Newton插值法詳解

1、數(shù)值分析實驗報告學號：姓名：_班級：_日期:題目：Newton插值法實驗環(huán)境:MATLAB 2014RaWin7旗艦版；實驗內(nèi)容與完成情況：1、編程實現(xiàn)求差商的算法。x 11 t22、設有函數(shù) 中(x) = f =e 2 dt的函數(shù)表如下 二2二x 0.00.10.20.30.4中(x) 0.50000.53980.57930.6179 0.7554(1)編程實現(xiàn)求Newton插值多項式的算法；(2)用Newton算法求邛(為)，為=0,0 + 0.01i, i =1,2,3L,40的近似值，統(tǒng)計計算時間，并與lagrange算法進行比較。1、算法:step 1 輸入x0,x1, xn,y1,

2、y2,丫川；step 2 對 j=0,1,2 n;di<-yi;step 3 對 k=0,1,2 n-1;dj=(dj-d(j-i)/(xi-x(j-k-1);step 4 輸出d0,d1,dn程序：(Matlab )建立自定義函數(shù)Chashang.mfunction f=Chashang(x,y,X)symst ;%定義符號變量t,進行公式的化簡和計算；n=length(x);%量向量x的長度，賦給n;m=length(y);%量向量y的長度，賦給m;ifm=n%ij斷m口 n是否相等，就是判斷 x與y是否一一對應；error('樣本數(shù)據(jù)中的x與y的對應個數(shù)不匹配);endA=

3、zeros(n,n);A(:,1)=y'forj=2:nfor i=1:(n-j+1)A(i,j)=(A(i+1,j-1)-A(i,j-1)/(x(i+j-1)-x(i); 公式，A(i,j)表示零矩陣的第j行，第i歹U;end end A編寫函數(shù)調(diào)用。x=;%y=;%X=;%f=Chashang(x,y,X)%2、利用1中的求差商的算法，%定義一個n行n列的零矩陣；咐巴向量y轉(zhuǎn)置，賦給零矩陣的第一列；%第一個循環(huán)，變量為j,用來表示第幾行；%第二個循環(huán)，變量為i ,用來表示第幾列；%差商%得到差商矩陣A;對應x值；已知的函數(shù)值；插值節(jié)點；調(diào)用函數(shù)得到差商值；把求出的差商矩陣的提取對角

4、線元素,利用秦九韶算法，得到牛頓插值多項式首先建立自定義函數(shù)Newton.mfunctionf=Newton(x,y,X)symst ;n=length(x);m=length(y);ifm=nerror('樣本數(shù)據(jù)中的end%定義符號變量t,進行公式的化簡和計算；%»量向量x的長度，賦給n;%»量向量y的長度，賦給m;%RJ斷mDn是否相等，就是判斷x與y是否，對應; x與y的對應個數(shù)不匹配,);A=zeros(n,n);A(:,1)=y'forj=2:nfor i=1:(n-j+1)%定義一個n行n列的零矩陣；咐巴向量y轉(zhuǎn)置，賦給零矩陣的第一列；%第一個

5、循環(huán)，變量為j,用來表示第幾行；%第二個循環(huán)，變量為i ,用來表示第幾列;%差商A(i,j)=(A(i+1,j-1)-A(i,j-1)/(x(i+j-1)-x(i);公式，A(i,j) 表示零矩陣的第j行，第i歹U;endendAf=A(1,1);forj=2:nT=1;for i=1:j-1T=T*(t-x(i);endf=f+A(1,j)*T;%輸出多項式；endf=vpa(f,2);%控制運算精度，不然結(jié)果的位數(shù)無法控制，或是輸出分數(shù);f=simplify(f)f=subs(f,'t',X)值函數(shù)值并輸出；然后編寫腳本文件ndcz.mclc%對表達式f進行化簡；%賦值函數(shù)

6、，用數(shù)值替代符號，用X代替t,計算差調(diào)用函數(shù)cleartic%始計時；x=0.0,0.1,0.2,0.3,0.4;y=0.5,0.5398,0.5793,0.6179,0.7554;X=0.0:0.01:0.4;f=Newton(x,y,X) toc%耗時間；調(diào)用程序運行時間消耗函數(shù)，第一次遇到tic開第一遇到toc利用之間的插值，求出程序運行消輸出結(jié)果:A=0.50000.53980.57930.39800.39500.3860-0.0150-0.04504.9450-0.100016.6333041.8333000.6179 1.37500.75540 f=41.8333333358168

7、60198974609375*tA4-25.20000000173943893364524038158 7*3+4.6166666671143805918639715309539*2+0.14649999995179435548593335003509*t + 0.5f=0.5,0.50190188499956100034371407515916,0.50458175999920145453262692962753,0.50790348499891182000280832282220.51174095999868315023677576806651,0.51597812499850709476

8、349453259003,0.52050895999837589915837763752862,0.52523748499828240504328585792434,0.53007775999822005008652772272554,0.5349538849981828680028595147868,0.53979999999816548855348527086896,0.5445602849981631375460567816391,0.54918895999817163683467359167056,0.55365028499818740431988299944293,0.5579185

9、5999820745394868005734203,0.56197812499822939571450757165996,0.56582335999825143565725610259505,0.56945868499827237586326396425189,0.57289855999829161446531722464131,0.57616748499830914564264970568039,0.57929999999832555962094298319248,0.58234068499834204267232638690715,0.585344159998360377115377000

10、46025,0.58837508499838294131511966139386,0.59150815999841270968302696115631,0.59482812499845325267701924510219,0.59842975999850873680146461249234,0.60241788499858392460717891649383,0.60690735999868417469142576418001,0.61202308499881544169791651653047,0.61789999999898427631681028843103,0.624683084999

11、19782528471394867379,0.63252735999946 383138468211995707,0.64159788499979063344621717888546,0.65206976000018716634526925596981,0.66412812500066296100423623562719,0.67796816000122814439196375618094,0.69379508500189343952374520986064,0.71182416000267016546132174280214,0.7322806850035702373128822550475

12、1,0.7554000000 046061662330634005451 時間已過0.269911 秒。拉格朗日插值程序自定義函數(shù) lagrange.mfunction y=lagrange(x0,y0,x);n=length(x0);m=length(x);fori=1:mz=x(i);s=0.0; for k=1:n p=1.0;for j=1:n if j=k p=p*(z-x0(j)/(x0(k)-x0Q);end end s=p*y0(k)+s;end y(i)=s;end腳本文件cz.mticx=0.0 0.1 0.2 0.3 0.4;y=0.5000 0.5398 0.5793

13、0.6179 0.7554;x0=0.0:0.01:0.4;y0=lagrange(x,y,x0) toc輸出結(jié)果：y0 = Columns 1 through 130.50000.5019 0.50460.5079 0.51170.51600.52050.5252 0.53010.53500.53980.5446 0.5492Columns 14 through 260.55370.55790.56200.56580.56950.57290.57620.57930.58230.58530.58840.59150.5948Columns 27 through 390.59840.60240.6

14、0690.61200.61790.62470.63250.64160.65210.66410.67800.69380.7118Columns 40 through 410.73230.7554時間已過0.003052秒。牛頓差值與拉格朗日插值所得插值函數(shù)值對比 程序：fND1=0.5,0.50190188499956100034371407515916,0.50458175999920145453262692962753, 0.50790348499891182000280832282220.51174095999868315023677576806651,0.5159781249985070

15、9476349453259003,0.52050895999837589915837763752862,0.52523748499828240504328585792434,0.53007775999822005008652772272554,0.5349538849981828680028595147868,0.53979999999816548855348527086896,0.5445602849981631375460567816391,0.54918895999817163683467359167056,0.55365028499818740431988299944293,0.557

16、91855999820745394868005734203,0.56197812499822939571450757165996,0.56582335999825143565725610259505,0.56945868499827237586326396425189,0.57289855999829161446531722464131,0.57616748499830914564264970568039,0.57929999999832555962094298319248,0.58234068499834204267232638690715,0.58534415999836037711537

17、700046025,0.58837508499838294131511966139386,0.59150815999841270968302696115631,0.59482812499845325267701924510219,0.59842975999850873680146461249234,0.60241788499858392460717891649383,0.60690735999868417469142576418001,0.61202308499881544169791651653047,0.61789999999898427631681028843103,0.62468308

18、499919782528471394867379,0.63252735999946383138468211995707,0.64159788499979063344621717888546,0.65206976000018716634526925596981,0.66412812500066296100423623562719,0.67796816000122814439196375618094,0.69379508500189343952374520986064,0.71182416000267016546132174280214,0.7322806850035702373128822550

19、4751, 0.7554000000046061662330634005451;fLA2= 0.50000.5019 0.50460.5079 0.5117 0.51600.52050.5252 0.53010.5350 0.5398 0.5446 0.5492 0.55370.55790.5620 0.56580.5695 0.57290.5762 0.57930.58230.58530.58840.59150.59480.59840.60240.60690.61200.61790.62470.63250.64160.65210.66410.6780 0.69380.7118 0.73230.7554;x=0.0:0.01:0.4;plot(x,fND1,'r*' ,x,fLA2,'ko')hh=ylabel( '差值函數(shù)值（）(x);hh=xlabel( '差值節(jié)點）；hh=title('拉格朗日差值與牛頓差值函數(shù)值對比圖);hh=legend( '牛頓差值函數(shù)值',拉格朗日差值函數(shù)值);強配專值與領(lǐng)全酗卻和比國愛國吊點結(jié)果分析：(1)通過時間比較