实习：Matlab作业hermite插值

实习：Matlab作业hermite插值

(总12页)

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题目：利用Matlab实现数据的Hermite插值和分段三次Hermite插值

小组成员：王晓波（38） 蔡明宇（20）

一、程序实现意义：

一般的，从各种试验得来的数据总有一定的数量，而利用插值技术能够从有限的数据中获取整体的状态。而Hermite插值不仅保证了插值函数与原函数在给定数据点处得拟合，同时保证了在相应点处导数的相同，从而在很大程度上保证了曲线的“光滑性”。因此，通过Matlab实现Hermite插值具有很普遍的意义。

二、实现过程：

1、Hermite插值

由于并不是所有的Matlab版本都提供现有的Hermite插值函数包，故我们首先编写了实现给定五个观测点的Hermite插值的M程序，代码如下： function [f,f0] = Hermite1(x,y,y_1) syms t; f = ;

if(length(x) == length(y))

if(length(y) == length(y_1)) n = length(x); else

disp('y和y的导数的维数不相等'); return; end else

disp('x和y的维数不相等！ '); return; end

for i=1:n h = ; a = ; for j=1:n

if( j ~= i)

h = h*(t-x(j))^2/((x(i)-x(j))^2); a = a + 1/(x(i)-x(j)); end end

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f = f + h*((x(i)-t)*(2*a*y(i)-y_1(i))+y(i));

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end

f0 = subs(f,'t');

其中x为给定点横坐标数组，y为给定点纵坐标数组，y_1为原函数在给定点处的导数数组。测试证明该程序可以实现，例如输入如下数组：

x=1::; y_1=[ ]; y=[1 ];

>> [f,f0] = Hermite1(x,y,y_1)； 运行结果如下：

f =

(390625*((3972231*t)/35 - 28321/0000)*(t - 1)^2*(t - 7/5)^2*(t - 8/5)^2*(t - 9/5)^2)/36 - (390625*(t - 1)^2*(t - 6/5)^2*(t - 7/5)^2*(t - 9/5)^2*((28557*t)/28 - 23/2000))/36 + (390625*((64*t)/3 - 61/3)*(t - 6/5)^2*(t - 7/5)^2*(t - 8/5)^2*(t - 9/5)^2)/576 + (390625*((763*t)/1984 + 043/6240000)*(t - 1)^2*(t - 6/5)^2*(t -

8/5)^2*(t - 9/5)^2)/16 - (390625*((77623*t)/28 - 931/60000)*(t - 1)^2*(t - 6/5)^2*(t - 7/5)^2*(t - 8/5)^2)/576 f0 =

．

利用matlab绘制图像：

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2、程序的窗口化：

利用Matlab提供的GUIDE工具以及callback函数实现相应函数的窗口化，GUI代码如下：

function varargout = untitled(varargin) % UNTITLED M-file for

% UNTITLED, by itself, creates a new UNTITLED or raises the existing

% singleton*. %

% H = UNTITLED returns the handle to a new UNTITLED or the handle to

% the existing singleton*. %

% UNTITLED('CALLBACK',hObject,eventData,handles,...) calls the local

% function named CALLBACK in with the given input arguments. %

% UNTITLED('Property','Value',...) creates a new UNTITLED or raises the

% existing singleton*. Starting from the left, property value pairs are

% applied to the GUI before untitled_OpeningFcn gets called. An % unrecognized property name or invalid value makes property application

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% stop. All inputs are passed to untitled_OpeningFcn via varargin.

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%

% *See GUI Options on GUIDE's Tools menu. Choose \"GUI allows only one

% instance to run (singleton)\". %

% See also: GUIDE, GUIDATA, GUIHANDLES

% Edit the above text to modify the response to help untitled

% Last Modified by GUIDE 15-Sep-2011 22:24:48

% Begin initialization code - DO NOT EDIT gui_Singleton = 1;

gui_State = struct('gui_Name', mfilename, ... 'gui_Singleton', gui_Singleton, ... 'gui_OpeningFcn', @untitled_OpeningFcn, ... 'gui_OutputFcn', @untitled_OutputFcn, ... 'gui_LayoutFcn', [] , ... 'gui_Callback', []); if nargin && ischar(varargin{1}) = str2func(varargin{1}); end

if nargout

[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:}); else

gui_mainfcn(gui_State, varargin{:}); end

% End initialization code - DO NOT EDIT

% --- Executes just before untitled is made visible.

function untitled_OpeningFcn(hObject, eventdata, handles, varargin) % This 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 untitled (see VARARGIN)

% Choose default command line output for untitled = hObject;

% Update handles structure guidata(hObject, handles);

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% UIWAIT makes untitled wait for user response (see UIRESUME) % uiwait;

% --- Outputs from this function are returned to the command line. function varargout = untitled_OutputFcn(hObject, eventdata, handles) % varargout 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 structure varargout{1} = ;

function 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(hObject,'String')) returns contents of edit1 as a double

guidata(hObject, handles);

% --- Executes during object creation, after setting all properties. function edit1_CreateFcn(hObject, eventdata, 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 && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end

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function edit2_Callback(hObject, eventdata, handles) % hObject handle to edit2 (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 edit2 as text % str2double(get(hObject,'String')) returns contents of edit2 as a double

guidata(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 until after all CreateFcns called

% Hint: edit controls usually have a white background on Windows. % See ISPC and COMPUTER.

if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end

function edit3_Callback(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')) returns contents of edit3 as a double

guidata(hObject, handles);

% --- Executes during object creation, after setting all properties. function edit3_CreateFcn(hObject, eventdata, handles) % 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

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% Hint: edit controls usually have a white background on Windows. % See ISPC and COMPUTER.

if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end

function edit4_Callback(hObject, eventdata, handles) % hObject handle to edit4 (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 edit4 as text % str2double(get(hObject,'String')) returns contents of edit4 as a double

% --- Executes during object creation, after setting all properties. function edit4_CreateFcn(hObject, eventdata, handles) % hObject handle to edit4 (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 && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end

% --- Executes on button press in pushbutton1.

function pushbutton1_Callback(hObject, eventdata, handles) % hObject handle to pushbutton1 (see GCBO)

% eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) x=str2num(get,'string')); y=str2num(get,'string')); y_1=str2num(get,'string')); x0=str2num(get,'string'));

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syms t; f = ;

if(length(x) == length(y)) if(length(y) == length(y_1)) n = length(x); else

disp('yoíyμ?μ?êyμ???êy2??àμè'); return; end else

disp('xoíyμ???êy2??àμè￡? '); return; end for i=1:n h = ; a = ; for j=1:n if( j ~= i)

h = h*(t-x(j))^2/((x(i)-x(j))^2); a = a + 1/(x(i)-x(j)); end end

f = f + h*((x(i)-t)*(2*a*y(i)-y_1(i))+y(i)); end

f0 = subs(f,'t',x0); plot,x,y,'*');

function edit5_Callback(hObject, eventdata, handles) % hObject handle to edit5 (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 edit5 as text % str2double(get(hObject,'String')) returns contents of edit5 as a double

guidata(hObject, handles);

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% --- Executes during object creation, after setting all properties. function edit5_CreateFcn(hObject, eventdata, handles) % hObject handle to edit5 (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 && isequal(get(hObject,'BackgroundColor'), get(0,'defaultUicontrolBackgroundColor')) set(hObject,'BackgroundColor','white'); end

程序运行结果：

其中左上方纵列的三个对话框从上到下分别输入给定点的横坐标x，纵坐标y以及导数值y_1，右侧空白框输入维数，下方坐标图显示插值函数图像，例如仍插入上面所给定的点列，得出结果：

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从图上看拟合程度还是比较不错的。

分段三次Hermite插值与普通Hermite插值类似，但在程序实现过程中有所改良，主函数代码如下：

function [f,f0] = SubHermite(x,y,y_1) syms t; f = ; f0 = ;

if(length(x) == length(y))

if(length(y) == length(y_1)) n = length(x); else

disp('y和y的导数的维数不相等！'); return; end else

disp('x和y的维数不相等！'); return;

end %维数检查 for i=1:n

if(x(i)<=x0)&& (x(i+1)>=x0)

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index = i; break; end

end %找到x0所在的区间 h = x(index+1) - x(index); %x0所在的区间长度

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fl = y(index)*(1+2*(t-x(index))/h)*(t-x(index+1))^2/h/h + ...

y(index+1)*(1-2*(t-x(index+1))/h)*(t-x(index))^2/h/h;

fr = y_1(index)*(t-x(index))*(t-x(index+1))^2/h/h + ...

y_1(index+1)*(t-x(index+1))*(t-x(index))^2/h/h; f = fl + fr; %x0所在区间的插值函数 f0 = subs(f,'t'); %x0处得插值 其余部分保持不变。

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