分段线性插值法

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1、数值分析实验报告实验序号:实验五实验名称: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9 (x)用来近似函数fx),用分段线性插值的方法来求解近似函数9 (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a二x x x . x二b和相应的函数0 1 2 n值y ,y，,y，求一个插值函数9(x)，满足以下条件：0 1 n

2、(1) 9 (x ) = y (j = 0,1,2,., n)；jj(2) 9(x)在每一个小区间x ,x 上是线性函数。j j+1对于给定函数f (x) -1,-1 x 1。在区间L1，1上画出fx)和分段线性插1 + 25x2值函数9 (x)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l (x)，然后再 j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：l0( x)=x - x, x x x x - x 01010, 其它x-xH, x x xx - xj -1jjj -1x

3、-xjL, x x xx - x jj+1jj+10,其它x - x1 (x)=nn 1 , xLoadIcon(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_SYSCOMMAN

4、D()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

5、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

6、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.DoMod

7、al();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); / de

8、vice 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() - cyI

9、con + 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();

10、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(T0,680);/x 箭头pDC-LineTo(

11、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

12、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(T0,680);/x 箭头pDC-LineTo(0,700);pDC-MoveTo(0,700);pDC-LineTo(10,680);pDC-Mo

13、veTo(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_fenmu

14、i=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_LOMETRI

15、C);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(T0,680);/x 箭头pDC-L

16、ineTo(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; iSetMapMode(MM_LO

17、METRIC);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(T0,680);/x 箭头p

18、DC-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

19、=-(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; qqx1; qq+=0.005)double pp= y0*(1+2*(qq-x0)/(x1-x0)*(qq-x1)/(x0-x1)*(qq-x1)/(x0-x1)+y1*(1+2*(qq-x1)/(x0-x1) * (qq-x0)/(x1-x0) * (qq-x0)/(x1-x0) +yd0*(qq-x0) * (qq-x1)/(x0-x1) * (qq-x1)/(x0-x1)+yd1*(qq

20、-x1) * (qq-x0)/(x1-x0) * (qq-x0)/(x1-x0); pDC-SetPixel(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。