數(shù)值分析課程設(shè)計-多項式插值的振蕩現(xiàn)象matlab(共23頁)

1、精選優(yōu)質(zhì)文檔-傾情為你奉上數(shù)值分析 課程設(shè)計多項式插值的振蕩現(xiàn)象（姓名）（學(xué)號）指導(dǎo)教師 學(xué)院名稱專 業(yè) 名 稱提交日期2012年6月一、 問題的提出考慮在一個固定區(qū)間上用插值逼近一個函數(shù)。顯然，Lagrange插值中使用的節(jié)點越多，插值多項式的次數(shù)就越高。我們自然關(guān)心插值多項式增加時，Ln(x)是否也更加靠近被逼近的函數(shù)。龍格(Runge)給出的一個例子是極著名并富有啟發(fā)性的。設(shè)區(qū)間-1,1上的函數(shù)考慮區(qū)間-1,1的一個等距劃分，節(jié)點為則拉格朗日插值多項式為其中的ai(x),i=0,1,2,n是n次Lagrange插值基函數(shù)。二、 實驗內(nèi)容研究以下三個函數(shù)在各自區(qū)間上運用不同的劃分1、2、3

2、、運用在區(qū)間-p,p上等距劃分（p>0）,節(jié)點為以x0,x1,xn為插值節(jié)點構(gòu)造上述各函數(shù)的Lagrange插值多項式。運用區(qū)間a,b上切比雪夫(Chebychev)點的定義為以x1,x2,xn+1為插值節(jié)點構(gòu)造上述各函數(shù)的Lagrange插值多項式，比較其結(jié)果。并分別比較兩種劃分方法，增加節(jié)點數(shù)，最大誤差的變化。三、 實驗結(jié)果及分析（一） 等距劃分對于函數(shù)來說，使用等距劃分其中綠色點線代表誤差，紅色劃線代表Lagrange插值多項式，藍色實線代表原函數(shù)?？梢妼τ诘染鄤澐謥碚f節(jié)點數(shù)越多，最大誤差越大，可是越靠近中間的誤差越少。越接近兩個端點的誤差越大。當節(jié)點數(shù)很大時，最大誤差的來源只與靠

3、近兩個端點的誤差有關(guān)。例如：n=20時對于，使用等距劃分當n=25時對于，使用等距劃分當n=30時，從以上三個函數(shù)圖像來看，可見節(jié)點越多，靠近端點處取得最大誤差，并且最大誤差越大。這就是龍格現(xiàn)象。（二） 切比雪夫(Chebychev)點首先研究 當n=20時可是當n=30同樣的n=40，50也出現(xiàn)了兩端誤差增大現(xiàn)象。 然后研究 當n=25時同樣的n=30，50也會出現(xiàn)龍格現(xiàn)象。最后研究當n=25時 此函數(shù)也不例外。由以上三個函數(shù)，通過不斷改變n的值，可得運用切比雪夫點來劃分，要使最大誤差小于0.1，n一般取12到20間的數(shù)。（三） 綜述對于Lagrange插值多項式，運用等距劃分取節(jié)點時，n要

4、不能取得太大，一般n=8或8左右。若要使n取得更大，那么需要使代入到函數(shù)的點盡量在區(qū)間的中間，這樣才能使真值與計算出來的值的誤差盡可能的小。運用切比雪夫點來劃分取節(jié)點時，n一般取12到20間的數(shù)。無論要代入到函數(shù)的點在區(qū)間的那個位置，都能使誤差盡可能的小。若要使n取得更大，那么需要使代入到函數(shù)的點盡量在區(qū)間的中間，這樣才能使真值與計算出來的值的誤差盡可能的小。若n取的較小，同樣誤差也是很大的。對于以上兩種取節(jié)點的方法，都不能避免龍格現(xiàn)象的出現(xiàn)。對于不同的選取節(jié)點的方法，只要n取得合適，同時代人函數(shù)的點適合，那么就可以使誤差盡可能的減少。四、 關(guān)于本設(shè)計的體會為了完成本設(shè)計，接觸了一個數(shù)學(xué)軟件m

5、atlab，并能初步運用本軟件，編寫程序，方便了以后對數(shù)學(xué)的研究。并且對于編寫過程中遇到的錯誤，顯示不正確等，通過網(wǎng)絡(luò)搜索查詢，能解決這些問題。例如運用fminbnd函數(shù)時，顯示的并不是正確的答案，后來百度了一下，明白了此函數(shù)的運作原理，并最終能顯示我想得到的結(jié)果。當然對于我所編寫的程序，我也就只能做到這一步，也是礙于時間關(guān)系，和對matlab不熟悉，并不能更好的完善下去。有時取值很大時，運行起來，運行得很慢，這是一大缺陷。主要是循環(huán)次數(shù)太多，和找不到更好的函數(shù)代替所造成的。其他的缺點也就不一一的列舉了。學(xué)習(xí)一個軟件需要耐心，細心。多尋找，多發(fā)現(xiàn)，多創(chuàng)作才能深入了解一個軟件。同樣的道理，這也是

6、做任何事所需要的素質(zhì)。五、 參考文獻數(shù)值分析（第三版）北京理工大學(xué)出版社六、 附錄所的運用軟件及編號MATLAB 7.0（圖略） 所用電腦的版本、型號與系統(tǒng)（圖略）用戶界面設(shè)計文件名：kcsjx.fig用戶界面程序，運行kcsjx.m來調(diào)用主要程序文件名：kcsjx.mfunction varargout = kcsjx(varargin)% KCSJX M-file for kcsjx.fig% KCSJX, by itself, creates a new KCSJX or raises the existing% singleton*.% H = KCSJX returns the ha

7、ndle to a new KCSJX or the handle to% the existing singleton*.% KCSJX('CALLBACK',hObject,eventData,handles,.) calls the local% function named CALLBACK in KCSJX.M with the given input arguments.% KCSJX('Property','Value',.) creates a new KCSJX or raises the% existing singleton

8、*. Starting from the left, property value pairs are% applied to the GUI before kcsjx_OpeningFunction gets called. An% unrecognized property name or invalid value makes property application% stop. All inputs are passed to kcsjx_OpeningFcn via varargin.% *See GUI Options on GUIDE's Tools menu. Cho

9、ose "GUI allows only one% instance to run (singleton)".% See also: GUIDE, GUIDATA, GUIHANDLES% Copyright 2002-2003 The MathWorks, Inc.% Edit the above text to modify the response to help kcsjx% Last Modified by GUIDE v2.5 04-Jun-2012 16:03:46% Begin initialization code - DO NOT EDITgui_Sin

10、gleton = 1;gui_State = struct('gui_Name', mfilename, . 'gui_Singleton', gui_Singleton, . 'gui_OpeningFcn', kcsjx_OpeningFcn, . 'gui_OutputFcn', kcsjx_OutputFcn, . 'gui_LayoutFcn', , . 'gui_Callback', );if nargin && ischar(varargin1) gui_State.g

11、ui_Callback = str2func(varargin1);endif nargout varargout1:nargout = gui_mainfcn(gui_State, varargin:);else gui_mainfcn(gui_State, varargin:);end% End initialization code - DO NOT EDIT% - Executes just before kcsjx is made visible.function kcsjx_OpeningFcn(hObject, eventdata, handles, varargin)% Thi

12、s function has no output args, see OutputFcn.% hObject handle to figure% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GUIDATA)% varargin command line arguments to kcsjx (see VARARGIN)% Choose default command line output for kcsjx

13、handles.output = hObject;% Update handles structureguidata(hObject, handles);% UIWAIT makes kcsjx wait for user response (see UIRESUME)% uiwait(handles.figure1);% - Outputs from this function are returned to the command line.function varargout = kcsjx_OutputFcn(hObject, eventdata, handles) % varargo

14、ut cell array for returning output args (see VARARGOUT);% hObject handle to figure% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GUIDATA)% Get default command line output from handles structurevarargout1 = handles.output;function

15、 edit1_Callback(hObject, eventdata, handles)% hObject handle to edit1 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GUIDATA)% Hints: get(hObject,'String') returns contents of edit1 as text% str2double(get(hObje

16、ct,'String') returns contents of edit1 as a doubleinput = str2num(get(hObject,'String');if (isempty(input) set(hObject,'String','0')endguidata(hObject, handles); % - Executes during object creation, after setting all properties.function edit1_CreateFcn(hObject, eventd

17、ata, handles)% hObject handle to edit1 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles empty - handles not created until after all CreateFcns called% Hint: edit controls usually have a white background on Windows.% See ISPC and COMPUTER.if ispc set(hObject,'

18、BackgroundColor','white');else set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor');endfunction edit2_Callback(hObject, eventdata, handles)% hObject handle to edit2 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles str

19、ucture with handles and user data (see GUIDATA)% Hints: get(hObject,'String') returns contents of edit2 as text% str2double(get(hObject,'String') returns contents of edit2 as a doubleinput = str2num(get(hObject,'String'); if (isempty(input) set(hObject,'String','0

20、') endguidata(hObject, handles); % - Executes during object creation, after setting all properties.function edit2_CreateFcn(hObject, eventdata, handles)% hObject handle to edit2 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles empty - handles not created unti

21、l after all CreateFcns called% Hint: edit controls usually have a white background on Windows.% See ISPC and COMPUTER.if ispc set(hObject,'BackgroundColor','white');else set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor');endfunction edit3_Callback(

22、hObject, eventdata, handles)% hObject handle to edit3 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GUIDATA)% Hints: get(hObject,'String') returns contents of edit3 as text% str2double(get(hObject,'String&#

23、39;) returns contents of edit3 as a doubleinput = str2num(get(hObject,'String'); if (isempty(input) set(hObject,'String','0') endguidata(hObject, handles); % - Executes during object creation, after setting all properties.function edit3_CreateFcn(hObject, eventdata, handles)%

24、 hObject handle to edit3 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles empty - handles not created until after all CreateFcns called% Hint: edit controls usually have a white background on Windows.% See ISPC and COMPUTER.if ispc set(hObject,'BackgroundColo

25、r','white');else set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor');end% - Executes on button press in pushbutton1.function pushbutton1_Callback(hObject, eventdata, handles)% hObject handle to pushbutton1 (see GCBO)% eventdata reserved - to be defined

26、in a future version of MATLAB% handles structure with handles and user data (see GUIDATA)tic;v=get(handles.edit1,'String');t=get(handles.edit2,'String');n=get(handles.edit3,'String');kcsj(str2num(t),str2num(v),str2num(n);toc;clear;% - Executes on button press in pushbutton2.f

27、unction pushbutton2_Callback(hObject, eventdata, handles)% hObject handle to pushbutton2 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GUIDATA)tic;v=get(handles.edit1,'String');t=get(handles.edit2,'String&#

28、39;);kcsj(str2num(t),str2num(v);toc;clear;以下為主要程序。并且可以直接在命令框調(diào)用例如：在命令框輸入>> kcsj(1,1,3) %代表選擇等距劃分，選擇f(x)=1/(1+25*x2)輸出結(jié)果為：以下有具體說明。主程序文件名：kcsj.mfunction kcsj(t,v,n)%t為對不同x選擇的判斷，t=0時為等距劃分，t=1時為Chebychev劃分%v=1是選擇f(x)=1/(1+25*x2);v=2是f(x)=x/(1+x4);v=3是f(x)=atan(x)% 函數(shù)的選取%if v=1 %題目要求不同函數(shù),x有不同的取值范圍 p

29、=1; %x取值范圍為-1,1 f='1/(1+25*x2)'else if v=2 p=5; %x取值范圍為-5,5 f='x/(1+x4)' else if v=3 p=5; f='atan(x)' end endendif t=0 P='等距劃分'else P='Chebychev'endF=inline(f); %inline為將用字符串表示的函數(shù)表達式化為函數(shù)表達式%以下為不輸入n值得情況%if nargin=2 for n=2:3:17 %節(jié)點數(shù)由2至11進行循環(huán)操作 X=zeros(n+1,1); %

30、有n+1個節(jié)點 Y=zeros(n+1,1); %對應(yīng)有n+1個插值% 選擇節(jié)點選取方式 % if t=0 for i=0:n X(i+1,1)=-p+2*p*i/n; %等距劃分 end % else % for i=1:n+1 % X(i,1)=p*cos(2*i-1)*pi/(2*(n+1);%Chebychev劃分 end end% 計算差值 % for i=0:n Y(i+1,1)=F(X(i+1,1); end% 通過差值與節(jié)點得出Lagrange公式 % Pn=Lagrange3(X,Y); %調(diào)用拉格朗日函數(shù)進行 Ln=inline(Pn);% 計算原函數(shù)與Ln兩個函數(shù)間的最大

31、誤差 % maxe='-abs(',char(Pn),'-',f,')' max=inline('abs(',char(Pn),'-',f,')'); maxe=fmin(maxe,X); %調(diào)用fmin.m查找函數(shù)最小值 maxe=abs(maxe); %得出最大誤差% 畫出函數(shù)圖像 % figure(v+t*3); %圖像窗口的編號 subplot(2,3,(n+1)/3); %subplot用于在同一窗口什么位置顯示圖像 fplot(f,-p,p,'c.-'); %畫出原函數(shù)

32、if (n+1)/3)=2 %添加適當?shù)臉祟} title(f,P); end hold on %多個函數(shù)在同一個圖像表達 fplot(Ln,-p,p); %畫出拉格朗日圖像 xlabel('n=',num2str(n),'最大誤差=',num2str(maxe);%在x軸標注n節(jié)點的取值與最大誤差 fplot(max,-p,p,'r:'); %畫出誤差圖像 hold off end%以下為輸入n值得情況%else if nargin=3 X=zeros(n+1,1); %有n+1個節(jié)點 Y=zeros(n+1,1); %對應(yīng)有n+1個插值% 選擇

33、節(jié)點選取方式 % if t=0 for i=0:n X(i+1,1)=-p+2*p*i/n; %等距劃分 end % else % for i=1:n+1 % X(i,1)=p*cos(2*i-1)*pi/(2*(n+1);%Chebychev劃分 end end% 計算差值 % for i=0:n Y(i+1,1)=F(X(i+1,1); end% 通過差值與節(jié)點得出Lagrange公式 % Pn=Lagrange3(X,Y); %調(diào)用拉格朗日函數(shù)進行 Ln=inline(Pn);% 計算原函數(shù)與Ln兩個函數(shù)間的最大誤差 % maxe='-abs(',char(Pn),'-',f,')' max=inline('abs(',char(Pn),'-',f,')'); maxe=fmin(maxe,X); %調(diào)用fmin.m查找函數(shù)最小值 maxe=abs(maxe); %得出最大誤差% 畫出函數(shù)圖像 % figure(v+t*3); %圖像窗口的編號 subplot(1,1,1); fplot(f,-p,