等距节点插值公式

2012-2013(1)专业课程实践论文等距节点插值公式柳希元，0818180127,R数学08-1班、算法理论将Newton差商插值多项式中各阶差商用相应差分替代，就可得到各种形式的等距节点插值公式。如果节点错误！未找到弓I用源。要计算错误！未找到引用源。附近点x的函数错误！未找到引用源。值，可令错误！未找到引用源。于是- 1)亍 t(t_1)... _Tl+I)N隽咒+班)=托+丛托+'f“斤+…+ . 护斤n!为Newton前插公式。其中=A+i-fk如果要求表示函数在错误！未找到引用源。附近的值错误！未找到引用源。此时应用Newton插值公式，插值点应按错误！未找到引用源。的次序排列,有叫(x)=fg)+ xn) f[xnfxn_lfxn_2](x一兀』(无一J』+■■■+f[xnfxn_it,x0](x-xn)...(x-xj做变换错误！未找到引用源。带入公式得凡。+班)凡。+班)=人+丹打+迫］1)+…+t(t—1〉...(t—n4-n\为Newton后插公式。其中=A+i-fk二、算法框图三、算法程序classInterpolation{public:Interpolation(intnum,doublex1,doublex2,doublefunc[]);doubleComputeForwardValue(doublex);//computeforwardinterpolationvalue~Interpolation();private:voidGetForwardTable();//gettheforwarddifferentialtableprivate:intm_num;//thenumberofinterpolationpointsdoublem_x1,m_x2;//thefirstpointm_x1andlastpointm_x2doublem_step;//theinterpolationstepdouble*m_func;//thefunctionvalueofinterpolationpointsdouble*m_ftable;//theforwarddifferentialtable};#include<iostream〉#include〈limits〉usingnamespacestd;#defineNUM11//上？面?输°?入“？需―要°&多“。少厂’个?样…本A?#defineMIN0//上？面?输°?入“？区？间？的i?最A?小?值#defineMAX10//上？面?输°?入「区？间？的i?最A?大汀®值iintmain(){//下?面?输°?入“？y的i?值“doublefunc[NUM]={0,1,4,9,16,25,36,49,64,81,100}；//上？面?输°?入“？y的i?值doublex1=MIN,x2=MAX,x;intnum=NUM;charflag='Y';Interpolationtest(num,x1,x2,func);while(flag=='Y'){cout〈〈"Inputx:";cin〉〉x;if(!cin)//checkingfailurestate{cin.clear();//clearfailuretagcin.ignore(numeric_limits<int〉：：max(),'\n');//clearinputbuffercontinue;}if(x〈x1||x〉x2){cout〈〈" Invalidinput:"〈〈x〈〈" "〈〈endl;cout〈〈"Onlythenumberbetween"〈〈x1〈〈"and"〈〈x2〈〈"isvalid..."〈〈endl;}else{cout〈〈"Forwardinterpolationvalue:"〈〈test.ComputeForwardValue(x+0.001)〈〈endl;}cout〈〈endl〈〈"Doyouwanttoprocess?pleaseinput(Y/N):"〈〈endl;cin〉〉flag;}return0;Interpolation::Interpolation(intnum,doublex1,doublex2,doublefunc[]){m_num=num;m_x1=x1;m_x2=x2;m_step=(m_x2-m_x1)/(numT);m_func=newdouble[m_num];m_ftable=newdouble[m_num];for(inti=0;i〈m_num;++i){m_func[i]=func[i];m_ftable[i]=func[i];}GetForwardTable();}Interpolation::~Interpolation(){deletem_func;deletem_ftable;}voidInterpolation::GetForwardTable(){//gettheforwarddifferentialtableinti,j;for(i=1;i〈m_num;++i)for(j=m_num-1;j>=i;—j)m_ftable[j]=m_ftable[j]-m_ftable[jT];}doubleInterpolation::ComputeForwardValue(doublex){//computeforwardinterpolationvaluedouble*coef;//coefficienttalbedoubleresult,t;inti;coef=newdouble[m_num];t=(x-m_x1)/m_step;for(i=1,coef[0]=1;i〈m_num;++i)//computethecoefficienttablecoef[i]=coef[iT]*(t-i+1)/i;for(i=0,result=0;i〈m_num;++i)result+=m_ftable[i]*coef[i];deletecoef;returnresult;}四、算法实现例1•当错误！未找到引用源。时求错误！未找到引用源。在范围错误！未找到引用源。内任意错误！未找到引用源。的值。解：在程序中输入11个样本：0,1,4,9,16,25,36,49,64,81,100、且输入MIN：0,MAX：10,样本容量：11。在运行程序中输入范围内的任意错误！未找到引用源。的值，如：1.2,1.4等。程序提示答案和提示是否继续，是则输入大写Y,否则输入大写N。例2.当给出15个错误！未找到引用源。：1.38,1.48,1.58,1.69,1.81,1.94,2.10,2.2&2.50,2.76,3.06,3.41,3.83,4.33,4.93,且给出范围：错误！未找到引用源。求出任意错误！未找到引用源。的值。解：给出15个题目上显示的值：1.38,1.48,1.58,1.69,1.81,1.94,2.10,2.2&2.50,2.76,3.06,3.41,3.83,4.33,4.93,且输入MIN：4000,MAX：11000，样本容量：15。在运行程序中输入范围内的任意错误！未找到引用源。的值，如：1001,5200等。程序提示答案和提示是否继续，是则输入大写Y，否则输入大写N。rawC:\Windows\system32\cTnd.exeInputx：4001Forwardinterpolationvalue:1.3747Doijouwanttoprocess?pleaseinput<¥/N>:VInputx：5200Forwardinterpolationvalue:1.61375Doijouwanttoprocess?p