Interpolation and extrapolation

This chapter deals with two related problems occurring frequently in the physical sciences: first, the problem of estimating the value of a function from a limited number of data points; and second, the problem of calculating its value from a series approximation. Numerical methods for interpolating and extrapolating data are presented. The famous Lagrange interpolating polynomial is introduced and applied to one-dimensional and multidimensional problems. Cubic spline interpolation is introduced and an implementation in terms of Eigen classes is given. Several techniques for improving the convergence of Taylor series are discussed, including Shank’s transformation, Richardson extrapolation, and the use of Padé approximants. Conversion between representations with the quotient-difference algorithm is discussed. The exercises explore public transportation, human vision, the wine market, and SU(2) lattice gauge theory, among other topics.

ON THE CONVERGENCE OF THE MFS–MPS SCHEME FOR 1D POISSON'S EQUATION

The method of fundamental solutions (MFS) has been an effective meshless method for solving homogeneous partial differential equations. Coupled with radial basis functions (RBFs), the MFS has been extended to solve the inhomogeneous problems through the evaluation of the approximate particular solution and homogeneous solution. In this paper, we prove the the approximate solution of the above numerical process for solving 1D Poisson's equation converges in the sense of Lagrange interpolating polynomial using the result of Driscoll and Fornberg [2002].

Shamir Threshold Based Encryption

This paper Shamir threshold scheme based on the protection of private keys, by constructing a Lagrange interpolating polynomial to achieve in the real environment using the key shared information systems, computation and communication in the case of less , the program can prevent the system key is lost, damaged, and from the enemy's attack, reduce the responsibility of the key holder, but also can reduce the success rate of an adversary to decipher the key. An example is the feasibility of the program.

New error inequalities for the Lagrange interpolating polynomial

A new representation of remainder of Lagrange interpolating polynomial is derived. Error inequalities of Ostrowski-Grüss type for the Lagrange interpolating polynomial are established. Some similar inequalities are also obtained.