Numerical Analysis 7.1. Numerical Differentiation скачать в хорошем качестве

Numerical Analysis 7.1. Numerical Differentiation

This video is a lecture on Numerical Differentiation, covering various methods to approximate derivatives and the associated error analysis. Core Topics Covered Introduction to Numerical Differentiation [00:02]: The lecture begins by reviewing the limit definition of a derivative and explaining why it is insufficient for numerical analysis because it lacks information about the approximation error. Lagrange's Method [01:36]: A technique where a function is replaced by its Lagrange interpolating polynomial. The derivative of the function is then approximated by the derivative of this polynomial. First-Order Difference Formulas [03:33]: Forward Difference: Used when the step size h is positive. Backward Difference: Used when h is negative. The lecture derives these using both Lagrange and Taylor expansion methods [13:40], showing they have a truncation error of order O(h). Higher-Order Formulas [19:01]: Three-Point Formulas [21:32]: Derived using second-degree Lagrange polynomials. Central Difference Formula [28:32]: A second-order formula (O(h 2 )) that uses symmetric points around x 0 ​ , providing much better accuracy than first-order methods [32:05]. Second Derivative Approximation [35:51]: The lecture demonstrates how to derive a formula for the second derivative using Taylor series expansions. The resulting formula is also second-order in terms of h [42:15]. Stability and Rounding Errors [45:52]: A critical discussion on why numerical differentiation is an unstable mathematical problem. It explains how small perturbations or rounding errors in function values can lead to large errors in the calculated derivative, especially as the step size h becomes very small [51:11]. Partial Derivatives [54:40]: A brief extension of these methods to multi-variable functions. Key Formulas and Concepts Forward Difference Error: Proportional to h times the second derivative [10:14]. Central Difference Error: Proportional to h 2 times the third derivative [28:32]. Rounding Error Sensitivity: Using a step size that is too small can actually increase the total error due to floating-point precision limits [54:05].