Numerical Differentiation & Integration

This topic covers the most common approaches for numerical integration—the Newton-Cotes formulas. These relationships are based on replacing a complicated function or tabulated data with a simple polynomial that is easy to integrate. Three of the most widely used Newton-Cotes formulas are discussed in detail: the trapezoidal rule, Simpson’s 1/3 rule, and Simpson’s 3/8 rule. All these formulas are designed for cases where the data to be integrated are evenly spaced. There is also a discussion on numerical integration of unequally spaced data. This is a very important topic because many real-world applications deal with data that are in this form. This topic also covers differentiation—high-accuracy finite difference formulas and the differentiation of unequally spaced data.