Page 173 - Applied Numerical Methods Using MATLAB

162 INTERPOLATION AND CURVE FITTING
(a) The above routine “rational_interpolation(x,y,xi)”uses the
Bulirsch–Stoer method to interpolate the set of data pairs (x,y)given
as its first/second input arguments over a set of intermediate points
xi given as its third input argument. Complete the routine and apply
it to interpolate the four data points {(−1,f(−1)), (−0.2,f(−0.2)),
2
(0.1,f (0.1)), (0.8,f(0.8))} on the graph of f(x) = 1/(1 + 8x ) for
xi = [-100:100]/100 and plot the interpolated curve together with the
graph of the true function f(x). Does it work well? How about doing
the same job with another routine “rat_interp()” listed in Section 8.3
of [F-1]? What are the values of yi([95:97]) obtained from the two
routines? If you come across anything odd in the graphic results and/or
the output numbers, what is your explanation?
(cf) MATLAB expresses the in-determinant 0/0 (zero-divided-by-zero) as NaN
(Not-a-Number) and skips the value when plotting it on a graph. It may,
therefore, be better off for the plotting purpose if we take no special
consideration into the case of in-determinant.
(b) Apply the Pade approximation routine “padeap()” (with M=2 & N=
2
2) to generate the rational function approximating f(x) = 1/(1 + 8x )
and compare the result with the true function f(x).
(c) To compare the rational interpolation method with the Pade approx-
imation scheme, apply the routines rational_interpolation() and
padeap() (with M=3 & N=2) to interpolate the four data points
{(−2,f (−2)), (−1,f(−1)), (1,f(1)), (2,f (2))} on the graph of
f(x) = sin(x) for xi = [-100:100]*pi/100 and plot the interpolated
curve together with the graph of the true function. How do you compare
the approximation/interpolation results?
3.7 Smoothness of a Cubic Spline Function

We claim that the cubic spline interpolation function s(x) has the smooth-
ness property of

x k+1 x k+1

2 2
(s (x)) dx ≤ (f (x)) dx (P3.7.1)
x k x k
for any second-order differentiable function f(x) matching the given grid
points and having the same first-order derivatives as s(x) at the grid points.
This implies that the cubic spline functions are not so rugged. Prove it by
doing the following.
(a) Check the validity of the equality

x k+1 x k+1

2
f (x)s (x) dx = (s (x)) dx (P3.7.2)
x k x k