Description
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Implement the following numerical methods for approximating integrals: (i) trapezoidal rule, (ii) Simpson’s rule, and (iii) Clenshaw-Curtis rule. Code for generating the points and weights of the Clenshaw-Curtis rule is available on Canvas.
Consider the function
f(x) = sin(2 x) + cos(3 x); x 2 [ 1; 1]:
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Using calculus, compute the de nite integral of f(x) on the interval [ 1; 1].
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For n = 2k + 1 with k = 1; : : : ; 20, use the three numerical integration methods to estimate the integral with n points. Plot the relative error as a function of n on a log-log scale.
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Identify the values of n that constitute the asymptotic regime. For each of the three methods, what convergence rate do you observe?
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Repeat the previous numerical study for the function
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f(x) = sign (x
0:2) + 1;
x 2 [ 1; 1]
where
8
0;
y = 0
sign (y) =
<
1;
y > 0
1;
y < 0
:
How do the observed convergence rates di er from the rst function? Why do they di er?
1
