Page 174 - Applied Numerical Methods Using MATLAB

PROBLEMS 163
where the left-hand and right-hand sides of this equation are

x k+1

LHS: f (x)s (x) dx

x k
x k+1

x k+1
= f (x)s (x)| − f (x)s (x) dx
x k
x k

= f (x k+1 )s (x k+1 ) − f (x k )s (x k ) − C(f (x k+1 ) − f(x k )) (P3.7.3a)

x k+1

RHS: s (x)s (x) dx
x k

= s (x k+1 )s (x k+1 ) − s (x k )s (x k ) − C(s(x k+1 ) − s(x k )) (P3.7.3b)
(b) Check the validity of the following inequality:
x k+1 2

0 ≤ (f (x) − s (x)) dx
x k
x k+1 x k+1 x k+1

2
2

= (f (x)) dx − 2 f (x)s (x) dx + (s (x)) dx
x k x k x k

x k+1 x k+1
(P3.7.2) 2 2
= (f (x)) dx − (s (x)) dx
x k x k

x k+1 x k+1
2 2
(f (x)) dx ≤ (s (x)) dx (P3.7.4)
x k x k
3.8 MATLAB Built-in Routine for Cubic Spline
There are two MATLAB built-in routines:
>>yi = spline(x,y,xi);
>>yi = interp1(x,y,xi,’spline’);
Both receive a set of data points (x,y) and return the values of the cubic
spline interpolating function s(x) for the (intermediate) points xi given as
the third input argument. Write a program that uses these MATLAB routines
to get the interpolation for the set of data points
{(0, 0), (0.5, 2), (2, −2), (3.5, 2), (4, 0)}
and plots the results for [0, 4]. In this program, append the statements that
do the same job by using the routine “cspline(x,y,KC)” (Section 3.5) with
KC = 1, 2, and 3. Which one yields the same result as the MATLAB built-
in routine? What kind of boundary condition does the MATLAB built-in
routine assume?