实验一插值计算-1240111118-武元甲.doc

实验一插值方法一 实验目的通过本次上机实习，能够进一步加深对各种插值算法的理解；学会使用用三种类型的插值函数的数学模型、基本算法，结合相应软件（如VC/VB/Delphi/Matlab/JAVA/Turbo C）编程实现数值方法的求解。并用该软件的绘图功能来显示插值函数，使其计算结果更加直观和形象化。二 实验内容通过程序求出插值函数的表达式是比较麻烦的，常用的方法是描出插值曲线上尽量密集的有限个采样点，并用这有限个采样点的连线，即折线，近似插值曲线。取点越密集，所得折线就越逼近理论上的插值曲线。本实验中将所取的点的横坐标存放于动态数组中，通过插值方法计算得到的对应纵坐标存放于动态数组中。本实验将Lagrange插值、Newton插值和三次样条插值实现为一个C+类CInterpolation，并在Button单击事件中调用该类相应函数，得出插值结果并画出图像。CInterpolation类为class CInterpolationpublic: CInterpolation();/构造函数CInterpolation(float *x1, float *y1, int n1);/结点横坐标、纵坐标、下标上限 CInterpolation();/析构函数int n, N;/结点下标上限，采样点下标上限float *x, *y, *X;/分别存放结点横坐标、结点纵坐标、采样点横坐标float *p_H,*p_Alpha,*p_Beta,*p_a,*p_b,*p_c,*p_d,*p_m;/样条插值用到的公有指针，分别存放，和;其中，有参数的构造函数为CInterpolation(float *x1, float *y1, int n1) /动态数组x1，y1中存放结点的横、纵坐标，n1是结点下标上限（即n1+1个结点） n=n1; N=x1n-x10; X=new floatN+1; x=new floatn+1; y=new floatn+1; for(int i=0;i=n;i+) xi=x1i; yi=y1i; for(int i=0;i=N;i+) Xi=x0+i;2.1 Lagrange插值，其中对于一个自变量，要求插值函数值，首先需要计算对应的Lagrange插值基函数值float l(float xv,int i) /求插值基函数的值 float t=1; for(int j=0;j=n;j+) if(j!=i) t=t*(xv-xj)/(xi-xj); return t;调用函数l(float x,int i)，可求出float p_l(float x) /求在一个点的插值结果 float t=0; for(int i=0;i=n;i+) t+=yi*l(x,i); return t;调用p_l(float x)可实现整个区间的插值float *Lagrange() /求整个插值区间上所有采样点的插值结果 float *Y=new floatN+1; for(int k=0;k=N;k+) Yk=p_l(x0+k*h); return Y;2.2 Newton插值，其中，对于一个自变量，要求插值函数值，首先需要计算出和float *f()/该函数的返回值是一个长度为n1的动态数组，存放各阶差商float w(float x, int i)/该函数计算在求的函数中调用*f()得到各阶差商，然后在循环中调用w(float x)可得出插值结果float p_n(float x)/该函数计算在一点的值调用p_n(float x)可实现整个区间的插值float *Newton()/该函数计算出插值区间内所有点的值2.3 三次样条插值三次样条插值程序可分为以下四步编写：（1） 计算结点间的步长、；（2） 利用、产生三对角方程组的系数矩阵和常数向量；（3） 通过求解三对角方程组，得出中间结点的导数；（4） 对自变量x，在对应区间上，使用Hermite插值；（5） 调用上述函数，实现样条插值。将每步写成函数：(1)void GetH(void)/该函数计算数组void GetAlpha(void)/该函数计算数组void GetBeta(void)/该函数计算数组(2)void Geta(void)/该函数计算数组下对角线 void Getb(void)/该函数计算数组主对角线 void Getc(void)/该函数计算数组上对角线void Getd(void)/该函数计算方程组右端常数项(3)float *Chasing(float *pa,float *pb,float *pc,float *pd,int n)/追赶过程，计算各点斜率(4)float F0(float x)/该函数计算函数的值float F1(float x)/该函数计算函数的值float P0(float x)/该函数计算函数的值float P1(float x)/该函数计算函数的值调用上述函数，实现三次样条插值float *Cubic_Spline()/该函数计算所有点的插值结果三 实验组织运行要求 实验前，由任课教师落实实验任务，每个学生事先编写好算法设计源程序代码。集中上机、调试并通过计算机图形可视化演示操作实例来测试、验证所学的数值分析理论。四 实验条件为每个学生提供一台具有WINDOWS 98/XP/NT/2000操作系统的计算机；同时提供VC+/VB/JAVA/TC等集成语言开发环境来编程设计计算方法的上机实验。五 实验内容拉格朗日插值/ TODO: 在此添加控件通知处理程序代码/首先从线性表中获取节点坐标double *x,*y;int n;n=m_SL.Length;x=new doublen;y=new doublen;for(int i=0;iMoveTo(mI.X0,mI.Y0);for(int i=0;iLineTo(mI.Xi,mI.Yi);ReleaseDC(pDC);*/void CInterpolationDlg:Draw(double * px, double * py, int n)/为了画图方便，可以将函数图像的绘制写成函数DrawCDC *pDC=GetDC();pDC-MoveTo(px0,py0);for(int i=0;iLineTo(pxi,pyi);ReleaseDC(pDC);/ InterpolationDlg.cpp : 实现文件/#include stdafx.h#include Interpolation.h#include InterpolationDlg.h#include afxdialogex.h#ifdef _DEBUG#define new DEBUG_NEW#endif/ 用于应用程序“关于”菜单项的 CAboutDlg 对话框class CAboutDlg : public CDialogExpublic:CAboutDlg();/ 对话框数据enum IDD = IDD_ABOUTBOX ;protected:virtual void DoDataExchange(CDataExchange* pDX); / DDX/DDV 支持/ 实现protected:DECLARE_MESSAGE_MAP();CAboutDlg:CAboutDlg() : CDialogEx(CAboutDlg:IDD)void CAboutDlg:DoDataExchange(CDataExchange* pDX)CDialogEx:DoDataExchange(pDX);BEGIN_MESSAGE_MAP(CAboutDlg, CDialogEx)END_MESSAGE_MAP()/ CInterpolationDlg 对话框CInterpolationDlg:CInterpolationDlg(CWnd* pParent /*=NULL*/): CDialogEx(CInterpolationDlg:IDD, pParent)m_hIcon = AfxGetApp()-LoadIcon(IDR_MAINFRAME);void CInterpolationDlg:DoDataExchange(CDataExchange* pDX)CDialogEx:DoDataExchange(pDX);BEGIN_MESSAGE_MAP(CInterpolationDlg, CDialogEx)ON_WM_SYSCOMMAND()ON_WM_PAINT()ON_WM_QUERYDRAGICON()ON_WM_LBUTTONDBLCLK()ON_BN_CLICKED(IDC_BUTTON3, &CInterpolationDlg:OnBnClickedButton3)END_MESSAGE_MAP()/ CInterpolationDlg 消息处理程序BOOL CInterpolationDlg:OnInitDialog()CDialogEx:OnInitDialog();/ 将“关于.”菜单项添加到系统菜单中。/ IDM_ABOUTBOX 必须在系统命令范围内。ASSERT(IDM_ABOUTBOX & 0xFFF0) = IDM_ABOUTBOX);ASSERT(IDM_ABOUTBOX AppendMenu(MF_SEPARATOR);pSysMenu-AppendMenu(MF_STRING, IDM_ABOUTBOX, strAboutMenu);/ 设置此对话框的图标。当应用程序主窗口不是对话框时，框架将自动/ 执行此操作SetIcon(m_hIcon, TRUE);/ 设置大图标SetIcon(m_hIcon, FALSE);/ 设置小图标/ TODO: 在此添加额外的初始化代码return TRUE; / 除非将焦点设置到控件，否则返回 TRUEvoid CInterpolationDlg:OnSysCommand(UINT nID, LPARAM lParam)if (nID & 0xFFF0) = IDM_ABOUTBOX)CAboutDlg dlgAbout;dlgAbout.DoModal();elseCDialogEx:OnSysCommand(nID, lParam);/ 如果向对话框添加最小化按钮，则需要下面的代码/ 来绘制该图标。对于使用文档/视图模型的 MFC 应用程序，/ 这将由框架自动完成。void CInterpolationDlg:OnPaint()if (IsIconic()CPaintDC dc(this); / 用于绘制的设备上下文SendMessage(WM_ICONERASEBKGND, reinterpret_cast(dc.GetSafeHdc(), 0);/ 使图标在工作区矩形中居中int cxIcon = GetSystemMetrics(SM_CXICON);int cyIcon = GetSystemMetrics(SM_CYICON);CRect rect;GetClientRect(&rect);int x = (rect.Width() - cxIcon + 1) / 2;int y = (rect.Height() - cyIcon + 1) / 2;/ 绘制图标dc.DrawIcon(x, y, m_hIcon);elseCDialogEx:OnPaint();/画坐标系CDC *pDC=GetDC();pDC-MoveTo(0,250);pDC-LineTo(500,250);/画x轴pDC-LineTo(490,240);pDC-MoveTo(500,250);pDC-LineTo(490,260);/画x轴箭头pDC-TextOutA(500,230,X);pDC-MoveTo(250,500);pDC-LineTo(250,0);/画y轴pDC-LineTo(240,10);pDC-MoveTo(250,0);pDC-LineTo(260,10);/画y轴箭头pDC-TextOutA(230,0,Y);pDC-TextOutA(230,255,O);ReleaseDC(pDC);/当用户拖动最小化窗口时系统调用此函数取得光标/显示。HCURSOR CInterpolationDlg:OnQueryDragIcon()return static_cast(m_hIcon);void CInterpolationDlg:OnLButtonDblClk(UINT nFlags, CPoint point)/ TODO: 在此添加消息处理程序代码和/或调用默认值/鼠标左键双击事件,将双击的点point作为差值节点/point的坐标为计算机坐标系，原点在对话框左上角，y轴向下m_SL.InsertAtTail(point);/将鼠标双击时的坐标插入线性表末尾CDC *pDC=GetDC();pDC-Ellipse(point.x-5,point.y-5,point.x+5,point.y+5);/在鼠标双击处画圆标记CString ms=(;CString mx,my;mx.Format(%d,point.x);my.Format(%d,point.y);ms+=mx;ms+=,;ms+=my;ms+=);pDC-TextOutA(point.x+5,point.y+5,ms);/显示点的坐标/请自行将计算机坐标转换为平面直角坐标系显示CDialogEx:OnLButtonDblClk(nFlags, point);void CInterpolationDlg:OnBnClickedButton3()/ TODO: 在此添加控件通知处理程序代码/首先从线性表中获取节点坐标double *x,*y;int n;n=m_SL.Length;x=new doublen;y=new doublen;for(int i=0;iMoveTo(mI.X0,mI.Y0);for(int i=0;iLineTo(mI.Xi,mI.Yi);ReleaseDC(pDC);*/void CInterpolationDlg:Draw(double * px, double * py, int n)/为了画图方便，可以将函数图像的绘制写成函数DrawCDC *pDC=GetDC();pDC-MoveTo(px0,py0);for(int i=0;iLineTo(pxi,pyi);ReleaseDC(pDC);:#include stdafx.h#include ZInterpolation.hZInterpolation:ZInterpolation(void)ZInterpolation:ZInterpolation(double *x,double *y,int n)this-n=n;this-x=new doublen+1;this-y=new doublen+1;for(int i=0;ixi=xi;this-yi=yi;N=xn-x0;X=new doubleN+1;Y=new doubleN+1;for(int i=0;i=N;i+)Xi=x0+i;ZInterpolation:ZInterpolation(void)double ZInterpolation:l(double mx, int i)/求拉格朗日差值基函数double t=1;for(int j=0;j=n;j+)if(j=i)continue;elset*=(mx-xj)/(xi-xj);return t;double ZInterpolation:L(double mx)/求一个点的拉格朗日插值结果double t=0;for(int i=0;i=n;i+)t+=yi*l(mx,i);return t;void ZInterpolation:Lagrange(void)/求整个差值区间上每个采样点（像素）的拉格朗日差值结果，并存放在数组Y中for(int i=0;iLoadIcon(IDR_MAINFRAME);void CInterpolation2Dlg:DoDataExchange(CDataExchange* pDX)CDialogEx:DoDataExchange(pDX);BEGIN_MESSAGE_MAP(CInterpolation2Dlg, CDialogEx)ON_WM_SYSCOMMAND()ON_WM_PAINT()ON_WM_QUERYDRAGICON()ON_WM_LBUTTONDBLCLK()ON_BN_CLICKED(IDC_BUTTON1, &CInterpolation2Dlg:OnBnClickedButton1)END_MESSAGE_MAP()/ CInterpolation2Dlg message handlersBOOL CInterpolation2Dlg:OnInitDialog()CDialogEx:OnInitDialog();/ Add About. menu item to system menu./ IDM_ABOUTBOX must be in the system command range.ASSERT(IDM_ABOUTBOX & 0xFFF0) = IDM_ABOUTBOX);ASSERT(IDM_ABOUTBOX AppendMenu(MF_SEPARATOR);pSysMenu-AppendMenu(MF_STRING, IDM_ABOUTBOX, strAboutMenu);/ Set the icon for this dialog. The framework does this automatically/ when the applications main window is not a dialogSetIcon(m_hIcon, TRUE);/ Set big iconSetIcon(m_hIcon, FALSE);/ Set small icon/ TODO: Add extra initialization herereturn TRUE; / return TRUE unless you set the focus to a controlvoid CInterpolation2Dlg:OnSysCommand(UINT nID, LPARAM lParam)if (nID & 0xFFF0) = IDM_ABOUTBOX)CAboutDlg dlgAbout;dlgAbout.DoModal();elseCDialogEx:OnSysCommand(nID, lParam);/ If you add a minimize button to your dialog, you will need the code below/ to draw the icon. For MFC applications using the document/view model,/ this is automatically done for you by the framework.void CInterpolation2Dlg:OnPaint()if (IsIconic()CPaintDC dc(this); / device context for paintingSendMessage(WM_ICONERASEBKGND, reinterpret_cast(dc.GetSafeHdc(), 0);/ Center icon in client rectangleint cxIcon = GetSystemMetrics(SM_CXICON);int cyIcon = GetSystemMetrics(SM_CYICON);CRect rect;GetClientRect(&rect);int x = (rect.Width() - cxIcon + 1) / 2;int y = (rect.Height() - cyIcon + 1) / 2;/ Draw the icondc.DrawIcon(x, y, m_hIcon);elseCDialogEx:OnPaint();/画坐标系CDC *pDC=GetDC();pDC-MoveTo(0,250);pDC-LineTo(500,250);/画X轴pDC-LineTo(490,240); pDC-MoveTo(500,250); pDC-LineTo(490,260);/画X轴箭头pDC-TextOutA(500,230,X); pDC-MoveTo(250,500);pDC-LineTo(250,0);/画Y轴pDC-LineTo(240,10); pDC-MoveTo(250,0); pDC-LineTo(260,10);/画Y轴箭头pDC-TextOutA(230, 0,Y); pDC-TextOutA(230,255,O);ReleaseDC(pDC);/ The system calls this function to obtain the cursor to display while the user drags/ the minimized window.HCURSOR CInterpolation2Dlg:OnQueryDragIcon()return static_cast(m_hIcon);void CInterpolation2Dlg:OnLButtonDblClk(UINT nFlags, CPoint point)/ TODO: Add your message handler code here and/or call default/鼠标左键双击事件，将双击的点point作为插值节点/point的坐标为计算机坐标系，原点在对话框左上角，y轴向下m_SL.InsertAtTail(point);/将鼠标双击时的坐标插入线性表末尾CDC *pDC=GetDC();pDC-Ellipse(point.x-5,point.y-5,point.x+5,point.y+5);/在鼠标双击处画圆标记CString ms=(;CString mx,my;mx.Format(%d,point.x-250);my.Format(%d,point.y*(-1)+250); ms+=mx;ms+=,;ms+=my;ms+=);pDC-TextOutA(point.x+5,point.y+5,ms);/显示点的坐标ReleaseDC(pDC);