中英文翻译 计算机毕业设计

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

to check if the results are the same.

3.4 PADE APPROXIMATION BY RATIONAL FUNCTION

Pade approximation tries to approximate a function f (x) around a point xo by a

rational function

(3.4.1)

where are known.

How do we find such a rational function? We write the Taylor series expansion

of f (x) up to degree M + N at x = xo as

130 INTERPOLATION AND CURVE FITTING

Assuming ＝0for simplicity, we get the coefficients of

The MATLAB routine “padeap()” implements this scheme to find the coefficient

vectors of the numerator/denominator polynomial of the

Pade approximation for a given function f (x). Note the following things:

ž The derivatives up to order (M + N) are

computed numerically by using the routine “difapx()”, that will be introduced

in Section 5.3.

ž In order to compute the values of the Pade approximate function, we substitute

for x in which has been obtained with the assumption

that ＝0.

PADE APPROXIMATION BY RATIONAL FUNCTION 131

Example 3.2. Pade Approximation for . Let’s find the Pade approximation

for around ＝0. We make the

MATLAB program “do_pade.m”, which uses the routine “padeap()” for this

Figure 3.6 Pade approximation and Taylor series expansion for f(x) = ex(Example

procedure. First, we write the Taylor series expansion at x = 0 up to degree

M + N = 5 for the given function as

(E3.2.1)

whose coefficients are

(E3.2.2)

We put this into Eq. (3.4.4b) with M = 3,N = 2 and solve it for di’s to get

(E3.2.3)

Substituting this to Eq. (3.4.4a) yields

INTERPOLATION BY CUBIC SPLINE 133

wiggle’), particularly near the ends of the whole interval as the number of data