分段线性插值法.docx

1、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

2、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 an

3、d 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 hand

4、lersBOOL 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); /

5、Set the icon for this dialog. The framework does this automatically / when the applications main window is not a dialog SetIcon(m_hIcon, TRUE); / Set big icon SetIcon(m_hIcon, FALSE); / Set small icon / TODO: Add extra initialization here return TRUE; / return TRUE unless you set the focus to a cont

6、rolvoid CLDlg:OnSysCommand(UINT nID, LPARAM lParam) if (nID & 0xFFF0) = IDM_ABOUTBOX) CAboutDlg dlgAbout; dlgAbout.DoModal(); else CDialog: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

7、/view model,/ this is automatically done for you by the framework.void CLDlg:OnPaint() if (IsIconic() CPaintDC dc(this); / device context for painting SendMessage(WM_ICONERASEBKGND, (WPARAM) dc.GetSafeHdc(), 0); / Center icon in client rectangle int cxIcon = GetSystemMetrics(SM_CXICON); int cyIcon =

8、 GetSystemMetrics(SM_CYICON); CRect rect; GetClientRect(&rect); int x = (rect.Width() - cxIcon + 1) / 2; int y = (rect.Height() - cyIcon + 1) / 2; / Draw the icon dc.DrawIcon(x, y, m_hIcon); else CDialog:OnPaint(); / The system calls this to obtain the cursor to display while the user drags/ the min

9、imized 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*50

10、0,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,

11、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-TextOu

12、t(-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,

13、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;

14、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(

15、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-

16、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

17、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,

18、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,

19、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; qqSetPixel(qq*500,pp*400,RGB(225,185,15); 4.实验截图 5. 实验结果分析：分析: 分段线性插值的方法克服了Lagrange插值法当节点不断加密时，构造的插值多项式的次数不断升高，高次多项式虽然是连续的，但是不一定都收敛到相应的被插函数而产生Runge现象。分段线性插值因为在每一段小区间上都是线性插值而极大地降低了插值多项式的次数，从几何图形上可以看出，当节点取得较多时插值函数的逼近效果还是很好的，但是所求函数是一条以型值点为顶点的折线，这也表现出了它的缺点就是所求得的插值函数的光滑性较差，这就要求一种更好的方法来克服这一缺点了。