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
. 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.



插值與擬合
我們能夠運(yùn)用這個(gè)公式直接得到給定函數(shù)f(x)的近似多項(xiàng)式估計(jì)，沒有必要采取拉格朗日或牛頓多項(xiàng)式。給定一個(gè)函數(shù)、多項(xiàng)式次數(shù)以及區(qū)間的左右邊界點(diǎn)，上面提到的MATLAB程序“cheby()”可以利用這個(gè)公式得到切比雪夫近似多項(xiàng)式。
以下的例子闡述了利用這個(gè)公式得到的相同的近似多項(xiàng)式，也可以利用切比雪夫節(jié)點(diǎn)的牛頓多項(xiàng)式得到。
例3.1．切比雪夫多項(xiàng)式近似值?？紤]找到函數(shù)
的二階多項(xiàng)式估計(jì)。寫出以下程序“do_cheby.m”，這個(gè)程序運(yùn)用了MATLAB程序“cheby()”解決這個(gè)問題，并且運(yùn)用了切比雪夫節(jié)點(diǎn)的牛頓或拉格朗日多項(xiàng)式來解決相同的問題，讀者可以運(yùn)行這個(gè)程序，檢查結(jié)果是否相同。