The boundary integral method is an efficient approach for solving time-harmonic acoustic obstacle scattering problems. The main computational task is the evaluation of an oscillatory boundary integral at each discretization point of the boundary. This paper presents a new fast algorithm for this task in two dimensions. This algorithm is built on top of directional low-rank approximations of the...

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or “first barycentric” formula dating to Jacobi in 1825. Thi...

A number of proofs have appeared since then, an especially elegant one given by Ganelius [5]. In most of the proofs, the indicator of A is approximated by its convolution with an appropriate (Fejér-type) kernel. We shall present another proof, based on the arguments developed by Chebyshev, Markov, and Stieltjes to prove the Central Limit Theorem (see Akhiezer [1, Ch. 3]). In this approach, the ...

Abstract Two efficient quadrature formulae have been developed for evaluating numerically certain singular integral equations of the first kind over the finite interval [-1,1]. Central to this work is the application of four special cases of the Jacobi polynomials P n (x), whose zeros served as interpolation and collocation nodes: (i) α = β = −2 , Tn(x), the first kind Chebyshev polynomials (ii...

Generalized matrix functions (GMFs) extend the concept of a matrix function to rectangular matrices via the singular value decomposition. Several applications involving directed graphs, Hamiltonian dynamical systems, and optimization problems with low-rank constraints require the action of a GMF of a large, sparse matrix on a vector. We present a new method for applying GMFs to vectors based on...

Approximating a hand-written curve by a Bézier curve can be performed by two main processes: (1) the selection of the significant points on the curve and (2) the curve approximation. This paper focuses on finding a set of featured points, which is an important component for curve fitting into a Bézier curve. This new approach is an extension of the authors’ previous work by reducing the weaknes...

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or “first barycentric” formula dating to Jacobi in 1825. Thi...