Laguerre–Galerkin methods with reduced sum-products

Abstract

The always stable solution of time-domain integral equations by plain implementation of the classical marching-on-in-degree (MOD) methods demands O(M2N3) operations, where M and N are, respectively, the spatial and temporal degrees of freedom. Diverse caching and basis reordering approaches are first explored to reduce the number of matrix–vector multiplications necessary to compute the right hand side of the MOD system. The new schemes primarily eliminate the existing O(N) innermost vector summations on all past polynomial orders. This complexity reduction facilitates the usage of FFT approaches to accelerate the calculation of existing temporal convolutions. In an additive secondary stage, the recursive time convolution products of the Toeplitz block aggregates of the retarded interaction matrices are expedited by array multiplications in spectral domain. When the translation invariancy on time-order indices are placed on the outermost possible nested Toeplitz levels due to the space shift invariancy, the overall computational complexities and memory requirements can be further reduced to O(MNlog(MN)) and O(MN), respectively.