Interpolating polynomial

Let us assume that values of functions f x) are known at a set of ( + I) values of the independent variables x  

These values are called the base points of the function. They are shown graphically in Fig. 3.3a. 

The genera] objective in developing interpolating polynomials is to choose a polynomial of the form 

For the given set of (n + 1) known base points, the polynomial must satisfy the equation 

Substitution of the known values of (x, f x,)) in Eq. (3.112) yields a set of ( + 1) simultaneous linear algebraic equations whose unknowns are the coefficients, . a of the polynomial equation. The. solution of this set of linear algebraic equations may be obtained using one of the algorithms discussed in Chap. 2. However, this solution results in an ill-conditioned linear system therefore, other methods have been favored in the development of interpolating polynomials. 

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Neville s algorithm constructs the same unique interpolating polynomial and improves the straightforward Lagrange implementation by the addition of an error estimate. 

J.M. Boone, T.R. FeweU and R.J. Jennings, Molybdenum, rhodium and tungsten anode spectral models using interpolated polynomials with application to mammography, Med. Phys. 24(12) (1997) 1863-1874. 

One may require additional solution points for purposes of presentation or analysis. In many cases, interpolating polynomials prove to be useful. Thus, one might approach our example problem using products of quartic equations of the general foxmX f (x) = + bjX + CjX + djX + ej and similarly for An appropri-... 

The module selects the four nearest neighbors of X and evaluates the cubic interpolating polynomial. 

The familiar formulas of numerical differentiation are the derivatives of local interpolating polynomials. All such formulas give bad estimates if there are errors in the data. To illustrate this point consider the case of linear... 

For small samples we can integrate the global interpolating polynomial. For larger samples the trapezium rule... 

In other words, the exact form of the interpolating polynomials varied from point to point. dyldt tt and dzyjdt2 ti for t5< tt< t499 were then calculated by evaluating the first and second derivatives of the appropriate interpolating polynomial, y ((t). 

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) ... 

For n + 1 experimental points th, (i = 1, 2,. . . , n + 1) given, it can be proved that there exists a single polynomial of order n interpolating the n + 1 points. The uniqueness of the interpolation polynomial does not imply any specific form. If a usual polynomial... 

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]. 

Fig. 1. An example of a possible behavior of an interpolation polynomial of high degree, as compared with the exact function. This is what we call an instability on the sides of the range of interpolation...

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... 

Savitzky Golay (1964) developed the algorithm for a filter to treat data measured in noisy processes, as spectroscopy. An experimental measurement series is filtered first by choosing a window size, n (which must be an odd number). Each window is interpolated using a polynomial of degree p, with p < n. Information obtained from the interpolated polynomial is less noisy. Thus, the first derivative of each polynomial at the central points is calculated and the value is used as a statistic for assessing the stationarity of the point. The parameters of the filter are the window size, n, and the polynomial degree, p. 

Finite difference methods (FDM) are directly derived from the space time grid. Focusing on the space domain (horizontal lines in Fig. 6.6), the spatial differentials are replaced by discrete difference quotients based on interpolation polynomials. Using the dimensionless formulation of the balance equations (Eq. 6.107), the convection term at a grid point j (Fig. 6.6) can be approximated by assuming, for example, the linear polynomial. 

If data at tn+i is included in the interpolation polynomial, implicit methods, known as Adams-Moulton methods, are obtained. The first order method coincides with the implicit Euler method, and the second order method coincides with the trapezoid rule. The third order method is written as ... 

The forward and the backward difference approximations give the derivatives of the Newton interpolation polynomial at the edges of the interpolation range. However, the central difference is derived from the Newton interpolation at the center of the range of interpolation. Accuracy of an interpolation formula based on equispaced points is highest at the center of the interpolation range. Therefore, the central difference interpolation formula is always more accurate than the forward or backward difference approximations. 

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