Question

Comparisons of numerical integration methods (80 points)

Compute the following integration.
∫₀¹4x³ex⁴dx.
Use 3 different methods, including (1) adaptive Simpson's rule, (2) Romberg's method and (3) Gaussian quadrature. Compare with the analytical results to plot the errors versus the number of intervals N in a log-log plot. Use the following values of N for all 3 methods, (4,8,16,32,64,128). (Use gaussxw .py in the folder for lecture 6 on Canvas to calculate the weights and integration points for Gauss quadrature.)

Answer 1

The student is tasked with computing an integral using adaptive Simpson's rule, Romberg's method, and Gaussian quadrature, then comparing these numerical solutions with the exact result.

Step-by-step explanation:

The student has been asked to compute the integral of 4x³e⁴ᴇ¹⁴ from 0 to 1 using three different numerical integration methods: adaptive Simpson's rule, Romberg's method, and Gaussian quadrature. They will compare the results of these methods with the exact analytical result by plotting the errors versus the number of intervals N in a log-log plot for N values of 4, 8, 16, 32, 64, and 128.

Using the provided gaussxw.py script from their course materials will allow them to calculate the weights and integration points necessary for the Gaussian quadrature.