Question

Numerical Integration Approximate the integral I=∫01​1+x2dx​ by using (a) Midpoint rule (d) Trapezoidal rule (f) Simpson's rule with the stepsizes h=hk​=2−k,k=1,2,…,16. Using the exact value I=4π​ of given integral, calculate the absolute errors in approximations for each stepsize hk​. Plot in one frame the errors against the stepsizes {hk​}k=116​ in log-log scale. What do you observe from the plots? Does this confirm the theory for order of convergence? If not, then explain why.

Answer 1

Numerical Integration involves using methods like the Midpoint, Trapezoidal, and Simpson's rules to approximate integrals, comparing these with the exact value to determine absolute errors, and examining their order of convergence by plotting on a log-log scale graph.

Step-by-step explanation:

The subject in question is Numerical Integration, which involves approximating the value of an integral using different numerical methods, such as the Midpoint rule, Trapezoidal rule, and Simpson's rule. To estimate the integral I = ∫01 √(1 + x2) dx with different stepsizes hk = 2-k where k ranges from 1 to 16, we would use these rules to calculate the approximate value of the integral for each stepsize, and then compare these results with the exact value I = π/4 to determine the absolute errors. By plotting these errors against the stepsizes on a log-log scale graph, we can analyze the order of convergence and check if it confirms the theoretical expectations for each numerical method.

Upon analyzing the plots, the order of convergence for each method can be observed, confirming the theoretical convergence rates (quadratic for Simpson's rule, linear for Trapezoidal and Midpoint rules) under ideal conditions. However, if the plot does not confirm the expected theory, possible explanations could include round-off errors or function behavior that doesn't align well with the assumptions of the numerical methods.

Answer 2

Approximate the integral using various numerical integration methods, compare with the exact value, calculate absolute errors, and observe the order of convergence from a log-log plot.

Step-by-step explanation:

The task is to approximate the integral I=∫_{0}^{1} (1+x^{2}) dx using various numerical integration methods: Midpoint Rule, Trapezoidal Rule, and Simpson's Rule. For each method, the step sizes are defined as hk = 2-k, where k is from 1 to 16.

To assess the accuracy of these approximations, one needs to compare them with the exact value of the integral, which is I = 4/π, and calculate the absolute errors. Then, representing these errors against the step sizes on a log-log scale plot will reveal the order of convergence. If the slope of error lines corresponds with the theoretical order of convergence for each method, the theory is confirmed.

From the log-log scale plot of errors against step sizes, one should observe the following:

1. The slope of the error lines for Midpoint and Trapezoidal rules should be approximately -2, indicating a second-order convergence.
2. For Simpson's rule, the slope should be approximately -4, indicating a fourth-order convergence.
3. Any deviations from these expected slopes need further investigation.