计算机科学与技术专业外文翻译--插值与拟合

外文原文：

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()1/(18)f x x =+. We make the following program “do_cheby.m ”, which uses the MATLAB routine “cheby()” for this job and uses Lagrange/Newton polynomial with the Chebyshev nodes to do the same job. Readers can run this program

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

00

,00020012002012()()()()()()

1()()()M M N N M M N N Q x x p x x D x x q q x x q x x q x x d x x d x x x x --=-+-+--+-+-+-＋＋＝＋d (3.4.1)

where 00(2)0()0(),'(),(),,()M N f x f x f x f x + 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

(2)0()000000200020012()()()()()'()()()()2()!()()()(3.4.2)

M N M N

M N M N M N f x f x f x T x x f x f x x x x x x x M N a a x x a x x a x x +++++≈-=+-+-++-+=+-+-+

+- Assuming 0x ＝0for simplicity, we get the coefficients of ()()N M D x andQ x such that

()()0()

M M N N Q x T x D x +-=011011()(1)()01M N N N M N N N N N a a x a x d x d x q q x q x d x d x ++++++++-++

+=+++

01101()(1)()M N N N M N N N a a x a x d x d x q q x q x ++++++++=+++ (3.4.3) by solving the following equations:

001

01121102211221

11212112212000

M M M M N N M M M M M N N M M M M N N M N M N M N M N a q a a d q a a d a d q a a d a d a d q a a d a d a d a a d a d a d a a d a d a d ---+--+++-++++=⎧⎪+=⎪⎪+=⎪⎪⎪⎨⎪=⎪+=⎪⎪⎪⎪=⎩＋＋＋＋＋＋＋＋＋＋＋＋（3.4.4a ）（3.4.4b ） Here, we must first solve Eq. (3.4.4b) for N d d d ,,,21 and then substitute di ’s

into Eq. (3.4.4a) to obtain M q q q ,,,10

The MATLAB routine “padeap()” implements this scheme to find the coefficient vectors of the numerator/denominator polynomial )(/)(x D x Q N M of the

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