13 - Numerical Analysis

Abstract

All measurements are subject to errors. Therefore, it is essential in any scientific endeavor to analyze the results of experiments and to estimate the reliability of the data obtained. In general, experimental errors can be classified as systematic, personal, and random. If systematic errors can be traced, and perhaps eliminated, and if the personal errors can be minimized, the remaining random errors can be analyzed by statistical methods. The chapter summarizes this procedure, elaborates the use of the Poisson distribution and Gaussian distribution to analyze the errors, and explains how the method of least squares can be applied to a simple problem to understand the basis. This procedure is illustrated for the simplest method of interpolating between two successive data points. A very simple example of interpolation is provided with the use of a linear function. Interpolation involving polynomials of higher degree, with more points on either side of the interpolated one is relatively complicated. The interpolation method outlined in the chapter can as well be applied to the “smoothing” of experimental data. This smoothing method has been used for a number of years by molecular spectroscopists, who generally refer to it as Savitzky and Golay's method. The two methods of numerical analysis of the Fourier transform as well as their application in interpolating and smoothing are discussed. The chapter provides the three methods for numerical integration used for the numerical evaluation of definite integrals—namely, the Trapezoid rule, the Simpson's rule, and the Romberg's method. The chapter examines the two methods of zeros of functions—such as, the Newton's method and the bisection method.