Newton’s interpolation

This is Newton s interpolation formula (Newton s Principia, 3, lem. 5, 1687) employed in finding or interpolating one or more terms when n particular values of the function are known. Let us write y0 in place of yx for the first term, then... 

For practical reasons it is convenient to use Newton s interpolative polynomial. It can be recommended for the merit, that when adding new interpolation nodes to the previous form of the polynomial a new term is added taking shape of higher degree polynomial. 

As mentioned earlier in this chapter, interpolation polynomials can be defined in different ways. Especially the definition based on finite differences, like in Newton s interpolation method, permits in an efficient way to vary the order of an interpolation polynomial by adding or taking away additional interpolation points. The Adams multistep code DE/STEP by Shampine and Gordon [SG75] and the BDF code DASSL by Petzold [BCP89] are based on a modification of Newton interpolation polynomials. 

If the accuracy afforded by a linear approximation is inadequate, a generally more accurate result may be based upon the assumption that fix) may be approximated by a polynomial of degree 2 or higher over certain ranges. This assumption leads to Newton s fundamental interpolation formula with divided differences... 

Determine the relative rates of convergence for (1) Newton s method, (2) a finite difference Newton method, (3) quasi-Newton method, (4) quadratic interpolation, and (5) cubic interpolation, in minimizing the following functions ... 

Throughout the Newton s law range, the separation ring continues to move forward as Re increases. At Re = 5000, separation moves in front of the equator towards a limit of 81 -83"" (A3, FI, M8, R4). Direct observations of the separation ring are scant for 800 < Re < 6 x lOA Several workers [e.g., B14, LIO, LI3, N3, Wl) have determined the point of minimum heat or mass transfer in this range, but, as discussed below, this occurs aft of separation. Seeley et ai (S7) report some flow visualization results, but they found separation closer to the rear than observed by other workers, perhaps due to wall effects. As shown in Fig. 5.6, a realistic interpolation is provided by... 

A comparison of the difference table, page 309, with Newton s formula will show that the interpolated term yx is built up by taking the algebraic sum of certain proportions of each of the terms employed. The greatest proportions are taken from those terms nearest the interpolated term. Consequently we should expect more accurate results when the interpolated term occupies a central position among the terms employed rather than if it were nearer the beginning or end of the given series of terms. 

Let us take the series y0, yv y2, ys, y4 so that the term, yx, to be interpolated lies nearest to the central term y2. Hence, with our former notation, Newton s expression assumes the form... 

Then, to find 74 = y( o + 4//), we substitute Newton s forward interpolation formula... 

There are two types of interpolating polynomials that can be used. These are open formulas which are used to predict the Xj+i value based on known information up to Xj, and closed formulas which are used to correct the Xi+i. In both cases backward difference interpolating polynomials are used since we are using previous time information to determine current or future time behavior. In order to develop the backward difference formulas, we use Newton s fundamental formula for interpolating polynomials,... 

TFM. And the particle phase motion is solved by tracking discrete parcels, each representing a number of particles with the same properties and following the Newton s law of motion. In the MP-PIC method, the coUisional interaction between particles is replaced by using the normal stress of solids (Snider, 2001), which is calculated on the grid points for gas phase and interpolated to the positions of parcels. The gas continuity equation is the same as Eq. (16), whereas the gas momentum equation reads... 

The determination of the switching point can be done, by using Newton s method, the secant rule or by inverse interpolation. We prefer derivative-free methods, because the numerical computation of the time derivative of q involves an extra evaluation of the right hand side function due to... 

The QM force/jM is computed by QM calculations for the atoms in the PS and the buffer zone, and the MM forceby force field calculations for the atoms in the buffer zone and SS. Note that only the forces acting on the buffer groups are interpolated, whereas the forces on the PS and SS are not treated. Clearly, such a treatment violates the Newton s Third Law of Motion and does not conserve momentum. 

A one-dimensional minimization process, component of many nonlinear optimization methods, performed via quadratic or cubic interpolation in combination with bracketing strategies. Newton s method... 

In this section, we will develop two interpolation methods for equally spaced data (I) the Gregory-Newton formulas, which are based on forward and backward differences, and (2) Stirling s interpolation formula, based on central differences. 

Stirling s interpolation formula is based on central differences. Its derivation is similar to that of the Gregory-Newton formulas and can be arrived at by using either the symbolic operator relations or the Taylor series expansion of the function. We will use the latter and expand the function fix + nh) in a Taylor series around jc ... 

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 ... 

When solving Eq. (5.8) the level shift parameter p must be chosen such that s(p) is the global minimizer on the boundary. From the discussion in Sec. Ill we know that p must be smaller than the lowest Hessian eigenvalue. Also, p must be negative since otherwise the step becomes longer than the Newton step. The exact value of p may be found by bisection or interpolation. 

These values give the best interpolation at t = 575°F rather than the Newton-Gregory s forward or backward interpolation formulae. 

THE PROGRAM USES THE NEWTON-GREGORY FORWARD AND BACKWARD INTERPOLATIONS AND STIRLING S CENTRAL DIFFERENCE METHOD. 

---