We present a new fast algorithm for computing the Boys function using nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are competitive with previously developed algorithms for the same purpose.
We present a fast method for evaluating expressions of the form $$ u_j = sum_{i = 1,i ot = j}^n frac{alpha_i}{x_i - x_j}, quad text{for} quad j = 1,ldots,n, $$ where $alpha_i$ are real numbers, and $x_i$ are points in a compact interval of $mathbb{R
}$. This expression can be viewed as representing the electrostatic potential generated by charges on a line in $mathbb{R}^3$. While fast algorithms for computing the electrostatic potential of general distributions of charges in $mathbb{R}^3$ exist, in a number of situations in computational physics it is useful to have a simple and extremely fast method for evaluating the potential of charges on a line; we present such a method in this paper, and report numerical results for several examples.
Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and another is
nonstandard diffusion including the use of fractional derivatives. A major obstacle for application of these methods is the high computational cost from the numerical implementation of the nonlocal operators. It is natural to consider fast methods of fast multipole or hierarchical matrix type to overcome this challenge. Unfortunately the relevant kernels do not satisfy the standard necessary conditions. In this work a new class of fast algorithms is developed and analyzed, which is some cases reduces the computational complexity of applying nonlocal operators to essentially the same order of magnitude as the complexity of standard local numerical methods.
The Cadzows algorithm is a signal denoising and recovery method which was designed for signals corresponding to low rank Hankel matrices. In this paper we first introduce a Fast Cadzows algorithm which is developed by incorporating a novel subspace p
rojection to reduce the high computational cost of the SVD in the Cadzows algorithm. Then a Gradient method and a Fast Gradient method are proposed to address the non-decreasing MSE issue when applying the Cadzows or Fast Cadzows algorithm for signal denoising. Extensive empirical performance comparisons demonstrate that the proposed algorithms can complete the denoising and recovery tasks more efficiently and effectively.
In this paper we introduce a family of rational approximations of the reciprocal of a $phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation
and properties of this family of approximations applied to scalar and matrix arguments are presented. Moreover, we show that the matrix functions computed by these approximations exhibit decaying properties comparable to the best existing theoretical bounds. Numerical examples highlight the benefits of the proposed rational approximations w.r.t.~the classical Taylor polynomials and other rational functions.
The paper is concerned with the three-dimensional electromagnetic scattering from a large open rectangular cavity that is embedded in a perfectly electrically conducting infinite ground plane. By introducing a transparent boundary condition, the scat
tering problem is formulated into a boundary value problem in the bounded cavity. Based on the Fourier expansions of the electric field, the Maxwell equation is reduced to one-dimensional ordinary differential equations for the Fourier coefficients. A fast algorithm, employing the fast Fourier transform and the Gaussian elimination, is developed to solve the resulting linear system for the cavity which is filled with either a homogeneous or a layered medium. In addition, a novel scheme is designed to evaluate rapidly and accurately the Fourier transform of singular integrals. Numerical experiments are presented for large cavities to demonstrate the superior performance of the proposed method.