Polynomial root interpolation

Fig. 2. Contour diagram of the CNDO/2 potential surface constructed by polynomial root interpolation. Energies given on the contours are in hartrees. The angle a is held fixed at its interpolated equilibrium value.

INTERPOLATION OF MULTIDIMENSIONAL POTENTIAL SURFACES BY POLYNOMIAL ROOTS... 

While it would be premature to claim that interpolation by polynomial roots has been demonstrated to be a viable procedure, it clearly shows promise. Among its attractive features are its general nature (the same functional form for all surfaces), its ability to readily accept values of derivatives as input and produce them as output, its ability to interpolate simultaneously on several branches of a multi-valued function, and its ability to produce contour... 

J. W. Downing, J. Michi, J. Cizek, and J. Paldus, Multidimensional interpolation by polynomial roots, Chem. Phys. Lett. 67 377 (1979). 

Possible solutions to this were given by Nesbet,29 30 36 who performed calculations on distances selected according to the roots of a Chebyshev polynomial, thus facilitating interpolation, and by Peyerimhoff55 and Mehler et a/.,53 who have fitted several different sets of points to different degrees of polynomial and averaged the results. 

False Position. The equation y(x) = 0 may have several roots [if y(x) is a polynomial of degree N, it will have roots, but some of them may be complex] we will seek only one root for this discussion. Assume that the function varies slowly enough that it does not change sign more than once between two adjacent data points. Let us say that (at ATj) and j2 (at a ) are of opposite sign. Then, by linear interpolation between the two points, to the first approximation (first iteration) the root is given by... 

Of practical interest in problem solving, the following situation often arises. Suppose Xy, X2,...,Xff are N roots of an iVth degree Jacobi polynomial Now we choose the (N + l)th point to be the end point of the domain (i.e., = 1). The interpolation polynomial passing through these N + 1 points is... 

Similarly, when the boundary point at x = 0 is added to the N interior interpolation points, the interior points must be chosen as roots of the following Nth degree polynomial... 

Again, our objective here is to calculate the overall reaction rate per unit volume, and from this to obtain the effectiveness factor. Therefore, the N + I interpolation points are chosen with the first N points being interior collocation points in the catalyst particle and the (N + l)th interpolation point being the boundary point (ma +i = D- The N interior points are chosen as roots of the Jacobian polynomial The optimal choice of N interior points in this... 

With the Lagrangian polynomials defined in part (b) of Problem 8.1, show that they are orthogonal to each other with respect to the weighting function 1V(x) = x (l — x) if the N+l interpolation points Xi,X2,Xi,--, Xpi,Xpi are chosen as roots of the Jacobi polynomial - 0 that is,... 

Therefore, to use the Radau quadrature with the exterior point ( = 1) included, the N interior collocation points are chosen as roots of the Jacobi polynomial with a = 1 and jS = 0. Once N -I- 1 interpolation points are chosen, the first and second order derivative matrices are known. 

The inverse polynomial interpolation method is never used in the BzzMalh library classes dedicated to root-finding. 

Description ( ) the column is separated into sections ( ) each column section is divided into smaller sub-domains (i.e. fine elements) (in) for each fine element a number of collocation points is specified, where the mass and energy balances are exclusively satisfied (iv) the collocation points are chosen as the roots of the discrete Hahn family of orthogonal polynomials and (w) Lagrange interpolation polynomials are used within each finite element to approximate the liquid- and vapor- component flow rates, the total stream flow rates and the liquid and vapor stream enthalpies. 

Another method for determining the characteristic values consists simply in computing the numerical value of the secular determinant for trial values of X. When enough calculations have been performed to yield both positive and negative values of the determinant, interpolation may be used to approximate a root. A modification of this procedure involves a method of constructing the characteristic polynomial from the values of the determinant evaluated at n - - 1 regularly spaced trial values of X. The details of this procedure are given by Hicks. ... 

Rational function models inherit the advantages of the polynomial family, despite a less simple form, and can take on an extremely wide range of shapes. They have better interpolation properties (typically smoother and less oscillatory) than polynomial models, and excellent extrapolation powers due to their asymptotic properties. Moreover, they can be used to model a complicated structure to a fairly low degree in both the numerator and denominator. On the other hand, because the properties of the rational function family are often not well understood, one might wonder which numerator and denominator degrees should be chosen. Unconstrained rational function fitting may also lead to undesired vertical asymptotes due to roots in the denominator polynomial. 

If the collocation points are chosen at equidistant intervals within the interval of integration, then the collocation method is equivalent to polynomial interpolation of equally spaced points and to the finite difference method. This is not at all surprising, as the development of interpolating polynomials and finite differences were all based on expanding the function in Taylor series (see Chap. 3). It is not necessary, however, to choose the collocation points at equidistant intervals. In fact, it is more advantageous to locate the collocation points at the roots of appropriate orthogonal polynomials, as the following discussion shows. 

We have been intrigued by the possibility that the matrix elements H (q) might be more amenable to approximation by a truncated Taylor Series. This approach leads to a new procedure, namely, interpolation by the roots of a polynomial whose coefficients are functions of q. We are now in the process of investigating the possibilities offered by this procedure and would like to report some preliminary observations and results. It should be noted that the notion that a root of a large matrix might be best approximated by a root of a small matrix did not originate with us but with Cizek and Paldus in a different context. ... 

Figure 4.1 (a) Lagrange polynomials for the support points 0, 0.5, 1 (b). Lagrange interpolation of the square root function on [0,1]. 

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