Lagrange Interpolation Polynomials

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... 

Let us first discuss in more detail Lagrange interpolation polynomials, labelled by the index a. In the following discussion we assume that the whole space of interest [Xa, xt,]... 

In addition to the equidistant distribution of the nodal points the Lagrange interpolation polynomial are defined via the wave-function not taking into account its derivative. Hermitian interpolation polynomials are in addition defined by the assumption that value and derivation of the wave-function are correct at the nodal points. Hence we get the following ansatz... 

The Lagrange interpolation polynomial is a useful building block. There are (N+l) building blocks, which are Nth degree polynomials. The building blocks are given as... 

The construction of the Lagrange interpolation polynomial proceeds as follows. First, the V -H 1 interpolation points are chosen, then the V -l- 1 building blocks /,(x) can be constructed (Eq. 8.90). If the functional values of y at those JV -I- 1 points are known, the interpolation polynomial is given in Eq. 8.89. Hence, the value of y at any point including the interpolation points, say x, is given by... 

In most practical problems, only the first two derivatives are required, so these are presented here. However, if higher derivatives are needed, the Lagrange interpolation polynomial can be differentiated further. 

Now, interpreting the last expression as a Lagrange interpolation polynomial the individual projectors can be approximated as (see Appendix A.4) ... 

Meyer, R., Cai, B. Perron, F. 2008. Adaptive rejection Metropolis sampling using Lagrange interpolation polynomials of degree 2. Computational Statistics and Data Analysis, 52(7) 3408-3423. 

Description ( ) the column is separated into sections ( ) each column section is divided into smaller sub-domains (i.e. fine elements) (in) for each fine element a number of collocation points is specified, where the mass and energy balances are exclusively satisfied (iv) the collocation points are chosen as the roots of the discrete Hahn family of orthogonal polynomials and (w) Lagrange interpolation polynomials are used within each finite element to approximate the liquid- and vapor- component flow rates, the total stream flow rates and the liquid and vapor stream enthalpies. 

The Lagrange interpolation polynomial is an exact representation if the value of the polynomial, r, (x), is known for at least i + 1 points. It is possible to rewrite the interpolation, polynomial more simply as... 

The form of these equations is the so called second-order Lagrange interpolation polynomial. Each equation is of second order as seen by inspection and reduces to the known data points at the required values of x. In fact Lagrange showed that an interpolation polynomial of any order can be written in the compact form ... 

To evaluate the integral over the remainder in (9.6.17). we note fliat Rr(Jf) may be written as a Lagrange interpolating polynomial... 

---