Interpolation 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. ... 

The higher order ODEs are reduced to systems of first-order equations and solved by the Runge-Kutta method. The missing condition at the initial point is estimated until the condition at the other end is satisfied. After two trials, linear interpolation is applied after three or more, Lagrange interpolation is applied. 

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)... 

An improved 0(h2) finite-difference representation of the boundary condition (8-44) results by approximating the solution in the vicinity of the boundary by the second order Lagrange interpolating polynomial passing through the points (xi,cj), (x2,cg), and (x3,cg) (equally spaced gridpoints are assumed) ... 

This expression is used as a trial-function expansion for T in much the same way as the Lagrange interpolation polynomial is in the polynomial collocation method of Villadsen and Stewart (10,1 1 There are four unknown constants associated with each node, giving a total of 4(n+l)(nH-1) unknowns in the expansion. 

The Lagrange interpolation polynomial was used to develop the spatial finite difference formulas used for the distance method of lines calculation. For example, the two point polynomial for the solids flux variable F(t,z) can be expressed by... 

The Lagrange interpolation polynomial was again used to develop the finite difference formulas. To avoid additional iterations, only upwind differences were used. The two point upwind formula for the solids stream concentration variable at any location z within the reactor for time t is given by... 

In the finite-element discretization, the sub.script i denotes the ith finite element, and the subscript j (or k) denotes the yth (or /cth) collocation point in any finite element. There are a total of N finite elements and K collocation points (i = 1, 7= 1, A. The state variable X is approximated over each finite element by Lagrange interpolation basis functions (L/.(a)) as ... 

The scheme illustrated here has its v-vertices at the original vertices, and each of its e-vertices at the place defined by a parametric cubic Lagrange interpolant through four points. The weights for an e-vertex are -1/16, 9/16, 9/16 and -1/16, and the scheme is therefore known as the four-point scheme. 

The end-conditions described above cover two distinct cases, those of interpolating schemes, which are likened to Lagrange interpolation, and those of B-splines, likened to the Bezier end-conditions. The schemes which interpolate when the data lies on a cubic or higher polynomial do not really fit either of these cases. They are almost interpolating (when the data is really smooth) but not quite. Somebody needs to play with these schemes to find out how they currently misbehave at the ends and what kinds of control are required to make them do what the curve designer wants. 

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... 

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