实习：Matlab作业hermite插值

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

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

<

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：

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

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

%

% *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);

% 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

￥

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

% 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'));

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);

{

% --- 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：

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)

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

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。