1 research outputs found
New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems
We present a new data structure to approximate accurately and efficiently a
polynomial of degree given as a list of coefficients. Its properties
allow us to improve the state-of-the-art bounds on the bit complexity for the
problems of root isolation and approximate multipoint evaluation. This data
structure also leads to a new geometric criterion to detect ill-conditioned
polynomials, implying notably that the standard condition number of the zeros
of a polynomial is at least exponential in the number of roots of modulus less
than or greater than .Given a polynomial of degree with
for , isolating all its complex roots or
evaluating it at points can be done with a quasi-linear number of
arithmetic operations. However, considering the bit complexity, the
state-of-the-art algorithms require at least bit operations even for
well-conditioned polynomials and when the accuracy required is low. Given a
positive integer , we can compute our new data structure and evaluate at
points in the unit disk with an absolute error less than in
bit operations, where means
that we omit logarithmic factors. We also show that if is the absolute
condition number of the zeros of , then we can isolate all the roots of
in bit operations. Moreover, our
algorithms are simple to implement. For approximating the complex roots of a
polynomial, we implemented a small prototype in \verb|Python/NumPy| that is an
order of magnitude faster than the state-of-the-art solver \verb/MPSolve/ for
high degree polynomials with random coefficients