MATH-UA-0252 - Gaussian Quadrature Rule - Mathematics Assignment Help

Assignment Task

1. [Quadratic approximation of sine] Compute explicitly the optimal L2 quadratic approximation of sin(x) on [0, π] for the standard L2 inner product. Compare it on the same plot to the sine function, as well as the quadratic interpolant of sin(x) with nodes x0 = 0, x1 = π/2, x2 = π.

2. [Chebyshev orthogonal polynomials] Obtain explicit formulas for the first three Chebyshev polynomials on [−1, 1] via the Gram-Schmidt orthogonalization process starting with the monomials {1, x, x2}. Note that unless you normalize these polynomials they are only defined up to a constant, so choose the coefficient in front of the highest power of x to be one.
Hint: By examining the integrands and using symmetry you can avoid computing several of the integrals, and only need to compute one integral in the end. The change of coordinates x = cos θ may be useful for this integral; if you cannot compute just write the final formulas down and leave the integral unevaluated.

3. [3-point Gaussian quadrature] Recall that Gauss-Legrendre quadrature (Gaussian quadrature for ω(x) = 1) is the unique quadrature rule over [−1, 1] with n points that is exact for polynomials of degree up to and including 2n − 1.
(a) Show that this statement is equivalent to saying that the quadrature rule is exact for the monomials x0, x1, x2, . . . , x2n−1.
(b) Compute the formula for the 3-point Gaussian quadrature rule on the interval (−1, 1). To do this, use symmetry and the fact 3 is an odd number to conclude that the nodes must be the points x = [−x1, 0, x1] where x1 ∈ (0, 1) is unknown, and the weights must be w = [w1, w2, w1]. Write down a system of 3 equations for the three unknowns {x1, w1, w2} and solve it. Hint: Solving the system does not require any complicated algebra, and the answer is on Wikipedia.

4. [Accuracy of quadrature rules, code required] Write a code to compute the definite integral of a function using the composite Simpson rule with a given number of points n; write your code so you can easily change the function and number of points. Use this code to approximate R 1 0 exp(x)2 sin(x)2 dx ≈ 1.2668.
(a) How many points n do you need to get 4 decimal places correctly, i.e., to get the answer 1.267?
Give details of how you computed this, and note that there are smarter ways to do this then just trying different values of n. In particular, think about how the error decays as you increase n and try to estimate the right n for an absolute error of ≈ 10−4.
(b) How many digits do you get for a 3-point Gaussian quadrature rule (see problem 3, lecture notes, or Wikipedia)?

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