Series acceleration

Abstract: An asymptotic theory for stochastic processes generated from nonlinear transformations of nonstationary integrated time series is developed. Various nonlinear functions of integrated series such as ARIMA time series are studied, and the asymptotic distributions of sample moments of such functions are obtained and analyzed. The transformations considered in the paper include a variety of functions that are used in practical nonlinear statistical analysis. It is shown that their asymptotic theory is quite different from that of integrated processes and stationary time series. When the transformation function is exponentially explosive, for instance, the convergence rate of sample functions is path dependent. In particular, the convergence rate depends not only on the size of the sample but also on the realized sample path. Some brief applications of these asymptotics are given to illustrate the effects of nonlinearly transformed integrated processes on regression. The methods developed in the paper are useful in a project of greater scope concerned with the development of a general theory of nonlinear regression for nonstationary time series. Nonstationary time series arising from autoregressive models with roots on the unit circle have been an intensive subject of recent research. The asymptotic behavior of regression statistics based on integrated time series (those for which one or more of the autoregressive roots are unity) has received the most attention, and a fairly complete theory is now available for linear time series regressions. The resulting limit theory forms the basis of much ongoing empirical econometric work, especially on the subject of unit root testing and cointegration model

Abstract: The classical algorithms require order n ~ operations to compute the first n terms in the reversion of a power series or the composition of two series, and order nelog n operations if the fast Founer transform is used for power series multiplication In this paper we show that the composition and reversion problems are equivalent (up to constant factors), and we give algorithms which require only order (n log n) ~/2 operations In many cases of practical importance only order n log n operations are required, these include certain special functions of power series and power series solution of certain differential equations Applications to root-finding methods which use inverse mterpolauon and to queuemg theory are described, some results on multivariate power series are stated, and several open questions are mentioned

Abstract: A method for improving the convergence of the series representing the doubly periodic free-space Green's function is presented. The method consists of successively applying three different transformations to the Green's function spectral representation. Kummer's transformation is first applied to convert the slowly converging spectral representation into the sum of a rapidly converging series and a slowly converging series. The latter series is recognized as the spectral representation of the original periodic source distribution radiating in a medium with an imaginary wavenumber. Application of the Poisson transformation to this series renders it exponentially convergent since it effectively represents propagation of point source contributions through a medium with imaginary wavenumber. Finally, Shanks' transform is plotted versus the number of terms taken in the series. Numerical results confirm that an improvement in the convergence rate of the series is achieved for a particular convergence criterion. >

Abstract: Analytic solutions of the Regge·Wheeler equation are presented in the form of a series of hypergeometric functions and Coulomb wave functions which have different regions of convergence. Relations among these solutions are established. The series solutions are given as the Post­ Minkowskian expansion with respect to the parameter €=2Mw, M being the mass of a black hole. This expansion corresponds to the post-Newtonian expansion when they are applied to the gravita­ tional radiation from a particle in a circular orbit around a black hole. These solutions can also be useful for numerical computations. In a previous work,l) we presented analytic solutions of the Regge-Wheeler (RW) equation in the form of a series of hypergeometric functions. We proved that recurrence relations among hypergeometric functions as given in Appendix A in this text and showed that coefficients of series are systematically determined in a power series of €=2Mw, M being the mass of black hole. We also presented analytic solutions in the form of a series of Coulomb wave functions which turn out to be the same as those given by Leaver. 2 ) We found that the series of solutions is character­ ized by the renormalized angular momentum which turns out to be identical. Then, we obtained a good solution by matching these two types of solutions. This method can be extended for the Teukolsky equation 3 ) in the Kerr geometry_ In this case, the coefficients of series of hypergeometric functions and also those of a series of Coulomb wave functions satisfy the three term recurrence relations. Con­ cerning these recurrence relations, Otchik 4 ) made the important observation that the recurrence relation for the two series are identical, which made it possible to relate these two series solutions_*) Following the discussion by Otchik,4) Mano, Suzuki and Takasugi 5 ) extended our analysis to the Teukolsky equation in the Kerr geometry and reported analytic solutions. We discussed the convergence regions of these series and the relation between two solutions of different regions of convergence. The series are expressed in the € expansion which corresponds to the Post-Minkowskian expansion and also to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. In this paper, we present analytic solutions of the RW equation and discuss the analytic properties of these solutions by reorganizing our previous work!) following *) In Otchik's paper, the relation between the series of hypergeometric functions and the series of Coulomb wave functions is studied in the intermediate region where both series converge, though the series which he treated are not the solutions of Teukolsky equation.