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1、Newton Interpolate牛頓插值方法Numerical Methods1應用2Newton Polynomials100()()nNNnknkPxaxx1010()()P xaa xx2010201()()()()P xaa xxaxxxx30102013012()()()()()()()P xaa xxaxxxxa xxxxxx101020130120.()()()()()()().()NNNkkPxaa xxaxxxxa xxxxxxaxxIs said to be Newton polynomial with N centers ，and 0121,.,Nx x xxHave

2、 the nodes 。0121,.,NNx x xxx如何計算Newton Polynomials101020130120()()()()()()().()NNNkkPxaa xxaxxxxa xxxxxxaxx4010201301240123433221100()()()()()()()()()()()()()()()()P xaa xxaxxxxa xxxxxxaxxxxxxxxaxxaxxaxxaxxa1110100().()NNNNNNSaSSxxaSSxxaNewton 插值數(shù)學問題Newton插值問題插值問題：已知在一組互異節(jié)點 上的函數(shù)值，求一個盡可能低的Newton多項式，使

3、得：即：()(0,1,2,)iip xyinbxxxan.10插值問題的解是唯一的，區(qū)別僅是表達方式的不同！Lagrange插值多項式的優(yōu)缺點1.當節(jié)點固定不變時，很容易計算多個不同點x出的Lagrange插值多項式的值。2.計算高階（n）插值多項式，不能利用已計算出的低階插值多項式。3.Newton插值方法是對Lagrange插值方法的一個補充。特別適合于計算一個點上的各種階數(shù)的插值多項式的值。低階Newton插值問題的解法n=0時：000()()Pxafxn=1時：1010()()Pxaaxx0001101()()()afxaaxxfx1010110()(),fxfxafxxxx20102

4、01()()()()Pxaaxxaxxxxn=2時：00011010120220212()()()()()()()afxaaxxfxaaxxaxxxxfx001010110()()(),af xf xf xaf x xxx低級Newton插值問題的解法 201202202110202010202120101020202110211010220()()()()()()()()()1()()()()()1()()()()(1f xaa xxaxxxxf xf xf xf xxxxxxxxxf xf xxxf xf xxxxxxxxxf xf xxxf xf xxxxx121101021202110

5、120101220)()()()()1,xxxxf xf xf xf xxxxxxxf x xf x xf x x xxxDivided difference00()f xf x100110,f xf xf x xxx120101220,f x xf x xf x x xxx120110120,.,.,.,kkkkf x xxf x xxf x x xxxx121112,.,.,.,kjkjkkjkjkkjkjkjkkkjf xxxf xxxf xxxxxx Newton Interpolate Polynomial012,.,kkaf x x xx101020130120()()()()()

6、()().()NNNkkPxaa xxaxxxxa xxxxxxaxx(),0,1,2,.,NkkPxy kNTheorem 3.6 定義則 滿足()NPxNewton Interpolate Polynomial300330030101331013012012332333()(),(),(),()()()f xf xf x xxxf x xf x xf x x xxxf x x xf x x xf x x x xxxf xP x我們以N=3為例來說明Theorem 3.6的證明思想。30010012010123012()(),(),()(),()()()P xf xf x xxxf x x

7、xxxxxf x x x xxxxxxx30031001101320012001220212330013001230310123303132()()()(),()()()(),(),()()()()(),(),()(),()()()(P xf xP xf xf x xxxf xP xf xf x xxxf x x xxxxxf xP xf xf x xxxf x x xxxxxf x x x xxxxxxxf x3)Exercise021010221010122120022000120012202132,(),()()(),()(),(),()()()f x xf x xf x x xxxf

8、x xf x x xxxf xf xf x xxxf xf x xxxf x x xxxxxP x誤差估計由于插值多項式的唯一性，按照Newton插值公式計算出來的多項式與按照Lagrangre插值公式計算出來的多項式相同，誤差也相同。(1)1()()()()()(1)!NNNNfExf xPxxN其中 。(,)a b均差與導數(shù)的關(guān)系(1)1()()()()()(1)!NNNNfExf xPxxN以N=3為例：000001011010120122012012301233()(),(),(),(),()f xf xf x xxxf x xf x xf x x xxxf x x xf x x xf

9、 x x x xxxf x x x xf x x x xf x x x x xxx00100120101230120123401234()(),(),()(),()()(),()(),()Nf xf xf x xxxf x x xxxxxf x x x xxxxxxxf x x x x xxPxf x x x x xx(4)0123(),4!ff x x x x x算法Example 3.12Example 3.13Chebyshev Polynomial(1)1()()()()()(1)!NNNNfExf xPxxN目標：調(diào)整節(jié)點，使得誤差估計達到最小！(1)111()()()()()(1)

10、!()(1)!NNNNNNfExf xPxxNMxN目標：調(diào)整節(jié)點，使得 最小！1()NxChebyshev PolynomialProperties of Chebyshev Polynomial定義：0112()1()()2()()kkkT xT xxT xxTxTxProperty 2：的首項系數(shù)為()kT x12kProperty 3（奇偶性）Property 3（三角表示）()cos(arccos()NTxNxcos(arccos()cos(2)arccos()2cos(1)arccos()cos(arccos()cos(arccos()2cos(1)arccos()cos(2)arccos()NxNxNxxNxxNxNxProperties of Chebyshev PolynomialMinMaxExample：等距節(jié)點的插值Example：Chebyshev節(jié)點的插值作業(yè)