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1、第2节牛顿插值法思考一、牛顿插值公式的基本思路二、新概念___均差及其性质.,)()()(],[20000的一阶（均差）关于点为函数称定义kkkkxxxfxxxfxfxxf--=均差计算表一阶均差二阶均差三阶均差四阶均差引入记号：称为k阶均差均差性质：三、Newton插值多项式的结构牛顿插值法的完整程序｛x[0],x[1],x[2],x[3],x[4]}={10,11,12,13,14};y[k_]:=Log[x[k]]Table[y[k],{k,0,4}]//N;MatrixForm[%]f[i_,j_]:=(y[j]-y[i])/(x[j]-x[i])Table[f[i,
2、i+1],{i,0,3}]//N;MatrixForm[%]f[i_,j_,k_]:=(f[j,k]-f[i,j])/(x[k]-x[i])Table[f[i,i+1,i+2],{i,0,2}]//N;MatrixForm[%]f[i_,j_,k_,l_]:=(f[j,k,l]-f[i,j,k])/(x[l]-x[i])Table[f[i,i+1,i+2,i+3],{i,0,1}]//N;MatrixForm[%]f[i_,j_,k_,l_,m_]:=(f[j,k,l,m]-f[i,j,k,l])/(x[m]-x[i])Table[f[i,i+1,i+2,i+3,i+4],{
3、i,0,0}]//N;MatrixForm[%]a[0]=y[0];a[1]=f[0,1];a[2]=f[0,1,2];a[3]=f[0,1,2,3];a[4]=f[0,1,2,3,4];N[x]=Sum[a[k]*Product[(x-x[m]),{m,0,k-1}],{k,0,4}]//NExpand[%]A={{y[0],y[1],y[2],y[3],y[4]},{0,f[0,1],f[1,2],f[2,3],f[3,4]},{0,0,f[0,1,2],f[1,2,3],f[2,3,4]},{0,0,0,f[0,1,2,3],f[1,2,3,4]},{0,0,0,0,f
4、[0,1,2,3,4]}};Transpose[A]//N;MatrixForm[%]例5.3解析一阶均差二阶均差三阶均差四阶均差五阶均差0.400.410750.550.578151.116000.650.696751.186000.280000.800.888111.275730.358930.197330.901.026521.384100.433480.213000.031341.051.253821.515330.524920.228630.03126-0.00012一般插值法计算结果程序如下：Clear[A,g1,g2]xx={0.40,0.55,0.65,0.8
5、0,0.90};yy={0.41075,0.57815,0.69675,0.88811,1.02652};A=Table[{xx[[k]],yy[[k]]},{k,1,5}];g1=ListPlot[A,Prolog->AbsolutePointSize[15]];Interpolation[A,InterpolationOrder->4]g2=Plot[%[x],{x,0.40,1.02652}]Show[g1,g2]N[%%%[0.596],20]0.6319175080796161x[0]=0.40;x[1]=0.55;x[2]=0.65;x[3]=0.80;x[4]=
6、0.90;y[0]=0.41075;y[1]=0.57815;y[2]=0.69675;y[3]=0.88811;y[4]=1.02652;f[i_,j_]:=(y[j]-y[i])/(x[j]-x[i])Table[f[i,i+1],{i,0,3}]//N;MatrixForm[%];f[i_,j_,k_]:=(f[j,k]-f[i,j])/(x[k]-x[i])Table[f[i,i+1,i+2],{i,0,2}]//N;MatrixForm[%];f[i_,j_,k_,l_]:=(f[j,k,l]-f[i,j,k])/(x[l]-x[i])Table[f[i,i+1,i
7、+2,i+3],{i,0,1}]//N;MatrixForm[%];f[i_,j_,k_,l_,m_]:=(f[j,k,l,m]-f[i,j,k,l])/(x[m]-x[i])Table[f[i,i+1,i+2,i+3,i+4],{i,0,0}]//N;MatrixForm[%];Newton插值法程序见MathematicaA={{y[0],y[1],y[2],y[3],y[4]},{0,f[0,1],f[1,2],f[2,3],f[3,4]},{0,0,f[0,1,2],f[1,2,3],f[2,3
