Comment #92 on issue 1598 by fredrik.johansson: New polynomials
manipulation module
http://code.google.com/p/sympy/issues/detail?id=1598
Great progress! Keep up the good work.
For complex root isolation, perhaps you can use mpmath to compute
approximate roots
r_k, then go back to exact arithmetic, translate the polynomial onto each
r_k and
bound the distance to the zero nearest to the origin of the translated
polynomial. If
the approximate roots have high accuracy then this should yield sharp bounds
immediately. That is, the polynomial translated onto an approximate root
will usually
be something like
eps + Ax + Bx^2 + ...
and with eps small, A large in comparison to eps (usually of unit
magnitude), |Bx^2 +
...| small, it should be possible to bound the error of the root (but I
don't know
the exact formula for this).
Obviously some care is required to do this robustly. Perhaps calling mpmath
with
repeatedly doubling precision, combined with a fully rigorous
implementation of the
verification step, will do the job.
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