Description
Prof. Boris Houska
1. Consider the function f(x) = x3. The goal of this exercies is to solve the Gauss approximation problem
where P2 denotes the set of polynomials of degree ≤ 2.
(a) We define the weighted L2 scalar product by
∀ϕ,ψ ∈ L2([0,1]),
Find an orthonormal basis of P2 with respect to this particular scalar product.
(b) Solve the above Gauss approximation problem by using the above orthonormal basis.
(c) Plot your results for p and the function f on the interval [0,1].
2. Explain the main difference between Newton Cotes formulas and Gauss’ quadrature.
3. Explain how to apply Simpson’s rule to find a numerical approximation of the integral
in R3 over the cylinder Ω = , where we use the notation
to denote the Euclidean norm. Also derive an upper bound on the
numerical approximation error.
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