20.4 Interpolation 571
Forx 1 = 1 0.832the sequence of approximations fore
0.832is
p
1(x) 1 = 1 2.29820177 p
2(x) 1 = 1 2.29790524
p
3(x) 1 = 1 2.29791008 p
4(x) 1 = 1 2.29790997
andp
4(x)is correct to all the figures quoted (see Examples 20.7 and 20.8).
0 Exercise 18
Although thex
iare evenly spaced in this example, Newton’s formula (20.23) is valid
for unevenly spaced points. Modern implementations on computers are essentially
adaptations of Newton’s method that find the best path through an interpolation
table such as Table 20.5 in order to give greatest accuracy.
Spline interpolation
Examples 20.7 to 20.9 show that increasing the degree of the interpolation polynomial
leads to increased accuracy for e
x, but this is not generally true when the nodes
exhibit maxima and minima as in Figure 20.5. Figure 20.7 shows the result of fitting
the polynomial of (maximum) degree 8 to the 9 points given in Table 20.4. This
demonstrates that increasing the degree nof the interpolation polynomial can
result in spurious oscillations between the nodes, and these oscillations can grow in
magnitude indefinitely as nincreases. Interpolation should therefore in general be
restricted to polynomials of low degree (not more than 4 or 5), but Figure 20.6 shows
that, in unfavourable cases, such interpolations can suffer from sharp discontinuities
of gradient at the nodes.
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.....................Figure 20.7 Figure 20.8
Both types of disadvantage are avoided in the method of splines.
4This is piecewise
polynomial interpolation, but with the polynomial in the interval between each pair
of nodes chosen to make the overall interpolation function smooth at the nodes; that
4Splines are flexible strips of wood long used by designers, engineers and shipwrights to draw smooth curves
through given points. The numerical method was described by I. J. Schoenberg in 1946.