《L數值分析ecture 7【教學類別】》由會員分享,可在線閱讀,更多相關《L數值分析ecture 7【教學類別】(23頁珍藏版)》請在裝配圖網上搜索。
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均差與導數的關系(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:調整節(jié)點,使得誤差估計達到最小!(1)111()()()()()(1)
10、!()(1)!NNNNNNfExf xPxxNMxN目標:調整節(jié)點,使得 最??!1()NxChebyshev PolynomialProperties of Chebyshev Polynomial定義:0112()1()()2()()kkkT xT xxT xxTxTxProperty 2:的首項系數為()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)