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 $n$-th term in a recurrent sequence of suitable type using $O(n^{1/2})$ "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 $n$ 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

Nearly Optimal Computations with Structured Matrices

We estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic most popular classes, that is, Toeplitz, Hankel, Cauchy and Van-der-monde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in \cite{kirrinnis-joc-1998}, except for rational interpolation, which we supply now. All known Boolean cost estimates for these problems rely on using Kronecker product. This implies the $d$-fold precision increase for the $d$-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representation of our tasks and algorithms in terms of both structured matrices and polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer's important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes.Comment: (2014-04-10

Computing Real Roots of Real Polynomials

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schoenhage in 1982. The main drawbacks of Pan's method are that it is quite involved and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted ANEWDSC, which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. ANEWDSC can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task.Comment: to appear in the Journal of Symbolic Computatio