49 research outputs found
Improved Polynomial Remainder Sequences for Ore Polynomials
Polynomial remainder sequences contain the intermediate results of the
Euclidean algorithm when applied to (non-)commutative polynomials. The running
time of the algorithm is dependent on the size of the coefficients of the
remainders. Different ways have been studied to make these as small as
possible. The subresultant sequence of two polynomials is a polynomial
remainder sequence in which the size of the coefficients is optimal in the
generic case, but when taking the input from applications, the coefficients are
often larger than necessary. We generalize two improvements of the subresultant
sequence to Ore polynomials and derive a new bound for the minimal coefficient
size. Our approach also yields a new proof for the results in the commutative
case, providing a new point of view on the origin of the extraneous factors of
the coefficients
Bounds for D-finite closure properties
We provide bounds on the size of operators obtained by algorithms for
executing D-finite closure properties. For operators of small order, we give
bounds on the degree and on the height (bit-size). For higher order operators,
we give degree bounds that are parameterized with respect to the order and
reflect the phenomenon that higher order operators may have lower degrees
(order-degree curves)
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache