7,806 research outputs found
On Polynomial Multiplication in Chebyshev Basis
In a recent paper Lima, Panario and Wang have provided a new method to
multiply polynomials in Chebyshev basis which aims at reducing the total number
of multiplication when polynomials have small degree. Their idea is to use
Karatsuba's multiplication scheme to improve upon the naive method but without
being able to get rid of its quadratic complexity. In this paper, we extend
their result by providing a reduction scheme which allows to multiply
polynomial in Chebyshev basis by using algorithms from the monomial basis case
and therefore get the same asymptotic complexity estimate. Our reduction allows
to use any of these algorithms without converting polynomials input to monomial
basis which therefore provide a more direct reduction scheme then the one using
conversions. We also demonstrate that our reduction is efficient in practice,
and even outperform the performance of the best known algorithm for Chebyshev
basis when polynomials have large degree. Finally, we demonstrate a linear time
equivalence between the polynomial multiplication problem under monomial basis
and under Chebyshev basis
Evaluating parametric holonomic sequences using rectangular splitting
We adapt the rectangular splitting technique of Paterson and Stockmeyer to
the problem of evaluating terms in holonomic sequences that depend on a
parameter. This approach allows computing the -th term in a recurrent
sequence of suitable type using "expensive" operations at the cost
of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either
naive evaluation or asymptotically faster algorithms for ranges of
encountered in applications. As an example, fast numerical evaluation of the
gamma function is investigated. Our work generalizes two previous algorithms of
Smith.Comment: 8 pages, 2 figure
Polynomial evaluation over finite fields: new algorithms and complexity bounds
An efficient evaluation method is described for polynomials in finite fields.
Its complexity is shown to be lower than that of standard techniques when the
degree of the polynomial is large enough. Applications to the syndrome
computation in the decoding of Reed-Solomon codes are highlighted.Comment: accepted for publication in Applicable Algebra in Engineering,
Communication and Computing. The final publication will be available at
springerlink.com. DOI: 10.1007/s00200-011-0160-
Faster polynomial multiplication over finite fields
Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two
polynomials in F_p[X] of degree less than n. For n large compared to p, we
establish the bound M_p(n) = O(n log n 8^(log^* n) log p), where log^* is the
iterated logarithm. This is the first known F\"urer-type complexity bound for
F_p[X], and improves on the previously best known bound M_p(n) = O(n log n log
log n log p)
A hybrid approach to Fermi operator expansion
In a recent paper we have suggested that the finite temperature density
matrix can be computed efficiently by a combination of polynomial expansion and
iterative inversion techniques. We present here significant improvements over
this scheme. The original complex-valued formalism is turned into a purely real
one. In addition, we use Chebyshev polynomials expansion and fast summation
techniques. This drastically reduces the scaling of the algorithm with the
width of the Hamiltonian spectrum, which is now of the order of the cubic root
of such parameter. This makes our method very competitive for applications to
ab-initio simulations, when high energy resolution is required.Comment: preprint of ICCMSE08 proceeding
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