Math 104A Homework #4 ∗
General Instructions: Please write your homework papers neatly, and follow TA’s instructions
(on Gauchoapace) to turn it in. Write your own code individually. Do not copy codes!
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).
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 ∈ [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.
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 coefficients of the splines and plot
the resulting (parametric) curve.
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Math 104A作业代写
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1
Table 1
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
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