Numerical Analysis 7.3. Newton-Cotes Formulas скачать в хорошем качестве

Numerical Analysis 7.3. Newton-Cotes Formulas

This video is a lecture on Numerical Analysis, specifically focusing on Newton-Cotes Formulas used for approximating the definite integral of a function. Key Concepts and Formulas The lecture explains how to approximate the integral of a function by replacing it with an interpolating polynomial, typically using Lagrange polynomials [06:55]. Quadrature Formulas: These are formulas that approximate an integral as a weighted sum of function values: ∑c i ​ f(x i ​ ) [06:29]. Closed vs. Open Formulas: Closed formulas include the interval's endpoints (a and b) as mesh points [07:31]. Open formulas use mesh points that fall only within the open interval (a,b) [07:50]. Major Integration Rules Covered Trapezoidal Rule (n=1): * Approximates the area under the function using a straight line between two points [16:08]. The Composite Trapezoidal Rule divides the interval into n smaller sub-intervals to improve accuracy, resulting in an error term of order O(h 2 ) [21:26]. Simpson’s Rule (n=2): * Uses three points (the endpoints and the midpoint) to fit a parabola [31:20]. The Composite Simpson's Rule is much more accurate, with an error term of order O(h 4 ) [35:16]. Higher-Order Rules: The lecture briefly mentions the Simpson’s 3/8 rule and formulas for n=4 which provide even higher-order accuracy [40:32]. Numerical Examples and Stability Example Problem: The instructor demonstrates these rules by approximating ∫ 0 1 ​ x 2 e x dx. He shows how the error significantly decreases as the step size h is reduced for both the Trapezoidal and Simpson’s rules [22:28], [36:00]. Stability: The lecture concludes by discussing the stability of these formulas. It is shown that if the coefficients c i ​ are positive, the quadrature formula is stable against small errors in function evaluations (such as rounding errors) [46:57].