$. Then, we set the initial error-part of each coefficient to $(5\sim 50) \varepsilon _\mathrm{init}\!\times \!|f|$; we set the error-part to $10^{-15} \times |f|$ for double-precision efloat, and to a much larger value for big-efloat. In our algebra system, the efloat $\#\mathrm{E}[f,\, e]$ with $|f| < e$ is set to 0 automatically. Therefore, our algebra system detects fully-erroneous terms and remove them automatically, unless the rounding errors accumulate to $(5\sim 50) \varepsilon _\mathrm{init}$ or more, which is extremely rare in practical computations. We note that the computation in Example 1 was done by using efloats; if we use the conventional floating-point numbers we will encounter many very small terms which are fully erroneous.