Interpolation Methods in Numerical Analysis: Trapezoidal Rule, Simpson's Rule, and Their Composite Forms

Interpolation methods in numerical analysis involve several important techniques, including the Trapezoidal Rule, Simpson's Rule, Composite Trapezoidal Rule, and Composite Simpson's Rule. These methods play a crucial role in numerical computation by providing approximations for estimating function values through numerical integration. The Trapezoidal Rule approximates the area under a curve using trapezoids, where the implementation typically involves calculating the average of function values at endpoints multiplied by the interval width. In code, this can be represented as (b-a)*(f(a)+f(b))/2 for a single interval. Simpson's Rule provides higher accuracy by using parabolic arcs instead of straight lines to approximate the curve. The algorithm employs quadratic interpolation, calculating the weighted average of function values at endpoints and midpoint. A standard implementation uses the formula (b-a)*(f(a)+4*f(midpoint)+f(b))/6. The Composite Trapezoidal Rule and Composite Simpson's Rule represent improved versions that enhance accuracy by dividing the entire interval into multiple subintervals. The Composite Trapezoidal Rule applies the basic trapezoidal formula to each subinterval and sums the results, while the Composite Simpson's Rule requires an even number of subintervals to apply Simpson's formula repeatedly. These composite methods significantly improve precision through iterative application across smaller segments, with implementation typically involving loop structures that accumulate results from each subdivision. The error reduction follows predictable patterns, with Composite Simpson's Rule generally providing O(h^4) accuracy compared to O(h²) for the Composite Trapezoidal Rule. These interpolation and numerical integration methods are widely applied in numerical analysis, providing effective tools for handling function approximation problems in scientific computing and engineering applications.