Polynomials Lagrange

In this section, we will develop two interpolation methods for unequally spaced data the Lagrange polynomials and spline interpolation. 

Consider a set of unequally spaced base points, such as those shown in Fig. 3.3a. Define the polynomial 

In order for the interpolating polynomial to fit the function exactly at all the base points, each particular weighting polynomial pf.(x) must be chosen so that it has the value of unity when X = a , and the value of zero at all other base points, that is, 

The interpolating polynomial P x) has a remainder term, which can be obtained from Eq. (3.6)  

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Interpolation with nonequally spaced data may be accomplished by the use of Lagrange Polynomials, defined as a set of n degree polynomials such that each one, P.(x) (j = 0, 1,. . n), passes through zero at each of the data points except... 

Recall that Lagrange polynomials L (x) are polynomials of degree n - 1, which are equal to the Kronecker delta at each of the points, and the function [Li (x)]2 is a polynomial of degree 2n - 2, which is also equal to the Kronecker delta at each of the points and whose derivative vanishes at Xj when i j. Therefore, because hi (x) and hi (x) are polynomials of degree 2n — 1, they can be written as... 

In order to use Hermite interpolation, we must first chosse the order for the interpolation of hi(x) and ht(x). For simplicity, let s use a first order interpolation, n = 2, for the Lagrange polynomials involved in these two terms. Using Hermite interpolation formula (eqn. (7.31)) and eqns. (7.32) and (7.33) we obtain... 

Cuthrell and Biegler (1989) considered the orthogonal collocation method which is described below. Two Lagrange polynomials one for the state variable (x) and one for the control variable (u) can be written as ... 

For higher degrees there can be imposed zero discontinuities of the relevant derivative at more internal knots, thus forcing the end of the curve to be controlled in a way which is more and more like a Lagrange polynomial. 

In order to construct higher-order approximations one must use information at more points. The group of multistep methods, called the Adams methods, are derived by fitting a polynomial to the derivatives at a number of points in time. If a Lagrange polynomial is fit to /(t TO, V )i. .., f tn, explicit method of order m- -1. Methods of this tirpe are called Adams-Bashforth methods. It is noted that only the lower order methods are used for the purpose of solving partial differential equations. The first order method coincides with the explicit Euler method, the second order method is defined by ... 

The Lagrange polynomial has a structure that does not allow us to calculate the first and second derivatives with any great ease. 

If the function y(jc) is known, we simply choose discrete positions x (for example, 2, 3, or 4 points) within the domain of integration (a approximating polynomial to pass through these points. We can then perform the integration of the approximating polynomial. If the discrete points are unequally spaced, the Lagrange polynomial developed in Chapter 8 can be used to fit the data, white if the points are equally spaced the Newton... 

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 ... 

All integrations may be performed symbolically. To avoid the implications of the effect, which might appear in the context of the description of the Timoshenko beam, a reduced integration scheme is applied, see for example Hughes [101]. Alternatively, one may start off from four nodes per element for the transverse displacements and rotations using cubic Lagrange polynomials for the interpolation and then reduce the degrees of freedom by means of a static condensation, see Knothe and Wessels [113]. 

Like in the explicit ODE case (see Sec. 4.3.1) the choice of collocation points defines an 5-stage Runge-Kutta method. We describe u and i) in terms of Lagrange polynomials and obtain by integrating... 

In order to construct higher-order approximations one must use information at more points. The group of multistep methods, called ht Adams methods, are derived by fitting a polynomial to the derivatives at a number of points in time. If a Lagrange polynomial is fit to/(f OT, V " "),/(fn-m-i-i, V ),and theresultis... 

Baroth J, Bresslotetle P, Chauviere C, Fogli M (2007) An efficient SEE method using Lagrange polynomials application to nonlinear mechanical problems with uncertain parameters. Comput Methods Appl Mech Eng 196 4419-4429... 

At the beginning, it checks the inputs and sets the order of interpolation if nece.ssary. If not introduced to the function, the interpolation is done by the first-order Lagrange polynomial (linear interpolation). 

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