Numerical Analysis Homework (Python code required, need it in 12 hrs)

Homework1

1. Write a code to compute a natural spline S(x) which interpolates a collection of given points (x0, y0), (x1, y1), . . . (xn, yn) where x0 < x1 < x2 < . . . < xn (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 2 [a, b]. Hence one needs to determine two splines interpolating (tj, xj) and (tj, yj) (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 tj’s are usually chosen as the distances of consecutive points:

t0 = 0, tj = tj�1 + q

(xj � xj�1)2 + (yj � yj�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 (xj, yj), j = 0, 1, . . . , 8 by finding the two natural cubic splines interpolating (tj, xj) and (tj, yj), j = 0, 1, . . . 8, respectively. Tabulate the coe�cients 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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