分段线性插值法

. 数值分析实验报告实验序号：实验五 实验名称: 分段线性插值法 1、 实验目的：随着插值节点的增加，插值多项式的插值多项式的次数也增加，而对于高次的插值容易带来剧烈的震荡，带来数值的不稳定（Runge现象）。为了既要增加插值的节点，减小插值的区间，以便更好的逼近插值函数，又要不增加插值多项式的次数以减少误差，可采用分段线性插值。2、 实验内容：求一个函数(x)用来近似函数f(x)，用分段线性插值的方法来求解近似函数(x)并画出近似函数图像及原函数图像。设在区间a,b上，给定n+1个插值节点和相应的函数值，求一个插值函数，满足以下条件：（1） ; （2） 在每一个小区间上是线性函数。对于给定函数。在区间上画出f(x)和分段线性插值函数的函数图像。1. 分段线性插值的算法思想：分段线性插值需要在每个插值节点上构造分段线性插值基函数，然后再作它们的线性组合。分段线性插值基函数的特点是在对应的插值节点上函数值取 1，其它节点上函数值取0。插值基函数如下： 设在节点ax0x1LoadIcon(IDR_MAINFRAME);void CLDlg:DoDataExchange(CDataExchange* pDX)CDialog:DoDataExchange(pDX);/AFX_DATA_MAP(CLDlg)/ NOTE: the ClassWizard will add DDX and DDV calls here/AFX_DATA_MAPBEGIN_MESSAGE_MAP(CLDlg, CDialog)/AFX_MSG_MAP(CLDlg)ON_WM_SYSCOMMAND()ON_WM_PAINT()ON_WM_QUERYDRAGICON()ON_BN_CLICKED(IDC_LARGRI, OnLargri)ON_BN_CLICKED(IDC_BUTTON2, OnButton2)ON_BN_CLICKED(IDC_HERMITE, OnHermite)/AFX_MSG_MAPEND_MESSAGE_MAP()/ CLDlg message handlersBOOL CLDlg: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 CLDlg: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 CLDlg: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 CLDlg:OnQueryDragIcon()return (HCURSOR) m_hIcon;void CLDlg:OnOK() int x00=300,y00=350,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=-700; 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);pDC-TextOut(-30,-10,0); pDC-TextOut(-30,430,1); pDC-TextOut(490,-10,1); pDC-TextOut(-490,-10,-1); pDC-MoveTo(-10,680); /x箭头pDC-LineTo(0,700);pDC-MoveTo(0,700);pDC-LineTo(10,680); pDC-MoveTo(680,10); /y箭头pDC-LineTo(700,0);pDC-MoveTo(700,0);pDC-LineTo(680,-10);pDC-TextOut(-30,700,y);pDC-TextOut(700,-10,x);void CLDlg:OnLargri() int x00=300,y00=350,i,j;double x;CDC *pDC=GetDC();pDC-SetMapMode(MM_LOMETRIC);pDC-SetViewportOrg(x00,y00);/画坐标轴for(i=-700; 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;pDC-TextOut(-30,-10,0); pDC-TextOut(-30,430,1); pDC-TextOut(490,-10,1); pDC-TextOut(-490,-10,-1); pDC-MoveTo(-10,680); /x箭头pDC-LineTo(0,700);pDC-MoveTo(0,700);pDC-LineTo(10,680); pDC-MoveTo(680,10); /y箭头pDC-LineTo(700,0);pDC-MoveTo(700,0);pDC-LineTo(680,-10);pDC-TextOut(-30,700,y);pDC-TextOut(700,-10,x);/ 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(132,112,225);void CLDlg:OnButton2() int x00=300,y00=350,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=-700; 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;double yy14;for(i=0; iTextOut(-30,-10,0); pDC-TextOut(-30,430,1); pDC-TextOut(490,-10,1); pDC-TextOut(-490,-10,-1); pDC-MoveTo(-10,680); /x箭头pDC-LineTo(0,700);pDC-MoveTo(0,700);pDC-LineTo(10,680); pDC-MoveTo(680,10); /y箭头pDC-LineTo(700,0);pDC-MoveTo(700,0);pDC-LineTo(680,-10);pDC-TextOut(-30,700,y);pDC-TextOut(700,-10,x);/ 线性分段差值的图像CPen pen;CPen*oldpen;pen.CreatePen(PS_SOLID,5,RGB(0,0,0);oldpen=pDC-SelectObject(&pen);for(i=0; iMoveTo(yxi*480,yyi*400); pDC-LineTo(yxi+1*480,yyi+1*400); void CLDlg:OnHermite() int x00=300,y00=350,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=-700; 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;double yy12;for(i=0; iTextOut(-30,-10,0); pDC-TextOut(-30,430,1); pDC-TextOut(490,-10,1); pDC-TextOut(-490,-10,-1); pDC-MoveTo(-10,680); /x箭头pDC-LineTo(0,700);pDC-MoveTo(0,700);pDC-LineTo(10,680); pDC-MoveTo(680,10); /y箭头pDC-LineTo(700,0);pDC-MoveTo(700,0);pDC-LineTo(680,-10);pDC-TextOut(-30,700,y);pDC-TextOut(700,-10,x);/分段三次Hermite差值的函数double x0,x1,yd1,yd0,y1,y0;for(i=0; i10; i+)x0=yxi,x1=yxi+1;y0=1.0/(1+25*x0*x0);y1=1.0/(1+25*x1*x1);yd0=-(50*x0)*1.0/(1+25*x0*x0)*(1+25*x0*x0);yd1=-(50*x1)*1.0/(1+25*x1*x1)*(1+25*x1*x1); for(double qq=x0; qqSetP