566 Chapter 20Numerical methods
20.4 Interpolation
Interpolation is the process of finding a function whose graph goes through a number
of given points. In Figure 20.3 the points represent the (n 1 + 11 ) number pairs
(x
0, y
0), (x
1, y
1), (x
2, y
2), = (x
n,y
n) (20.15)
and the dashed curve represents a continuous functiony 1 = 1 f(x)such thaty
i1 = 1 f(x
i)
for the number pairs (20.15).
The numbers themselves may, for example, be the results of measurements of
concentration and time in a kinetics experiment, or of pressure and temperature in
a study of the thermodynamic properties of a fluid. Alternatively, they may be the
tabulated values of a function that cannot be expressed in simple functional form.
The object of interpolation is to find a ‘simple’ function that can be used to find
intermediate points in the interval (x
0, y
0)to(x
n, y
n) (outside this interval the process
is extrapolation). Important uses of interpolation are in numerical integration (Section
20.5), whereby a function that cannot be integrated by the methods described in
Chapters 5 and 6 is approximated by a simpler function that can be so integrated, and
in the numerical solution of differential equations (Section 20.9).
Polynomial interpolation
Given (n 1 + 11 ) points, it is possible to find a unique polynomialp
nof degree n,
p
n(x) 1 = 1 a
01 + 1 a
1x 1 + 1 a
2x
21 +1-1+ 1 a
nx
n(20.16)
that passes through all the points. For example, through any two points there is a
unique straight line (n 1 = 11 ), and through any three points there is a unique quadratic
(n 1 = 12 ).
2.................
...
..
...
...
..
...
...
...
...
..
...
...
..
...
...
................
........
............................y
x
f(x)?
x
0x
1x
2x
3x
ny
0y
1y
2y
3y
n···
.......................................................................................................................................................................................................................................Figure 20.3
2The theoretical basis for polynomial interpolation, and one of the important theorems of modern analysis,
is Weierstrass’ theorem: ‘Iff(x)is an arbitrary continuous function defined in the intervala 1 ≤ 1 x 1 ≤ 1 b, it is always
possible to approximatef(x)over the whole interval as closely as we please by a power polynomial of sufficiently
high degree’. The essential requirement is continuity; the function may have infinitely many maxima and minima,
and need not have a derivative at any point in the interval. Karl Weierstrass (1815–1897), professor at Berlin, made
important and influential contributions to analysis.