Lagrange插值法.doc

学院：计算机与信息工程学院 班级：11级计算机科学与技术 学号：20111105461 姓名：郭雁军数值分析实验报告任课教师：马季骕 班级：11级计算机科学与技术1 实验目的及要求2 程序的源代码3 实验操作4 实验结果及分析1 实验目的及要求使用Lagrange插值的方法求原函数的逼近函数。在数学分析中，用y=f（x）来描述一条平面直线，但是在实际问题中，函数y=f（x）往往是通过观测得到的一组数据来给出的，只是已知个别点的函数值，而非在整个区间上，插值法是应用十分广泛的一种方法。本实验是用拉格朗日法来逼近被逼近函数，并画出其图像。当给出了n+1个节点上f(x)的一张函数表后，用Lagrange插值法求一个函数(x)，并满足以下两个条件：（1）(x)是一个不超过n次的多项式；（2）在给定点上与取相同值，即=（i=0,1,2n）。当插值节点取的足够多时逼近函数(x)能够很好的逼近被逼近函数f(x)。而插值函数(x)的次数就会相应地升高，高次的插值多项式就不一定收敛到相应的被逼近函数，就会产生Runge现象，本实验可以从函数的图像上清楚地看到这一现象。2 程序的源代码/ 数值分析Dlg.cpp : implementation file/#include stdafx.h#include 数值分析.h#include 数值分析Dlg.h#ifdef _DEBUG#define new DEBUG_NEW#undef THIS_FILEstatic char THIS_FILE = _FILE_;#endif/ CAboutDlg dialog used for App Aboutclass CAboutDlg : public CDialogpublic:CAboutDlg();/ Dialog Data/AFX_DATA(CAboutDlg)enum IDD = IDD_ABOUTBOX ;/AFX_DATA/ ClassWizard generated virtual function overrides/AFX_VIRTUAL(CAboutDlg)protected:virtual void DoDataExchange(CDataExchange* pDX); / DDX/DDV support/AFX_VIRTUAL/ Implementationprotected:/AFX_MSG(CAboutDlg)/AFX_MSGDECLARE_MESSAGE_MAP();CAboutDlg:CAboutDlg() : CDialog(CAboutDlg:IDD)/AFX_DATA_INIT(CAboutDlg)/AFX_DATA_INITvoid CAboutDlg:DoDataExchange(CDataExchange* pDX)CDialog:DoDataExchange(pDX);/AFX_DATA_MAP(CAboutDlg)/AFX_DATA_MAPBEGIN_MESSAGE_MAP(CAboutDlg, CDialog)/AFX_MSG_MAP(CAboutDlg)/ No message handlers/AFX_MSG_MAPEND_MESSAGE_MAP()/ CMyDlg dialogCMyDlg:CMyDlg(CWnd* pParent /*=NULL*/): CDialog(CMyDlg:IDD, pParent)/AFX_DATA_INIT(CMyDlg)/ NOTE: the ClassWizard will add member initialization here/AFX_DATA_INIT/ Note that LoadIcon does not require a subsequent DestroyIcon in Win32m_hIcon = AfxGetApp()-LoadIcon(IDR_MAINFRAME);void CMyDlg:DoDataExchange(CDataExchange* pDX)CDialog:DoDataExchange(pDX);/AFX_DATA_MAP(CMyDlg)/ NOTE: the ClassWizard will add DDX and DDV calls here/AFX_DATA_MAPBEGIN_MESSAGE_MAP(CMyDlg, CDialog)/AFX_MSG_MAP(CMyDlg)ON_WM_SYSCOMMAND()ON_WM_PAINT()ON_WM_QUERYDRAGICON()ON_BN_CLICKED(IDC_NEW, OnNew)ON_BN_CLICKED(IDC_LAG, OnLag)ON_BN_CLICKED(IDC_LINE, OnLine)ON_BN_CLICKED(IDC_HER, OnHer)/AFX_MSG_MAPEND_MESSAGE_MAP()/ CMyDlg message handlersBOOL CMyDlg:OnInitDialog()CDialog: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 CMyDlg:OnSysCommand(UINT nID, LPARAM lParam)if (nID & 0xFFF0) = IDM_ABOUTBOX)CAboutDlg dlgAbout;dlgAbout.DoModal();elseCDialog: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 CMyDlg:OnPaint() if (IsIconic()CPaintDC dc(this); / device context for paintingSendMessage(WM_ICONERASEBKGND, (WPARAM) 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);elseCDialog:OnPaint();/ The system calls this to obtain the cursor to display while the user drags/ the minimized window.HCURSOR CMyDlg:OnQueryDragIcon()return (HCURSOR) m_hIcon;void CMyDlg:OnNew() / TODO: Add your control notification handler code here int x00=225,y00=225,i,j;double x;CDC *pDC=GetDC();pDC-SetMapMode(MM_LOMETRIC);pDC-SetViewportOrg(x00,y00);/画坐标轴与原函数for(i=-650; iSetPixel(i,0,RGB(0,0,0);pDC-SetPixel(0,i,RGB(0,0,0);for(x=-1; xSetPixel(x*500,j,RGB(255,0,0);void CMyDlg:OnLag() / TODO: Add your control notification handler code hereint x00=225,y00=225,i,j;double x;CDC *pDC=GetDC();pDC-SetMapMode(MM_LOMETRIC);pDC-SetViewportOrg(x00,y00);/画坐标轴for(i=-650; iSetPixel(i,0,RGB(0,0,0);pDC-SetPixel(0,i,RGB(0,0,0);double yx=-1,-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,0.6,0.8,1;/ 拉格朗日差值的函数double yy12,lx12,ly12;double l_fenzi12,l_fenmu12; double l_x,l_y;for(i=0; i=10; i+)yyi=1.0/(1+25*yxi*yxi);for(i=0; i=10; i+)l_fenmui=1.0;for(j=0; j=10; j+)if(i!=j)l_fenmui=l_fenmui*(yxi-yxj); double qq,pp;for(qq=-1; qq=1; qq+=0.0003)for(i=0; i=10; i+)l_fenzii=1.0; for(j=0; j=10; j+)if(i!=j) l_fenzii=l_fenzii*(qq-yxj);pp=0;for(i=0; iSetPixel(qq*500,pp*390+5,RGB(150,255,0);3 实验操作算法描述：先从特殊情况入手，研究线形插值和二次插值。在线性插值中，若逼近函数用两点式表示出来，则有F（x）=y0*(x-x1)/(x0-x1)+y1*(x-x0)/(x1-x0);过三个点的二次插值可以确定一条抛物线，经过化简之后，被逼近函数可以表示称F(x)=y0*(x-x1)*(x-x2)/(x0-x1)/(x0-x2)+y1*(x-x0)*(x-x2)/(x1-x0)/(x1-x2)+y2*(x-x0)*(x-x1)/(x2-x0)/(x2-x1);经过分析对比可以总结出：对于具有n+1个节点的函数，都可以表示成该节点的函数值与其对应基函数的乘积的线形组合。其中要求逼近函数是n次多项式，并且逼近函数过所有的型值点。（1） 原函数图像的绘制： 对于给定函数。在区间上画出f(x)的函数图像。（2） Lagrange插值函数图像的绘制： 根据题目要求取11个等距插值节点，求出每一个