1、課程設計實驗報告 學院：數學與信息科學與學院 專業：信息與計算科學專業姓名：宋寶學號：201014412一、 理論基礎：本次課程設計做兩個方面的設計：1. 插值1.1：參考數值分析（第5版）上面關于拉格朗日、牛頓、三次樣條插值算法的介紹結合自己所學的MATLAB程序設計，將其程序化；通過相關matlab GUI資料的介紹設計出可視化界面進行插值的相關操作。2. 解線性方程組2.1：參考數值分析（第5版）和有關接線性方程組的資料研究里面關于列主元消元法、多利特勒分解法、改進平方根法、QR分解法、雅可比迭代法、高斯-賽德爾迭代法、超松弛迭代法解線性方程組的具體步驟，通過matlab語言編程實現，并

2、通過GUI設計實現可視化操作。二、 算法結構1. 插值：拉格朗日插值算法描述：首先求出插值節點數據的長度，然后將賦值為0，然后計算 ，通過for循環將相加賦值給，得到拉格朗日插值函數，最后通過MATLAB的化簡多項式相關的函數對進行化簡。牛頓插值算法描述：首先求出一階差商，然后以上一次差商的結果作為，求出二階、三階等更高階差商，運用下面公式即可求出牛頓插值多項式三次樣條插值算法描述：首先構造滿足條件的三次樣條插值函數，將數據結點的值帶入到插值函數中得到該函數。2. 解線性方程組：列主元消元法算法描述：使用高斯消元法，每次消元前，先求出到的絕對值最大的，然后矩陣的第行和第行進行交換，然后消元，最

3、后解系數矩陣是上三角矩陣的方程組，得出解。直接三角分解法算法描述：將 分解為，其中 1. 2.計算的第行和的第列元素3. 解系數矩陣是下三角矩陣的方程組，得出解，然后解系數矩陣是上三角矩陣的方程組，解出方程組的解改進平方根法算法描述：將分解為，其中為下三角矩陣，對角元素為1，為對角矩陣，通過矩陣之間的乘法求出和；然后解系數矩陣是下三角矩陣的方程組，得出解，然后方程組，解出方程組的解QR分解法算法描述：對線性方程組的系數矩陣進行QR分解：，方程變為其為三角方程，很容易求解解法求解。雅可比迭代法算法描述：首先將的系數矩陣進行分解：，利用公式，；解的計算公式為：高斯-賽德爾迭代法算法描述：首先將的系

4、數矩陣進行分解：，利用公式，；解的計算公式為：超松弛迭代法算法描述：首先將的系數矩陣進行分解：，利用公式，；解的計算公式為：三、 程序代碼插值MATLAB代碼：function varargout = chazhiGUI(varargin)gui_Singleton = 1;gui_State = struct('gui_Name', mfilename, . 'gui_Singleton', gui_Singleton, . 'gui_OpeningFcn', chazhiGUI_OpeningFcn, . 'gui_OutputFcn

5、', chazhiGUI_OutputFcn, . 'gui_LayoutFcn', , . 'gui_Callback', );if nargin && ischar(varargin1) gui_State.gui_Callback = str2func(varargin1);end if nargout varargout1:nargout = gui_mainfcn(gui_State, varargin:);else gui_mainfcn(gui_State, varargin:);end% End initializatio

6、n code - DO NOT EDITglobal a x y t xx yy ff% - Executes just before chazhiGUI is made visible.function chazhiGUI_OpeningFcn(hObject, eventdata, handles, varargin)handles.output = hObject; % Update handles structureguidata(hObject, handles); % UIWAIT makes chazhiGUI wait for user response (see UIRESU

7、ME)% uiwait(handles.figure1);% - Outputs from this function are returned to the command line.function varargout = chazhiGUI_OutputFcn(hObject, eventdata, handles) varargout1 = handles.output; function edit1_shuju_Callback(hObject, eventdata, handles)% - Executes during object creation, after setting

8、 all properties.function edit1_shuju_CreateFcn(hObject, eventdata, handles)if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end % - Executes on selection change in listbox1.funct

9、ion listbox1_Callback(hObject, eventdata, handles)% - Executes during object creation, after setting all properties.function listbox1_CreateFcn(hObject, eventdata, handles)if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,

10、'BackgroundColor','white');end % - Executes on selection change in listbox2.function listbox2_Callback(hObject, eventdata, handles)% - Executes during object creation, after setting all properties.function listbox2_CreateFcn(hObject, eventdata, handles)% Hint: listbox controls usuall

11、y have a white background on Windows.% See ISPC and COMPUTER.if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end% - Executes on button press in pushbutton1. function x,y,a=pushb

12、utton1_Callback(hObject, eventdata, handles)%獲取文本框內容a = get(handles.edit1_shuju,'String');b=str2num(a);x=b(1,:);y=b(2,:);handles.a=a;handles.x=x;handles.y=y;t=handles.t;val1=t;if val1=1 f=Language(x,y);else if val1=2 f=Newton(x,y); else if val1=3 n=length(x); xx=x(1):0.01:x(n); yy=interp1(x,

13、y,xx,'spline'); end endendif val1=3ff=fhandles.ff=ff;n=length(x);xx=x(1):0.01:x(n);t=xx;yy=subs(f,t);endhandles.xx=xx;handles.yy=yy;guidata(hObject, handles); % - Executes on button press in pushbutton3.function pushbutton3_Callback(hObject, eventdata, handles)% hObject handle to pushbutton3

14、 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GUIDATA)t=handles.t;val1=t;if val1=3ff=handles.ff;gg=sym2poly(ff);hh=num2str(gg);set(handles. edit3_jieguo,'String',hh); else set(handles. edit3_jieguo,'String

15、','太復雜，在此不給出！'); end guidata(hObject, handles); %更新結構體% - Executes on button press in pushbutton4.function pushbutton4_Callback(hObject, eventdata, handles)% hObject handle to pushbutton4 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with

16、 handles and user data (see GUIDATA)% - Executes on button press in pushbutton5.function pushbutton5_Callback(hObject, eventdata, handles)% hObject handle to pushbutton5 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GU

17、IDATA)close;% - Executes on button press in pushbutton6.function pushbutton6_Callback(hObject, eventdata, handles)function edit3_jieguo_Callback(hObject, eventdata, handles)% - Executes during object creation, after setting all properties.function edit3_jieguo_CreateFcn(hObject, eventdata, handles)%

18、 hObject handle to edit3_jieguo (see GCBO)% See ISPC and COMPUTER.if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end % - Executes on selection change in popupmenu1.function pop

19、upmenu1_Callback(hObject, eventdata, handles)% hObject handle to popupmenu1 (see GCBO)function popupmenu1_CreateFcn(hObject, eventdata, handles)% See ISPC and COMPUTER.if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'

20、;BackgroundColor','white');end % - Executes on selection change in popupmenu2.function t=popupmenu2_Callback(hObject, eventdata, handles)% hObject handle to popupmenu2 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user da

21、ta (see GUIDATA)list=get(handles.popupmenu2,'String');val1=get(handles.popupmenu2,'Value');t=val1;handles.t=t;selectedval=str2num(listval1);guidata(hObject, handles);function popupmenu2_CreateFcn(hObject, eventdata, handles)% See ISPC and COMPUTER.if ispc && isequal(get(hObje

22、ct,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end %拉格朗日插值法function f=Language(x,y)syms t;if(length(x)=length(y) n=length(x);else disp('x和y的維數不相等！'); return;endf=0.0;for(i=1:n) l=y(i); for(j=1:i-1) l=l*(t

23、-x(j)/(x(i)-x(j); end for(j=i+1:n) l=l*(t-x(j)/(x(i)-x(j); end f=f+l; simplify(f);end%simple(f)k=expand(f);f=k;f=vpa(k); %牛頓插值function f=Newton(x,y)syms t;if(length(x)=length(y) n=length(x); c(1:n)=0.0;else disp(' x和y的維數不相等！'); return;endf=y(1);y1=0;l=1;for(i=1:n-1) for(j=i+1:n) y1(j)=(y(j)-

24、y(i)/(x(j)-x(i); end c(i)=y1(i+1); l=l*(t-x(i); f=f+c(i)*l; simplify(f); y=y1;endk=expand(f);f=k;f=vpa(k);% - Executes on button press in pushbutton7.function pushbutton7_Callback(hObject, eventdata, handles)% hObject handle to pushbutton7 (see GCBO)x=handles.x;y=handles.y;plot(x,y,'o');hold

25、 on;xx=handles.xx;yy=handles.yy;plot(xx,yy,'b-');% - Executes on button press in pushbutton8.function pushbutton8_Callback(hObject, eventdata, handles)% hObject handle to pushbutton8 (see GCBO)hold off;newplot;解線性方程組MATLAB代碼：function varargout = jiefangchengGUI(varargin)gui_Singleton = 1;gui

26、_State = struct('gui_Name', mfilename, . 'gui_Singleton', gui_Singleton, . 'gui_OpeningFcn', jiefangchengGUI_OpeningFcn, . 'gui_OutputFcn', jiefangchengGUI_OutputFcn, . 'gui_LayoutFcn', , . 'gui_Callback', );if nargin && ischar(varargin1) gui_S

27、tate.gui_Callback = str2func(varargin1);endif nargout varargout1:nargout = gui_mainfcn(gui_State, varargin:);else gui_mainfcn(gui_State, varargin:);end% End initialization code - DO NOT EDITglobal t% - Executes just before jiefangchengGUI is made visible.function jiefangchengGUI_OpeningFcn(hObject,

28、eventdata, handles, varargin)handles.output = hObject;% Update handles structureguidata(hObject, handles); function varargout = jiefangchengGUI_OutputFcn(hObject, eventdata, handles) varargout1 = handles.output; function edit1_juzhen_Callback(hObject, eventdata, handles)function edit1_juzhen_CreateF

29、cn(hObject, eventdata, handles)% hObject handle to edit1_juzhen (see GCBO)if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end function edit2_Callback(hObject, eventdata, handles

30、)function edit2_CreateFcn(hObject, eventdata, handles)% hObject handle to edit2 (see GCBO)% See ISPC and COMPUTER.if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end function ed

31、it3_Callback(hObject, eventdata, handles)function edit3_CreateFcn(hObject, eventdata, handles)% hObject handle to edit3 (see GCBO)if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white')

32、;end function edit4_chuzhi_Callback(hObject, eventdata, handles)% hObject handle to edit4_chuzhi (see GCBO)function edit4_chuzhi_CreateFcn(hObject, eventdata, handles)% hObject handle to edit4_chuzhi (see GCBO)% See ISPC and COMPUTER.if ispc && isequal(get(hObject,'BackgroundColor'),

33、 get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end function edit5_Callback(hObject, eventdata, handles)% - Executes during object creation, after setting all properties.function edit5_CreateFcn(hObject, eventdata, handles)% See ISPC and COMPU

34、TER.if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end function edit6_songci_Callback(hObject, eventdata, handles)function edit6_songci_CreateFcn(hObject, eventdata, handles)if

35、 ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end function edit7_Callback(hObject, eventdata, handles)function edit7_CreateFcn(hObject, eventdata, handles)if ispc && ise

36、qual(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end function edit8_jingdu_Callback(hObject, eventdata, handles)% - Executes during object creation, after setting all properties.function edit8_jingdu_

37、CreateFcn(hObject, eventdata, handles)% See ISPC and COMPUTER.if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end function edit10_result_Callback(hObject, eventdata, handles)fun

38、ction edit10_result_CreateFcn(hObject, eventdata, handles)if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end % - Executes on selection change in listbox1.function listbox1_Call

39、back(hObject, eventdata, handles)function listbox1_CreateFcn(hObject, eventdata, handles)% See ISPC and COMPUTER.if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor') set(hObject,'BackgroundColor','white');end% - Executes

40、on button press in pushbutton4_jisuan.function pushbutton4_jisuan_Callback(hObject, eventdata, handles)% hObject handle to pushbutton4_jisuan (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GUIDATA)% 獲取文本框內容a = get(handl

41、es.edit1_juzhen,'String');b = get(handles.edit4_chuzhi,'String');c = get(handles.edit6_songci,'String');d = get(handles.edit8_jingdu,'String');aa=str2num(a);x0=str2num(b);x0=x0'w=str2num(c);eps=str2num(d);n=length(aa);A=aa(:,1:n-1);b=aa(:,n);t=handles.t;val1=t;swi

42、tch(val1) case 1 x,XA=GaussXQLineMain(A,b); case 2 x,L,U=Doolittle(A,b); case 3 msgbox( '方程組的系數矩陣必須對稱正定' ,'warning!','help'); x,L,D=Sympos3 (A,b); case 4 x,Q,R=qrxq(A,b); case 5 x,n=jacobi(A,b,x0,eps); case 6 x,n=gauseidel(A,b,x0,eps); case 7 x,n=SOR(A,b,x0,w,eps); endhh=num2

43、str(x');set(handles. edit10_result,'String',hh);if(val1=5|val1=6|val1=7) hh=num2str(n);set(handles. edit11_cishu,'String',hh);else set(handles. edit11_cishu,'String',' '); end guidata(hObject, handles); %更新結構體 % - Executes on button press in pushbutton5.function p

44、ushbutton5_Callback(hObject, eventdata, handles)% hObject handle to pushbutton5 (see GCBO)% eventdata reserved - to be defined in a future version of MATLAB% handles structure with handles and user data (see GUIDATA) msgbox('作者：宋寶',. '學院：數學與信息科學學院', .'專業：信息與計算科學專業', . '學號

45、：201014412', . '水平有限，程序簡陋', . '不足之處，敬請原諒' ,'作者信息','help'); % - Executes on button press in pushbutton6.function pushbutton6_Callback(hObject, eventdata, handles)% hObject handle to pushbutton6 (see GCBO)% eventdata reserved - to be defined in a future version of M

46、ATLAB% handles structure with handles and user data (see GUIDATA)close;%列主元消元法function x,XA=GaussXQLineMain(A,b)N=size(A);n=N(1);index=0;for i=1:(n-1) me=max(abs(A(1:n,i); for k=i:n if(abs(A(k,i)=me) index=k; end end temp=A(i,1:n); A(i,1:n)=A(index,1:n); A(index,1:n)=temp; bb=b(index); b(index)=b(i)

47、; b(i)=bb; for j=(i+1):n if(A(i,i)=0) disp('對角元素為0！'); return; end l=A(j,i); m=A(i,i); A(j,1:n)=A(j,1:n)-l*A(i,1:n)/m; b(j)=b(j)-l*b(i)/m; endendfor i=n:-1:1 if(i<n) s=A(i,(i+1):n)*x(i+1):n,1); else s=0; end x(i,1)=(b(i)-s)/A(i,i);endXA=A;%直接法function x,L,U=Doolittle(A,b)N=size(A);n=N(1);

48、L=eye(n,n);U=zeros(n,n);U(1,1:n)=A(1,1:n);L(1:n,1)=A(1:n,1)/U(1,1);for k=2:n for i=k:n U(k,i)=A(k,i)-L(k,1:(k-1)*U(1:(k-1),i); end for j=(k+1):n L(j,k)=(A(j,k)-L(j,1:(k-1)*U(1:(k-1),k)/U(k,k); endendN=size(L);n=N(1); for i=1:n if(i>1) s=L(i,1:(i-1)*y(1:(i-1),1); else s=0; end y(i,1)=(b(i)-s)/L(i,

49、i); end N=size(U);n=N(1);for i=n:-1:1 if(i<n) s=U(i,(i+1):n)*x(i+1):n,1); else s=0; end x(i,1)=(y(i)-s)/U(i,i);end%改進平方根法function x,L,D=Sympos3 (A,b)N=size(A);n=N(1);L=eye(n,n);d=zeros(n,1);AA=zeros(n,n);for k=1:n d(k,1)=A(k,k)-AA(k,1:(k-1)*transpose(L(k,1:(k-1); for i=(k+1):n AA(i,k)=A(i,k)-AA(i

50、,1:(k-1)*transpose(L(k,1:(k-1); L(i,k)=AA(i,k)/d(k,1); endendD=diag(d);N=size(L);n1=N(1);for i=1:n1 if(i>1) s=L(i,1:(i-1)*y(1:(i-1),1); else s=0; end y(i,1)=(b(i)-s)/L(i,i);endfor i=1:n y(i,1)=y(i,1)/d(i,1);endt=transpose(L);N=size(t);n=N(1);for i=n:-1:1 if(i<n) s=t(i,(i+1):n)*x(i+1):n,1); els

51、e s=0; end x(i,1)=(y(i)-s)/t(i,i);end%QR分解法function x,Q,R=qrxq(A,b)N=size(A);n=N(1);B=A;A(1:n,1)=A(1:n,1)/norm(A(1:n,1);for i=2:n for j=1:(i-1) A(1:n,i)=A(1:n,i)-dot(A(1:n,i),A(1:n,j)*A(1:n,j); end A(1:n,i)=A(1:n,i)/norm(A(1:n,i);endQ=A;R=transpose(Q)*B;k=transpose(Q)*b;N=size(R);n=N(1);for i=n:-1:1

52、 if(i<n) s=R(i,(i+1):n)*x(i+1):n,1); else s=0; end x(i,1)=(k(i)-s)/R(i,i);end%雅可比迭代法function x,n=jacobi(A,b,x0,eps,M)if nargin=3 eps=1.0e-6; M=10000;else if nargin=4 M=10000; endend D=diag(diag(A); L=-tril(A,-1); U=-triu(A,1); B=D(L+U); f=Db; x=x0; n=0; tol=1; while tol>=eps x=B*x0+f; n=n+1; t

53、ol=norm(x-x0); x0=x; if(n>=M) disp('Waring:迭代次數太多，可能不收斂！'); return; end end %高斯-賽德爾迭代法 function x,n=gauseidel(A,b,x0,eps,M)if nargin=3 eps=1.0e-6; M=10000;else if nargin=4 M=10000; endend D=diag(diag(A); L=-tril(A,-1); U=-triu(A,1); G=(D-L)U; f=(D-L)b; x=x0; n=0; tol=1; while tol>=eps x=G*x0+f; n=n+1; tol=norm(x-x0); x0=x; if(n>=M) disp('Waring: 迭代次數太多，可能不收斂！'); return; end end %超松弛迭代法 function x,n=SOR(A,b,x0,w,eps,M)if nargin=4 eps=1.0e-6; M=10000;else if nargin=5 M=10000; endendif(w<=0|w>=2) error; return;endD=diag(diag(A);L=-tril(A,