Lagrange formula

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. 

The classical Lagrange formula is not efficient numerically. One can derive more efficient, but otherwise naturally equivalent interpolation formulas by introducing finite differences. The first order divided differences are defined by... 

With the above equidistant Lagrange formula, the integral approximation explicitly casts in the so-called Newton-Cdtes numerical integration formula ... 

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 ... 

Applying Lagrange s formula to the last integral reveals the remainder term... 

Since g2 s) > 0, Lagrange s formula provides support for the representation 1... 

With this choice of constraint functions and Lagrange multipliers, we can rewrite formula (6) and express the MaxEnt distribution of electrons as... 

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. 

Associate the Lagrange multiplier ji (chemical potential) with the normalization condition in Eq. (6), the set of Hermitian-Lagrange multipliers X[ with orthonormality constraints in Eq. (4), and define the auxiliary functional Q, by the formula... 

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... 

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

In the last equation X is the Lagrange multiplier and can be interpreted, analogously to the Slater transition state formula [26], as the effective electronegativity, or the negative of the chemical potential. Using the set of equations 31, the response of the charge deviation with respect to the external potential (u) measured relative to... 

Therefore, one has recourse to other interpolation polynomials associated with the names of Lagrange, Newton, Stirling, Hermite, etc. Let us give the following formulae, for equally spaced points [136]. 

To measure the rate of weight loss, readings were made of the microbalance and temperature every 2 minutes in the temperature range of 800° to 970°C. Large-scale plots of the readings were made as a function of time and the slopes were evaluated from a smoothed curve by means of Lagrange s formula. 

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... 

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