Revision sheet 3

Question 1

monomial basis

Answer 1

1) get into matrix
x =
(x_1^0 x_1^1 … x_1^n-1
x_2^0 x_2^1 … x_2^n-1
.
.
.
x_n^0…..x_n^n-1)
2) solve x(a1,…,an) = y (guass)
3) put into form a1 + a2x +a3x^2+…+anx^n-1

Question 2

Lagrange polynomial

Answer 2

sum of (for each point aka for point 1 do the following, for point 2 swap points 1 and 2)
y1 * ((x-x2)(x-x3))/(x1-x2)(x1-x3)

Question 3

newton nested table

Answer 3

xi| f[xi] = yi|f[x_i,x_(i+1)] = (f[x_(i+1)]-f[x_(i)])/(x_(i+1)-x_i)) | f[x_1,x_2,x_3] = (f[x2,x3] - f[x1,x2])/(x3-x1)

P2 (X) =f[x_1]+f(x_1, x_2) (x-x1) +f[x1,x2,x3] (x-x1)(x-x2)

Question 4

second derivative of natural cublic spine

Answer 4

second derivitve = 0

Question 5

find an upper bound on the error
|𝑓(𝑥) – 𝑝2(𝑥)| on the interval [a,b] using the error bound
formula for equally spaced nodes
at x = x1,…,xn

Answer 5

h = (b-a)/(n-1)

(h^n)/4n * max_(yE[a,b]) (f^n(y))