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Newton Interpolate牛顿插值方法Numerical Methods1应用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如何计算Newton Polynomials101020130120()()()()()()().()NNNkkPxaa xxaxxxxa xxxxxxaxx4010201301240123433221100()()()()()()()()()()()()()()()()P xaa xxaxxxxa xxxxxxaxxxxxxxxaxxaxxaxxaxxa1110100().()NNNNNNSaSSxxaSSxxaNewton 插值数学问题Newton插值问题插值问题：已知在一组互异节点 上的函数值，求一个尽可能低的Newton多项式，使得：即：()(0,1,2,)iip xyinbxxxan.10插值问题的解是唯一的，区别仅是表达方式的不同！Lagrange插值多项式的优缺点1.当节点固定不变时，很容易计算多个不同点x出的Lagrange插值多项式的值。2.计算高阶（n）插值多项式，不能利用已计算出的低阶插值多项式。3.Newton插值方法是对Lagrange插值方法的一个补充。特别适合于计算一个点上的各种阶数的插值多项式的值。低阶Newton插值问题的解法n=0时：000()()Pxafxn=1时：1010()()Pxaaxx0001101()()()afxaaxxfx1010110()(),fxfxafxxxx2010201()()()()Pxaaxxaxxxxn=2时：00011010120220212()()()()()()()afxaaxxfxaaxxaxxxxfx001010110()()(),af xf xf xaf x xxx低级Newton插值问题的解法 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 定义则 满足()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我们以N=3为例来说明Theorem 3.6的证明思想。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误差估计由于插值多项式的唯一性，按照Newton插值公式计算出来的多项式与按照Lagrangre插值公式计算出来的多项式相同，误差也相同。(1)1()()()()()(1)!NNNNfExf xPxxN其中 。(,)a b均差与导数的关系(1)1()()()()()(1)!NNNNfExf xPxxN以N=3为例：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目标：调整节点，使得误差估计达到最小！(1)111()()()()()(1)!()(1)!NNNNNNfExf xPxxNMxN目标：调整节点，使得 最小！1()NxChebyshev PolynomialProperties of Chebyshev Polynomial定义：0112()1()()2()()kkkT xT xxT xxTxTxProperty 2：的首项系数为()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：等距节点的插值Example：Chebyshev节点的插值作业
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