Polynomials of Higher Degree

The form of the symmetric matrix of coefficients in Eq. 3-20 for the normal equations of the quadratic is very regular, suggesting a simple expansion to higher-degree equations. The coefficient matrix for a cubic fitting equation is a 4 x 4 

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A very simple example of interpolation was provided above with the use of a linear function. However, interpolation involving polynomials of higher degree, with more points on either side of the interpolated one is relatively complicated. In effect, the matrix A is then not easily found by inspection. 

Polynomials of higher degree can be fitted in this case only if double precisian is used for the computations. 

Transfer functions involving polynomials of higher degree than two and decaying exponentials (distance-velocity lags) may be dealt with in the same manner as above, i.e. by the use of partial fractions and inverse transforms if the step response or the transient part of the sinusoidal response is required, or by the substitution method if the frequency response is desired. For example, a typical fourth-order transfer function ... 

The equations (1.207) can be solved for b0, b and b2. Extensions to polynomials of higher degree are obvious and the solution follows in the same manner. 

Using polynomials of higher degree a considerable reduction of points can be expected. 

It is sometimes necessary to obtain the content of a harmonic of the degree / in a homogeneous polynomial/j+2n of the degree I 2 n. In this case one has to annihilate the terms of higher degree than I by operating on the polynomial fi+zn with the totally symmetric operator F2 . This can be done because... 

Irving and Zwanzig that the theorem expressed by Eq. 1.11 also holds in quantum-statistical mechanics if the function f is the Wigner distribution function and if the dynamical variable a is a polynomial of a degree not higher than the second in the momenta. 

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. 

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

For this reason, the reader will find another very interesting exercise to compute the sums of the squares of the coefficients for several of the sets of coefficients, to extend these results to both higher order derivatives and higher degree polynomials, to ascertain their effect on the variance of the computed derivative for extended versions of these tables. Hopkins [8] has performed some of these computations, and has also coined the term RSSK/Norm for the 2((coefl7Normalization factor)2) in the S-G tables. Since here we pre-divide the coefficients by the normalization factors, and we are not taking the square roots, we use the simpler term SSK (sum squared coefficients) for our equivalent quantity. Hopkins in the same paper has also demonstrated how the two-point... 

The prototype application is the fitting of the np linear parameters, a, ...,a p defining a higher order polynomial of degree np-1. The generalisation of equation (4.5) reads as ... 

Since there is no direct mathematical way to write down general formulas for the roots of general polynomials of degree larger than 4, the roots of such higher degree polynomials can only be computed iteratively by numerical procedures, giving both birth and need to Numerical Analysis. 

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