Page 140 - Applied Numerical Methods Using MATLAB

PADE APPROXIMATION BY RATIONAL FUNCTION 129
We can apply this formula to get the polynomial approximation directly for
a given function f(x), without having to resort to the Lagrange or Newton
polynomial. Given a function, the degree of the approximate polynomial, and the
left/right boundary points of the interval, the above MATLAB routine “cheby()”
uses this formula to make the Chebyshev polynomial approximation.
The following example illustrates that this formula gives the same approximate
polynomial function as could be obtained by applying the Newton polynomial
with the Chebyshev nodes.

Example 3.1. Approximation by Chebyshev Polynomial. Consider the problem
of finding the second-degree (N = 2) polynomial to approximate the function
2
f(x) = 1/(1 + 8x ). We make the following program “do_cheby.m”, which uses
the MATLAB routine “cheby()” for this job and uses Lagrange/Newton polyno-
mial with the Chebyshev nodes to do the same job. Readers can run this program
to check if the results are the same.

%do_cheby.m
N=2;a=-2;b=2;
[c,x1,y1] = cheby(’f31’,N,a,b) %Chebyshev polynomial ftn
%for comparison with Lagrange/Newton polynomial ftn
k = [0:N]; xn = cos((2*N + 1 - 2*k)*pi/2/(N + 1));%Eq.(3.3.1a):Chebyshev nodes
x = ((b-a)*xn +a + b)/2; %Eq.(3.3.1b)
y = f31(x); n = newtonp(x,y), l = lagranp(x,y)

>>do_cheby
c = -0.3200 -0.0000 1.0000

3.4 PADE APPROXIMATION BY RATIONAL FUNCTION

o
Pade approximation tries to approximate a function f(x) around a point x by a
rational function
o
Q M (x − x )
o
p M,N (x − x ) = with M = N or M = N + 1
o
D N (x − x )
o
o M
o 2
q 0 + q 1 (x − x ) + q 2 (x − x ) +· · · + q M (x − x )
=
o N
o 2
o
1 + d 1 (x − x ) + d 2 (x − x ) +· · · + d N (x − x )
(3.4.1)
o

o
o
o
where f(x ), f (x ), f (2) (x ), . ..,f (M+N) (x ) are known.
How do we find such a rational function? We write the Taylor series expansion
o
of f(x) up to degree M + N at x = x as