Polynomials Newton

In 1687, three years after describing a root finder for a polynomial, Newton described in Principia Mathematica an application of his procedure to a nonpolynomial equation. That equation originated from the problem of solving Kepler s equation determining the position of a planet moving in an elliptical orbit around the sun, given the time elapsed since it was nearest the sun. Newton s procedure was nonetheless purely algebraic and not even iterative, as the solution process at each step was not the same. 

Carry out the first two iterations of the Newton-Raphson solution of the polynomial Eq. (1-10). 

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... 

Equation 5-197 is a polynomial of the third degree, and by employing either a numerieal method or a spreadsheet paekage sueh as Mierosoft Exeel, the roots (C ) of the equation ean be determined. A developed eomputer program PROGS 1 using the Newton-Raphson method to determine was used. The Newton-Raphson method for the roots of Equation 5-197 is... 

Equation 5-247 is a polynomial, and the roots (C ) are determined using a numerical method such as the Newton-Raphson as illustrated in Appendix D. For second order kinetics, the positive sign (-r) of the quadratic Equation 5-245 is chosen. Otherwise, the other root would give a negative concentration, which is physically impossible. This would also be the case for the nth order kinetics in an isothermal reactor. Therefore, for the nth order reaction in an isothermal CFSTR, there is only one physically significant root (0 < C < C g) for a given residence time f. 

The most successful strategy for approximating the Liouville-von Neumann propagator is to interpolate the operator with polynomial operators. To this end, Newton and Faber polynomials have been suggested to globally approximate the propagator,126,127,225,232-234 as in Eq. [95]. For short-time propagation, short-iterative Arnoldi,235 dual Lanczos,236 and Chebyshev... 

Computational Algorithm for Green s Functions Fourier Transform of the Newton Polynomial Expansion. 

NEWTON-RAPHSON ITERATIVE TECHNIQUE, THE FINAL ITERATIONS ON EACH ROOT ARE PER,FORMED USING THE ORIGINAL FOLYNOMIAL RATHER THAN THE RF-DUCEU POLYNOMIAL TO AVOID ACCUMULATED ERRORS IN THE REDUCED POLYNOMIAL. 

M60 Newton interpolations computation of polynomial coefficients and interpolated values 6000 6054... 

Newton iteration starts with an approximation xo of a root x of / and goes along the tangent line to the curve at (xo, /(xo)) until it intersects the x axis. This intersection is labeled xi. ft is an improved iterative approximation for the actual root, and the process continues leading from xi to X2 via the tangent to / at xi, etc. until the difference between successive iterates becomes negligible, see Figure 1.1 for our trial polynomial equation p x) = x3 — 2x2 +4 = 0 and the start xo = —2. 

The following experiments validate our assessment of troubles with Newton or bisection root finders for multiple roots. First we use the bisection method based MATLAB root finder f zero, followed by a simple Newton iteration code, both times using the chosen polynomial p x) of degree 9 in its extended form (1.6). 

Output Newton iterate and number of iterations used 7. Computes the zero of the polynomial... 

In order to call newtonpoly (start, n) implicitly on a command line with success, the two implicit inputs start and n must have been declared previously and must be available in the current workspace. Of course, one can also call newtonpoly9 explicitly by entering newtonpoly (21,42), for example, on the command line if one wants to see the list of 41 Newton iterates for the same polynomial-root problem, starting from start = xo = 21. 

Looking at the three shallow intersections of the horizontal axis with the graph of / in Figure 3.4, we are reminded of the problems encountered in Chapter 1 on p. 30 and 31 with both the bisection and the Newton root finder for polynomials with repeated roots. The common wisdom is that the shallower these intersections become, the worse the roots will be computed by standard root-finding methods (see the exercises below), and multiple roots will easily be missed. 

Berna, T. J. Westerberg, A. W., "Polynomial, Chao-Seader and Newton Raphson - The Use of Partially Ordered Pivot Sequences" DEC 06-l-79 Dept, of Chem. Eng., Carnegie-Mellon University, Pittsburgh, Penn. 15213 (January 1979). 

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]. 

Explicit combinatorial expressions are known for the first few coefficients of (f>(B, x) [2, 21, 33, 39, 40]. We will skip these results because in the subsequent paragraph the spectral moments are discussed at due length. Using the Newton identities [24] it is easy to compute the coefficients of the characteristic polynomial from spectral moments and vice versa. 

W. Huisinga, et al, Faber and Newton polynomial integrators for open-system density matrix propagation, /. Chem. Phys. 110 (12) (1999) 5538-5547. 

NewtRaph.xls illustrates the use of the Newton-Raphson method to find the roots of a polynomial. 

Equation 9.51 is a polynomial that cannot be solved exphcitly, but its roots can be evaluated easily by a numerical technique such as Newton s method. The characteristic parameters, cu, employed by Rhee and Amimdson [10] which were discussed earlier in this chapter are analogous to the reciprocal of the hi values. 

An alternative closed Newton-Cotes quadrature formula of second order can be obtained by a polynomial of degree 1 which passes through the end points. This quadrature formula is called the trapezoid rule. In 2D this surface integral approximation requires the integrand values at the GCV corners. 

When the data points are available at equal intervals of the independent variable, we can use the Newton-Gregory forward polynomial. 

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. 

THIS PROGRAM PERFORMS THE NEWTON-GREGORY FORWARD INTERPOLATION OF POLYNOMIALS... 

When we have too few points to justify linearizing the function between adjacent points (as the trapezoidal integration does) we can use an algorithm based on a higher-order polynomial, which thereby can more faithfully represent the curvature of the function between adjacent measurement points. The Newton-Cotes method does just that for equidistant points, and is a moving polynomial method with fixed coefficients, just as the Savitzky-Golay method used for smoothing and differentation discussed in sections 8.5 and 8.8. For example, the formula for the area under the curve between x, and xn, is... 

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