DVR3D calculates rotationless (J = 0) vibrational energy levels and wavefunctions for triatomic systems using scattering (Jacobi) coordinates, or optionally unsymmetrised Radau coordinates, for a given potential energy surface. The program uses a discrete variable representation (DVR) based on Gauss—Legendre and Gauss—Laguerre quadrature for all 3 internal coordinates and thus yields a fully po...

In the present paper, we develop a modified pseudospectral scheme for solving an optimal control problem which is governed by a switched dynamical system. Many real-world processes such as chemical processes, automotive systems and manufacturing processes can be modeled as such systems. For this purpose, we replace the problem with an alternative optimal control problem in which the switching t...

In this paper, we introduce an efficient Legendre-Gauss collocation method for solving nonlinear delay differential equations with variable delay. We analyze the convergence of the single-step and multi-domain versions of the proposed method, and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner. We also make numerical comparison with other ...

Abstract In this work, three different integration techniques, which are the numerical, semi-analytical and exact integration techniques are briefly reviewed. Numerical integrations are carried out using three different Quadrature rules, which are the Classical Gauss Quadrature, Gauss Legendre and Generalized Gaussian Quadrature. Line integral method is used to perform semi-analytical integrati...

In this article we study the implementation of the Nonlinear Galerkin method as a multiresolution method when a two-level Chebyshev-collocation discretization is used. A fine grid containing an even number of Gauss-Lobatto points is considered. The grid is decomposed into two coarse grids based on half as many Gauss-Radau points. This splitting suggests a decomposition of the unknowns in low mo...

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss-Lobatto-Legendre-Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a useroriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach al...

Newton, in an unauthorized textbook, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method — that Newton did not want to publish, that Euler did not recommend, that Legendre called “ordinary,” and that Gauss called “common” — is now named after Gauss: “Gaussian” elimination. (One suspects, he would not be amused.) Gauss’s...

We investigate candidates of finite random variables for c-secure random fingerprinting codes, in viewpoints of both code lengths and required memories. We determine, under a natural assumption, the random variables with the minimal number of outputs (i.e. optimal in a viewpoint of memory) among all candidates, by revealing their deep relation with theory of Gauss-Legendre quadrature (a famous ...

Consider the problem of deciding whether a trajectory pair (u∗(t), x∗(t)), t ∈ [0, T ] of a generally nonlinear system ẋ = F (x, u), x ∈ Mn is a time-optimal solution connecting the endpoints x(0) and x(T ), or whether the system is locally controllable about this trajectory. The classical approach analyzes the endpoint map u 7→ x(T, u) (for fixed T and x(0)) and determine whether or not it is ...

Boundary integral equations and Nyström discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weaklysingular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarith...