Interpolation polynomial

The Savitzky-Golay algorithm could readily be adapted for polynomial interpolation. The computations are virtually identical to smoothing. In smoothing, a polynomial is fitted to a range of (x,y)-data pairs arranged around the x-value that needs to be smoothed. For polynomial smoothing, the polynomial is evaluated for a set number of data points around the desired x-value and the computed y-value at that x is the interpolated value. 

Because we can evaluate flie integral of a polynomial analytically, integration techniques typically employ polynomial approximations of the integrand. The general problem of polynomial interpolation is as follows. Let us say that we have sampled some function /(x) at the W -I- 1 support points xo, xi, X2,. . . , x to obtain the function values yb. fi, , /n, fj = /(JCj). We wish to construct a polynomial of degree N 

The coefficients of the polynomial therefore must satisfy the hnear system 

From the drawbacks of solving (4.22) by elimination, it would be nice to have a direct way to construct p(x) from the function values. In fact, it is possible to express p(x) as the linear combination 

One disadvantage of Lagrange interpolation is that if we add another support pointxjv+i, we have to recompute all of the Lagrange polynomials again from scratch. Newton interpolation avoids this difficulty, and expresses the interpolating polynomial as 

We define divided differences of higher orders recursively 

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Evaluation of the integral in equation 2.102 may be performed graphically or by polynomial interpolation. 

Selected entries from Methods in Enzymology [vol, page(s)] Aspartate transcarbamylase [assembly effects, 259, 624-625 buffer sensitivity, 259, 625 ligation effects, 259, 625 mutation effects, 259, 626] baseline estimation [effect on parameters, 240, 542-543, 548-549 importance of, 240, 540 polynomial interpolation, 240, 540-541,549, 567 proportional method for, 240, 541-542, 547-548, 567] baseline subtraction and partial molar heat capacity, 259, 151 changes in solvent accessible surface areas, 240, 519-520, 528 characterization of membrane phase transition, 250,... 

Global polynomial interpolation is restricted to small samples of fairly good data. If there are many grid points, the resulting higher order polynomial tends to oscillate wildly between the tabulated values as shown in Fig. 4.3. 

Spline interpolation is a global method, and this property is not necessarily advantageous for large samples. Several authors proposed interpolating formulas that are "stiffer" than the local polynomial interpolation, thereby reminding spline interpolation, but are local in nature. The cubic polynomial of the form... 

An intermediate approach between table lookups and polynomial approximations is to use interpolated table lookups. Typically, linear interpolation is used, but higher order polynomial interpolation can also be considered [Laakso et ah, 1996],... 

Finally sine-wave analysis/synthesis is also suitable for extrapolation of missing data [Maher, 1994], Situations occur, for example, where a data segment is missing from a digital data stream. Sine-wave analysis/synthesis can be used to extrapolate the data across the gap. In particular, the measured sine-wave amplitude and phase are interpolated using the linear amplitude and cubic phase polynomial interpolators, respectively. In this way, the slow variation of the amplitude and phase function are exploited, in contrast with rapid waveform oscillations. 

Block SS gives the user the possibility of choosing between these algorithms. When statistical data on snow layer thickness exist, function g(t,

, A) can be reconstructed for (ip, A) e Q by means of the approximation algorithm at the time of polynomial interpolation in space (Krapivin, 2000a, b Nitu et al., 2000b). 

According to eqns. (7.6), (7.7) and (7.12), we can generalize for an arbitrary polynomial interpolation of order n — 1 (n number of points), by saying that the function u (x) is approximately equal to a linear combination of the known values Ui multiplied by interpolating functions, Niy... 

First, second and 5th order polynomial interpolation for the specific heat capacity of a semi-crystalline thermoplastic (PA6). When performing a heat transfer simulation (heating or cooling) for a thermoplastic, the complete course of the specific heat capacity as a function of temperature is needed. A common way to do this... 

A first, second and 5th order polynomial interpolation will be performed in order to obtain the specific heat for three different temperatures. 

The results for each interpolation are shown in Table 7.2 and Fig. 7.4 For 190°C and 250°C, the specific heat is locally a smooth function, therefore the three different schemes give very accurate results. The problem is for the 220°C, this temperature is right in the middle of the melting. The linear interpolation can use only two points, whereas the 2nd and 5th interpolation use more points following the behavior closer. For this particular case the 5th order interpolation is the one that mimics best the complete curve obtained from the equation. However, as already mentioned, we must be careful because a polynomial interpolation of a very high order can lead to oscillations [19]. 

In the previous discussion of interpolation we stated the problem for known values of a function at n points within a domain. In some cases, values of both, the function / (x) and its derivative / (x) are available. Hence, it is required to find an interpolation formula which utilizes 2n data points. One way to solve this problem is to use polynomial interpolation since for 2n known values, a 2n — 1 order polynomial must exist. However, we must find this polynomial, y (x), in a way that it has the same value and the same derivative as / (x) at all n points. One function that fulfills this is [10]... 

For many applications, interpolations of functions of two or three variables defined in two-and three-dimensional domains must be considered. For example, global interpolations in two- and three-dimensional systems are analogous to polynomial interpolation in onedimensional systems however, global interpolants do not exist in 2- and 3D. This is a big drawback in numerical analysis because a basic tool available for one variable is not available for multivariable approximation [21], The best developed aspect of this theory is that of piecewise polynomial approximation, associated with finite element and finite volume approximations for partial differential equations, which will be examined in detail in Chapters 9 and 10. 

Pascal s triangle is often used to generate piecewise polynomial interpolations for various domains (triangles and rectangles in 2D and tetrahedrons, cubes and shells in 3D). In fact, most of the families of elements that are commonly used in finite elements, finite volumes and boundary elements, come from expansions of this triangle (more detail can be found in [67, 68]). 

The major drawback with such methods is the requirement of the Cooley -Cashion method of a very large number of points on the curve, to obtain the necessary accuracy. Thus some form of interpolation is inevitable. The calculation of Kolos and Wolniewicz gave values of the derivative as well Wolniewicz was therefore able to use an interpolation method utilizing these. However, comparison with a simple polynomial interpolation showed that the errors introduced by the latter were very small. 

The curves in Figs. 2, 3, and 4 were obtained by a quadratic polynomial interpolation. 

The use of online data together with steady-state models, as in Real Time Optimization applications, requires the identification of steady-state regimes in a process and the detection of the presence of gross errors. In this paper a method is proposed which makes use of polynomial interpolation on time windows. The method is simple because the parameters in which it is based are easy to tune as they are rather intuitive. In order to assess the performance of the method, a comparison based on Monte-Carlo simulations was performed, comparing the proposed method to three methods extracted from literature, for different noise to signal ratios and autocorrelations. 

Figure 2. Performance of Polynomial Interpolation Test for window size=51 and first level of noise...

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