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...

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 ...

The idea of Gauss–Kronrod quadrature, in a germinal form, is traced back to an 1894 paper of R. Skutsch. The idea of inserting n+1 nodes into an n-point Gaussian quadrature rule and choosing them and the weights of the resulting (2n+1)-point quadrature rule in such a manner as to maximize the polynomial degree of exactness is generally attributed to A.S. Kronrod [2], [3]. This is entirely justi...

The cornerstone of nodal spectral element methods is the co-location of the interpolation and integration points, yielding a diagonal mass matrix that is efficient for time-integration methods. On quadrilateral elements Legendre-Gauss-Lobatto points are both good interpolation and integration points but on triangles analogous points have not yet been found. In this paper we use a promising set ...