Reference no: EM131002909
Question 1 (a) Show that
0∫1x-x dx = n=1Σ∞ n-n
(b) Use the sum in (a) to evaluate the integral in (a) to 12-digit accuracy.
(c) Evaluate the integral in (a) by Romberg integration. Estimate how many function evaluations Romberg integration will require to achieve 12-digit accuracy. Explain the agreement or disagreement of your results with theory.
Question 2 In class we proved the Euler-Maclaurin summation formula
0∫1 f(x) dx = ½ (f(0) + f(1)) + m=1Σ∞ bm (f(2m-1) (1) - f(2m-1) (0))
for some unknown constants bm independent of f.
(a) Find a formula for bm by evaluating both sides for f(x) = eλx where λ is a parameter.
(b) Compute b1, b2, b3,....., b10.
Question 3 (a) Use the Euler-Maclaurin formula to show that
j=1Σn jk = Pk+1(n)
is a degree-(k + 1) polynomial in n. Example:
j=1Σn j= n(n + 1) / 2.
b) Use the results of question 2 to find Pk+1 for 2 ≤ k ≤ 10.
(c) Use polynomial interpolation to find Pk+1 for 2 ≤ k ≤ 10 and compare with the results from (b).
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