Numerical Integration

The trapezoid rule

Simpson's rule

Choosing rules and checking

Error bound computation

In the trapezoid rule T_n, how are the sample values weighted?

Simpson's rule requires which condition on the strip count n?

How do the trapezoid and Simpson error orders compare?

The trapezoid error bound needs M bounding which derivative of f?

For the integral from 0 to 1 of e^(-x^2) dx, Simpson's rule with n = 4 gives about 0.74683. The true value is about 0.74682. Roughly how many digits agree?

For the integral from 0 to 2 of f dx with n = 2, the samples are f(0), f(1), f(2), and Delta x = 1. The trapezoid rule gives (1/2)(f(0) + 2 f(1) + f(2)). For f(x) = x^2 this equals what number? Answer with a number.

In Simpson's rule the interior weights alternate starting with 4. As a number, what weight follows the first interior 4? Answer with a number.

Simpson's error behaves like 1 over n to which power? Answer with a number.

Why do both trapezoid and Simpson rules struggle near an integrand's singularity?

When no antiderivative is known, agreement between S_n and S_2n to the target digits certifies the value. As a single word, what kind of check is this? Answer in lowercase.