A class of quadrature formulae is presented applicable to both nonsingular and singular functions, generalizing the classical endpoint corrected trapezoidal quadrature rules. While the latter rules are usually derived by means of the Euler-Maclaurin formula, their generalizations are obtained as solutions of certain systems of linear algebraic equations. A procedure is developed for the construction of very high-order quadrature rules, applicable to functions with a priori specified singularities, and relaxing the requirements on the distribution of nodes. The scheme applies not only to nonsingular functions but also to a wide class of functions with monotonic singularities. Numerical experiments are presented demonstrating the practical usefulness of the new class of quadratures. Tables of quadrature weights are included for singularities of the form s(x) = |x| λ for a variety of values of λ, and s(x) = log |x|.

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The Method of Moments MoM is often e ectively used i n the extraction of passive components in modeling integrated circuits and MCM packaging. MoM extraction, however, involves solving a dense system of linear equations, and using direct factorization methods can be p r ohibitive for large problems. In this paper, we present a Fast Method of Moments Solver FMMS for the rapid solution of such linear systems. Our algorithm exploits the fact that the integral equation kernels are l o c ally smooth" and can be d r amatically compressed via the singular value decomposition SVD. This greatly speeds up the matrix-vector products in a Krylov-subspace iterative algorithm e.g., GMRES. We demonstrate the e ciency and exibility of our scheme for the modeling of embedded inductors in MCM-D. Results are presented to show that the method i s a c curate and can be two orders of magnitude faster than Gaussian elimination and one order of magnitude faster than standard iterative schemes.

We describe a new method for accurate large-scale capacitance calculations. The algorithm uses an integral equation formulation, but with a new representation for charge distributions that decouples charge variation from conductor geometry. This separation significantly reduces the problem size compared to a traditional discretization, resulting in a large speed increase. The full capacitance matrix of typical interconnect problems with thousands of nets can be computed in a few hours.