On Sep 4, 12:31 pm, Robert Miller <[EMAIL PROTECTED]> wrote: > Using numpy, reasonably accurate means up to only a few places, not > the full double precision. This leads to badness when you try to > factor anything. If it finds that 1.0000112451357 is a root of > (x-1)^3, you will likely get a quadratic irreducible as a factor...
Multiple roots are a very bad case for floating-point root finding. Unless the solver includes special-purpose heuristics to detect multiple roots, it's basically impossible to get an "accurate" result; if your input is given to n bits of precision, you should not expect more than n/k good bits for a k-fold root. (As William points out, you can get solutions that make the polynomial evaluate to a number very close to zero; basically the problem is that with a multiple root, there are many such numbers, and it's difficult to choose between them.) To see why this is true, consider the naïve floating-point error analysis model where you just pretend that all floating-point numbers are somewhat imprecise -- a little "fuzzy", if you will. Then the graph of a floating-point polynomial will be a fuzzy line. Consider the graph of (x-1)^3; this will be a fuzzy line with a horizontal tangent at x=1, y=0. If the fuzziness extends up and down by about j, then it will extend left and right by about cube_root(j). Carl Witty --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
