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Numerical Analysis 7. Numerical Differentiation and Integration

This video presents an overview of numerical differentiation and integration, corresponding to Chapter 7 of a numerical analysis course. Numerical Differentiation [00:03] Methods: Derivatives are approximated using Lagrange interpolation or Taylor series expansions [01:36]. Key Formulas: First-order: Forward and backward difference formulas [11:42]. Second-order: Central difference formulas, which provide higher accuracy by using symmetric mesh points [28:22]. Higher Derivatives: The video demonstrates deriving second derivative approximations using Taylor series [35:59]. Stability Issues: Numerical differentiation is inherently unstable and sensitive to rounding errors [47:47]. Small step sizes (h) can paradoxically increase the error due to arithmetic precision limits [51:51]. Richardson Extrapolation [56:03] This general technique improves the order of an approximation by combining results from different step sizes (e.g., h and h/2) to cancel out lower-order error terms [59:41]. Numerical Integration (Quadrature) [01:08:44] Newton-Cotes Formulas: Based on integrating Lagrange polynomials over equidistant points [01:15:41]. Trapezoidal Rule: First-order polynomial approximation [01:18:28]. Simpson’s Rule: Second-order (parabolic) approximation, offering higher accuracy [01:40:04]. Composite Rules: To reduce error, the integration interval is divided into n sub-intervals, applying the basic rules to each [01:26:01]. Gaussian Quadrature [01:56:25]: Unlike Newton-Cotes, this method optimizes both the weights and the location of the mesh points (roots of Legendre polynomials) [02:21:04]. It provides maximum precision for polynomials up to degree 2n−1 [02:21:57].