Description
1. Given that each of the following sequences {pn}∞
n=0 converges to p
∗
, show that it converges
linearly:
(a) The sequence is (
pn+1 =
1
2
ln(pn + 1)
p0 = 1
and the limit is p
∗ = 0;
(b) The sequence is (
pn = 1 + 21−n +
1
(n+2)n
p0 = 4
and the limit is p
∗ = 1;
2. Show that the following sequences {pn := 10(−2
n)}∞
n=0 converges to p
∗
, show that it converges quadratically.
3. (a) Use the Lagrange interpolation method to find a polynomial f such that
f(1) = 2, f(2) = 1, f(3) = 4, f(4) = 3.
(b) Use the Neville’s Method instead to find the same polynomial f.
4. Programming problem: Consider the following function f : [−1, 1] → R
f(x) = |x|
(a) Plot the graph of the function f.
(b) Given n ∈ N\{0}, define x
k
n = −1 + 2k
n
for 0 ≤ k ≤ n.
Let gn(x) be the unique polynomial of degree n which results by interpolating the
n + 1 data {(x
k
n
, f(x
k
n
))}0≤k≤n, i.e. gn(x
k
n
) = f(x
k
n
) for all 0 ≤ k ≤ n. Plot the
functions f, g2, g3, g4 and g5 on the same graph.
(c) Plot the sequence {gn(0.3)}1≤n≤20.
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