Comment #3 on issue 3295 by [email protected]: nroots and RootOf.evalf
duplicate code
http://code.google.com/p/sympy/issues/detail?id=3295
Mateusz will have to confirm, but I believe so. polyroots is a numerical
algorithm. It keeps track of errors, but at the end of the day it requires
floating point precision and suffers from the usual problems of floating
point numbers.
The RootOf interval isolation algorithm is a rational algorithm, meaning it
only uses operations like polynomial division and multiplication, and is
thus infallible. See for example,
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.5163 (note: I do
not know if this is the exact algorithm implemented, but the principle is
the same).
And anyway, it's not too hard to find an example where nroots gives wrong
answers, simply because it uses floating point numbers always, even when
the coefficients are rational:
In [47]: nroots(expand((x - S(1000000000000)/1000000000001)*(x - 1)))
Out[47]: [0.9999999999995 - 1.05345638729316e-8⋅ⅈ, 0.9999999999995 +
1.05351934154826e-8⋅ⅈ]
In [48]: roots(expand((x - S(1000000000000)/1000000000001)*(x - 1)))
Out[48]:
⎧1000000000000 ⎫
⎨─────────────: 1, 1: 1⎬
⎩1000000000001 ⎭
In [49]: RootOf(expand((x - S(1000000000000)/1000000000001)*(x - 1)), 0)
Out[49]:
1000000000000
─────────────
1000000000001
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