Application of vector-valued rational approximations to the matrix eigenvalue problem and connections with Krylov subspace methods

Let F(z) be a vectored-valued function F: C approaches C sup N, which is analytic at z=0 and meromorphic in a neighborhood of z=0, and let its Maclaurin series be given. We use vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N X N matrix that may be diagonalizable or not. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory provides a set of completely new results and constructions for these Krylov subspace methods. This theory suggests at the same time a new mode of usage for these Krylov subspace methods that were observed to possess computational advantages over their common mode of usage

Fast algorithms for computing isogenies between elliptic curves

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree $\ell$ ($\ell$ different from the characteristic) in time quasi-linear with respect to $\ell$. This is based in particular on fast algorithms for power series expansion of the Weierstrass $\wp$-function and related functions

An Extended and More Sensitive Search for Periodicities in RXTE/ASM X-ray Light Curves

We present the results of a systematic search in approximately 14 years of Rossi X-ray Timing Explorer All-Sky Monitor data for evidence of periodicities not reported by Wen et al. (2006). Two variations of the commonly used Fourier analysis search method have been employed to achieve significant improvements in sensitivity. The use of these methods and the accumulation of additional data have resulted in the detection of the signatures of the orbital periods of eight low-mass X-ray binary systems and of ten high-mass X-ray binaries not listed in the tables of Wen et al.Comment: 20 pages, 22 figures, in emulateapj format; submitted to ApJ