Chapter 22 - Summing Riemann Theta Functions over the N-Ellipsoid

Abstract

Publisher Summary This chapter describes the procedure for summing Riemann theta functions over the N-ellipsoid. The number of terms in the summation of the theta function occurs over a hypercube in lattice space, a number that increases exponentially with the number of degrees of freedom N. The chapter presents how to make these computations polynomial rather than exponential in character, thus rendering the approach practical from a computer-time perspective. The hypercube circumscribes an N-sphere or hypersphere. For the purposes of numerical computation, the number of terms in the partial theta-function summation in the hypercube N cube = (2M +1) N to the number of terms inside the enclosed hypersphere, N sphere can be compared. The chapter also presents a summary of theta-function summation over hyperellipsoid.