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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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