Newton Interpolate牛頓插值方法Numerical Methods1應(yīng)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 the nodes 。0121,.,NNx x xxx如何計(jì)算Newton Polynomials101020130120()()()()()()().()NNNkkPxaa xxaxxxxa xxxxxxaxx4010201301240123433221100()()()()()()()()()()()()()()()()P xaa xxaxxxxa xxxxxxaxxxxxxxxaxxaxxaxxaxxa1110100().()NNNNNNSaSSxxaSSxxaNewton 插值數(shù)學(xué)問(wèn)題Newton插值問(wèn)題插值問(wèn)題：已知在一組互異節(jié)點(diǎn) 上的函數(shù)值，求一個(gè)盡可能低的Newton多項(xiàng)式，使得：即：()(0,1,2,)iip xyinbxxxan.10插值問(wèn)題的解是唯一的，區(qū)別僅是表達(dá)方式的不同！Lagrange插值多項(xiàng)式的優(yōu)缺點(diǎn)1.當(dāng)節(jié)點(diǎn)固定不變時(shí)，很容易計(jì)算多個(gè)不同點(diǎn)x出的Lagrange插值多項(xiàng)式的值。2.計(jì)算高階（n）插值多項(xiàng)式，不能利用已計(jì)算出的低階插值多項(xiàng)式。3.Newton插值方法是對(duì)Lagrange插值方法的一個(gè)補(bǔ)充。特別適合于計(jì)算一個(gè)點(diǎn)上的各種階數(shù)的插值多項(xiàng)式的值。低階Newton插值問(wèn)題的解法n=0時(shí)：000()()Pxafxn=1時(shí)：1010()()Pxaaxx0001101()()()afxaaxxfx1010110()(),fxfxafxxxx2010201()()()()Pxaaxxaxxxxn=2時(shí)：00011010120220212()()()()()()()afxaaxxfxaaxxaxxxxfx001010110()()(),af xf xf xaf x xxx低級(jí)Newton插值問(wèn)題的解法 201202202110202010202120101020202110211010220()()()()()()()()()1()()()()()1()()()()(1f xaa xxaxxxxf xf xf xf xxxxxxxxxf xf xxxf xf xxxxxxxxxf xf xxxf xf xxxxx121101021202110120101220)()()()()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()()()()()()().()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(lái)說(shuō)明Theorem 3.6的證明思想。30010012010123012()(),(),()(),()()()P xf xf x xxxf x x 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 x xf x x xxxf xf xf x xxxf xf x xxxf x x xxxxxP x誤差估計(jì)由于插值多項(xiàng)式的唯一性，按照Newton插值公式計(jì)算出來(lái)的多項(xiàng)式與按照Lagrangre插值公式計(jì)算出來(lái)的多項(xiàng)式相同，誤差也相同。(1)1()()()()()(1)!NNNNfExf xPxxN其中 。(,)a b均差與導(dǎo)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 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(biāo)：調(diào)整節(jié)點(diǎn)，使得誤差估計(jì)達(dá)到最??！(1)111()()()()()(1)!()(1)!NNNNNNfExf xPxxNMxN目標(biāo)：調(diào)整節(jié)點(diǎn)，使得 最?。?()NxChebyshev PolynomialProperties of Chebyshev Polynomial定義：0112()1()()2()()kkkT xT xxT xxTxTxProperty 2：的首項(xiàng)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(diǎn)的插值Example：Chebyshev節(jié)點(diǎn)的插值作業(yè)