Hello I need help with questions 3 and 4 of this problem set. I am weak at proofs and honestly for now I just need the solution. Thank you
MA/CSC 427 Numerical Analysis ? HW5 November 9, 2016 1. Consider the function f (x) = sin2 (10?x + 1).
(a) Determine the exact value (on paper, without numerical methods) of R1
?1 f (x)dx. (b) Use Matlab?s quad function, which implements adaptive integration with Simpson?s rule, to determine
the approximate value of the integral. Set the optimal tolerance level to 10?10 , and check how many
function evaluations this method needed. Using the correct value of the integral, find the error of
the value returned by quad. (Hint: check the Matlab documentation to find out how to specify the
tolerance and determine the number of function evaluations.)
(c) Use chebfun to determine a very close polynomial approximation of f on [?1, 1]. What is the degree
of the polynomial returned by chebfun? Based on this degree, how manyRfunction evaluations would
you expect we need to determine a very precise value of the integral ?1 f (x)dx using Gaussian
quadrature? Compare your number to the number of evaluations used by quad.
(d) For n = 10, 20, . . . , 90, 100, compute the value of the n-point Gaussian quadrature formula applied to
f . (Use the legpts function in chebfun to obtain the Legendre points and the weights, do not try
and compute them yourself.) Around what value of n does the error become very small (smaller than
the error of quad)? Do the results agree with what you predicted in the previous problem?
2. (a) How can we transform a quadrature formula for integration over [?1, 1] to a formula for integration
over [a, b]? Describe how the points and the weights need to be changed. Argue that the transformation
(b) Determine the points and weights of the 6-point Gaussian quadrature formula for the interval [1, 5].
Use legpts to find the Legendre points and weights for [?1, 1], then use your transformation from
above. Test your formula by computing the approximate value of 1 ex and comparing it to the
3. Let Ln be the Legendre polynomial of degree n. Prove that Ln has n distinct real roots, all in [?1, 1].
(Hint: Of all the interesting properties Legendre polynomials have, we learned about two: (1) Ln has
degree n, and (2) Ln is orthogonal to all polynomials of degree less than n. That is all you should need in
4. Extra credit: Suppose we have an n-point quadrature formula for approximate integration over [?1, 1] that
we wish to extend by adding p additional points to it. (The weights of the resulting n + p points can be
completely different, but we want to reuse the computed function values when doing adaptive integration.)
Let F be a degree-n polynomial whose roots are the points of the starting formula, and suppose that there
exists a polynomial G of degree p satisfying
F (t)G(t)ti dt = 0 i = 0, . . . , p ? 1.
?1 Assume further that the roots of this polynomial G are all real, distinct, and different from the roots
of F . Prove that in this case there exists a quadrature formula using the roots of F G whose degree of
exactness is n + 2p ? 1. (Hint: this exercise is very difficult if you don?t follow the details behind Gaussian
quadrature, but becomes very easy if you follow the proof of Gaussian quadrature we did in class.)
Use this result to find an extension of the 7-point Gaussian quadrature formula over [?1, 1]. Find a small
value of p for which a polynomial G with all the required properties exists, and determine the new points
to add to the formula. Due on November 18, before class. Turn in the physical copy of your work, including the printed code
and computational results.
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