Interpolation formula Lagrange

Lagrange Interpolation Formulas A global polynomial is defined over the entire region of space... 

Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula (see Refs. 143 and 18.5) may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange s formulas must be used. By differentiating three- and five-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points ... 

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

Method of Lines. The method of lines is used to solve partial differential equations (12) and was already used by Cooper (I3.) and Tsuruoka (l4) in the derivation of state space models for the dynamics of particulate processes. In the method, the size-axis is discretized and the partial differential a[G(L,t)n(L,t)]/3L is approximated by a finite difference. Several choices are possible for the accuracy of the finite difference. The method will be demonstrated for a fourth-order central difference and an equidistant grid. For non-equidistant grids, the Lagrange interpolation formulaes as described in (15 ) are to be used. 

The above equation can be written in terms of Lagrange interpolation formula (Chapter 7) Te = ATiTi + N2T2 (9.8)... 

Choosing the generating function UNg as the polynomial, UNf (q) = (q - qx)(q - 2) " (q qi) (q q ) leads to the well-known Lagrange interpolation formula. Figure 4 shows the expansion function g (q) which is based on the zeros of the Cheby-chev orthogonal polynomial of order Ng. Another choice appropriate for evenly distributed sampling points is based on the global function NNg(q) = sin(2 Tr /A ). It is closely related to the Fourier method described in the next section. 

The Lagrange interpolation formula expresses the value y(x) in terms of polynomials (up to degree iV - 1 for iV pairs of data). The general equation is... 

Example 2.2. Irrational transfer functions have been approximated in the literature using truncated infinite partial fraction expansions (Partington et al., 1988) and the Lagrange interpolation formula (Olivier, 1992). Here, we will illustrate that this class of linear systems can be efficiently approximated by a Laguerre model based on the minimization of the fi-equency domain loss function in Equation (2.34). We will consider the following system (Partington et al, 1988)... 

Olivier, P. D. (1992), Approximating irrational transfer functions using Lagrange interpolation formula , lEE Proceedings-D 139, 9-12. 

From Eq. (3.139) it can be concluded that the second derivative of the interpolating polynomial at any point in the interval [Xf. j, j ,] can be given by the first-order Lagrange interpolation formula... 

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