Description

• Write in a neat and legible handwriting or use LATEX
• Clearly explain the reasoning process
• Write in a complete style (subject, verb, and object)
• Be critical on your results
Questions preceded by a * are optional. Although they can be skipped without any deduction, it is important to know and understand the results they contain.
Ex. 1 — Legendre polynomials
Let (Qn)n∈N be the sequence of polynomials defined by Qn over [−1,1]. This sequence defines the Legendre polynomials.
1. Using the constant weight function w(x) = 1 over (−1,1), show that (Qn)n∈N defines a sequence of orthogonal polynomials.
2. Show that Qn(−x) = (−1)nQn(x).
3. Show that for any x ∈ [−1,1], the Legendre polynomials obey the recurrence relation
(n + 1)Qn+1(x) = (2n + 1)xQn(x) − nQn−1(x).
Hint: use elements from the proof of proposition 2.53. * 4. Prove that
ni
Qn(x) = ∑(−1)i.
i=0
Ex. 2 — Interpolation
Let f be a continuous function for which we know the eight points deifned in the following table.
x −5 −1 0 1 3 5 10 12
f (x) 781 5 1 1 61 521 9091 19141
Determine f (2).
Ex. 3 — Newton’s form of the interpolation polynomial
Let f be a continuous function and Pn be its interpolation polynomial in the points x0,··· ,xn.
1. Let P0(x) = f (x0) be the interpolation polynomial in a single point x0.
b) a) Show that for two pointsDetermine a polynomial Rxof degree at most two, such that0 and x1, P1(x) = P0(x) + f (xx11)−−Pfx0(x20()x() =x −Px01)(.x)+R(x), for three
nodes x0, x1, and x2.
c) Prove by induction that
j−1
Pj(x) = Pj−1(x) + aj ∏(x − xk), where aj only depends on x0,··· ,xj.
k=0
2. Show that
n j−1
Pn(x) = f (x0) + ∑aj ∏(x − xk).
j=1 k=0
* 3. Denoting aj by f [x0,··· ,xj], prove that for any k > 0,
f (xk)
,xk] = ,···,xk−1]
4. Write the pseudocode of a clear algorithm to compute Pn(x) when given n + 1 nodes x0,··· ,xn and the value of f at those nodes.
We now consider the case of equidistant nodes, i.e. xi = x0 + ih, for any 0 ≤ i ≤ n, and some h .
Denoting f [xi] by fi, 0 ≤ i ≤ n, we recursively define the operator ∇ such that
 ∇∇0fi = fi ∇ − ∇ k+1fi = kfi+1 kfi.
5. Show that for all i,k ∈ N, f .
* 6. Observing that , prove that Pn f0, where s = x−hx0.
7. Write the pseudocode of an algorithm which takes a step h as input, a number of nodes, the value of f at each of those nodes, and a value x. The algorithm should return Pn(x) ≈ f (x).