M371 - Alexiades
                    Lab 6:   Quadrature
            [ For each lab, create a new folder (Lab#) and work in it ]

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Let L denote the integral of the log function F(x) = log(x) ( = ln(x) ) over the interval (0,1).

1. Find the exact value of the integral L by hand.

2. Clearly, cannot use any (closed) rule that evaluates F at x=0. Try an open rule:
  Implement the Midpoint Rule for integrals of the form I = ∫_a^bf(x)dx using M subintervals (user-specified M), in a code MidRule.
  a. Inputs should be a, b, M
  b. The integrand f(x) should be coded as a function subprogram that returns y=f(x).
  c. The code should print out: M, value of MR = Midpoint Rule approximation of I, and the error I − MR , nicely.
  d. Debug on ∫₀¹ (2x−1) dx with M=4.
  e. Test it on ∫₀¹ 3(1−x²)dx with M=1000.
  f. Now apply it to L = ∫₀¹ log(x)dx with M=10 and M=1000. What are the MR values and errors?

3. Another good option, especially for singular integrals like L, is Gaussian quadrature.
  a. To apply Gauss-Legendre, the integral L must be transformed over [−1, 1].
    Any interval a≤x≤b can be mapped to −1≤t≤1 with change of variable: x = (a+b)/2 + (b-a)/2 t
    Change the variable in L, on paper, so that Gauss-Legendre can be applied.
  b. Implement the 2-point Gauss-Legendre rule for integrals of the form ∫₋₁¹g(t)dt in a code gLegendre2.
    The evaluation points are ±1/√3 and weights=1.
  c. Use it to approximate L. What is the value and error ?

4. Make a change of variable (on paper) to transform L to an integral over (0, Inf), of the form: ∫₀^∞ g(t)e^-t dt
  for which Gauss-Laguerre quadrature is appropriate. What is g(t) here?

5. Implement the 2-point Gauss-Laguerre rule for integrals of the form ∫₀^∞ g(t)e^-t dt in a code gLaguerre2.
  The evaluation points and weights are: t1 = 2−√2 , t2 = 2+√2 , A1 = 0.853553390593 , A2 = 0.146446609407
  Use it to approximate L. What is the value and error ?

6. Compare the results from 3 and 5. Can you explain why one of them does so much better than the other on this integral ?

Submit, in a text file Lab6.txt   (please include separators and labels A,B, ...) :
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Lab6: Name, Date
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A. Exact value of L   (from 1.)
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B. Midpoint Rule: results and errors (from 2.f.) for M=10 and M=1000.
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C. Gauss-Legendre 2 pt (from 3.c): result and error
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D. Gauss-Laguerre 2 pt (from 5.): result and error. What is g(t) here?
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E. Answer to 6.
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F. Your MidRule code.
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G. Your gLegendre2 code.
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H. Your gLaguerre2 code, including formula for g(t).
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