Write a code to compute a natural spline

Question

Assignment: Python Code Required

1.  Write a code to compute a natural spline S(x) which interpolates a collection of given points (x_o, y_o), (x₁, y₁), ... (x_n, Y_n) where x₀ < x₁ < x₂ < ... < x_n (do not assume they are equidistributed). Your code should have a triadiagonal solver for the resulting linear system of equations (you're not allowed to use Matlab's \ operator to solve the linear system).

2.  One important application of spline interpolation is the construction of smooth curves that are not necessarily the graph of a function but that have a parametric representation x = x(t) and y = y(t) for t ε [a, b]. Hence one needs to determine two splines interpolating (t_j,x_j) and (t_j, y_j) (j = 0, 1, ... n).

The arc length of the curve is a natural choice for the parameter t. However, this is not known a priori and instead the t_j's are usually chosen as the distances of consecutive points:

t_o = 0, t_j = t_j-i + √[(xj - xj-1)² + (yj-uj-1)²], j= 1, 2, ...n.                                  (1)

Use the values in Table 1 to construct a smooth parametric representation of a curve passing through the points (x_j, y_j), j = 0, 1, ... , 8 by finding the two natural cubic splines interpolating (t_i, x_j) and (t_i,y_j), j = 0, 1, ... 8, respectively. Tabulate the coefficients of the splines and plot the resulting (parametric) curve.

j	tj	xj	yj
0	0	1.50	0.75
1	0.618	0.90	0.90
2	0.935	0.60	1.00
3	1.255	0.35	0.80
4	1.636	0.20	0.45
5	1.905	0.10	0.20
6	2.317	0.50	0.10
7	2.827	1.00	0.20
8	3.330	1.50	0.25

Table 1: Table of data for Problem 2.

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