Lagrange interpolation function

No readily useful analytical solutions are available for the system (7) and we resort to an expansion of the amplitude functions in a basis set. The discrete variable representation is a convenient means and we chose to employ a localized basis associated with the Lobatto quadrature rule. It is convenient then to choose units and displacement such that the interval [r,p] equals the standard one, [-1,1]. A basis of n+1 Lagrange interpolation functions is defined from the Legendre polynomial Pniq) as follows... 

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. 

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 ... 

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. 

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 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... 

A.4 LAGRANGE INTERPOLATION AND NUMERICAL INTEGRATION APPLICATION ON ERROR FUNCTION... 

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. 

By abandoning the warping effect, the beam torsion problem may also be treated with linear Lagrange polynomials. For the interpolation functions of Eqs. (9.15) and (9.16), the element coordinate Xi is introduced with its origin at the center of the element and the element length /j. Thus, the continuous blade coordinate x can be expressed with the aid of the distance L to the element coordinate origin ... 

Method of Solution The Lagrange interpolation is done based on Eqs. (3.132) and (3.136). The order of interpolation is an input to the function. The cubic spline interpolation is done based on Eq. (3.143). The values of the second derivatives at base points, assuming a natural spline, are calculated from Eq. (3.147). 

Program Description The general MATLAB function Lagrange.m performs the nth-order Lagrange interpolation. This function consists of the following three parts ... 

YI = Lagrange(X,Y,XI,N) applies the Nth-order Lagrange interpolation to find YI, the values of the underlying function Y at the points in the vector XI. The vector X specifies the points at which the data Y is given. 

Figure 4.1 (a) Lagrange polynomials for the support points 0, 0.5, 1 (b). Lagrange interpolation of the square root function on [0,1]. 

Extrapolation is required if f(x) is known on the interval [a,b], but values of f(x) are needed for x values not in the interval. In addition to the uncertainties of interpolation, extrapolation is further complicated since the function is fixed only on one side. Gregory-Newton and Lagrange formulas may be used for extrapolation (depending on the spacing of the data points), but all results should be viewed with extreme skepticism. 

H - wi) = (xo xi) - H - wi) ( 2 i)-When the intervals between the two terms are large, or the differences between the various members of the series decrease rapidly, simple proportion cannot be used with confidence. To take away any arbitrary choice in the determination of the intermediate values, it is commonly assumed that the function can be expressed by a limited series of powers of one of the variables. Thus we have the interpolation formulas of Newton, Bessel, Stirling, Lagrange, and Gauss. 

To calculate y(x) at any point x, Eq. 11.9 uses three points Xo,x,X2 with X() < X < X2, and is called the Lagrange three-point interpolation formula. The three-point formula amounts to a parabolic representation of the function y(x) between any three points. 

Example 11.3 Calculate the value of the function /(x) at x = 11.8 from the table below using the Lagrange two- and three-point interpolation formulas. The data are plotted in Fig. 11.2. 

Up to now the basis functions Ni x) are still arbitrary and not restricted to a finite element approach. In the finite element frame a suitable approximation is given by Lagrange- or Hermite-interpolation polynomials. 

---