We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the interpolation formula to handle all poly...

The accurate calculation of the sound field is one most concerning issues in hydroacoustics. one-dimensional spectral method has been used to correctly solve simplified underwater acoustic propagation models, but it difficult actual ocean fields using this model due its application conditions and approximation error. Therefore, necessary develop a direct solution for two-dimensional Helmholtz e...

This paper reviews the notion of interpolation of a smooth function by means of Chebyshev polynomials, and the well-known associated results of spectral accuracy when the function is analytic. The rate of decay of the error is proportional to ρ−N , where ρ is a bound on the elliptical radius of the ellipse in which the function has a holomorphic extension. An additional theorem is provided to c...

In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn(z,N) as the degree grows to infinity. Global asymptotic formulas are obtained as n → ∞, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.

Deep unfolding is a promising deep-learning technique, whose network architecture based on expanding the recursive structure of existing iterative algorithms. Although convergence acceleration remarkable advantage deep unfolding, its theoretical aspects have not been revealed yet. The first half this study details analysis in deep-unfolded gradient descent (DUGD) trainable parameters are step s...

By using complex variable methods (steepest descent and residues) to asymptotically evaluate the coefftcient integrals, the numerical analysis of Hermite function series is discussed. There are striking similarities and differences with the author’s earlier work on Chebyshev polynomial methods (J. Comp. Phys. 45 (1982), 45-49) for infinite or semi-infinite domains. Like Chebyshev series, the He...

Bratu's problem, which is the nonlinear eigenvalue equation Au + 2 exp(u)= 0 with u = 0 on the walls of the unit square and 2 as the eigenvalue, is used to develop several themes on applications of Chebyshev pseudospectral methods. The first is the importance of symmett:v: because of invariance under the C 4 rotation group and parity in both x and y, one can slash the size of the basis set by a...

‎It is commonly accepted that fractional differential equations play‎ ‎an important role in the explanation of many physical phenomena‎. ‎For‎ ‎this reason we need a reliable and efficient technique for the‎ ‎solution of fractional differential equations‎. ‎This paper deals with‎ ‎the numerical solution of a class of fractional differential‎ ‎equation‎. ‎The fractional derivatives are described...

Abstract. This paper discusses a method based on Laguerre polynomials combined with a Filtered Conjugate Residual (FCR) framework to compute the product of the exponential of a matrix by a vector. The method implicitly uses an expansion of the exponential function in a series of orthogonal Laguerre polynomials, much like existing methods based on Chebyshev polynomials do. Owing to the fact that...