样条插值的算例

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,样条插值的算例,三次样条的概念,用一阶导数表示的样条,三次样条的极性,数值分析,16,引,例,.,sin,x,在区间,0,，,上的插值逼近,1.,二次插值,2.,两点埃尔米特插值,3.,分段埃尔米特插值,x,0,/2,Sin,x,010,Cos,x,101,x,0,Sin,x,00,Cos,x,11,2/18,x=-5:5;,y=1./(1+x.2);,plot(x,y,x,y,o),x=-5:5;,y=1./(1+x.2);,xi,=-5:.05:5;,yi,=,spline,(x,y,xi,);,plot(,xi,yi,b,x,y,ro,),被插值,函数:,-,5,x,5,3/18,x=0,0.0155,0.1485,0.3493,0.6480,1.0547,2.0;,y=0,0.1242,0.3654,0.4975,0.5472,0.4781,0;,n=length(x);,t=0:n-1;,tt,=0:.25:n-1;,xx,=,spline,(t,x,tt,);,yy,=,spline,(t,y,tt,);,plot(,xx,yy,x,y,o),4/18,定义,5.4,给定区间,a,b,上的一个分划,:,a=x,0,x,1,x,n,=b,已知,f,(,x,j,)=,y,j,(,j=,0,1,n,),如果,满足,:(1),S,(,x,),在,x,j,，,x,j,+,1,上为三次多项式,;,(2),S”,(,x,),在区间,a,，,b,上连续,;,(3),S,(,x,j,)=,y,j,(,j=,0,1,n,).,则,称,S,(,x,),为三次样条插值函数.,5/18,当,x,x,j,x,j,+,1,(,j=,0,1,n-,1),时,S,j,(,x,)=,a,j,+,b,j,x+,c,j,x,2,+,d,j,x,3,插值条件,:,S,(,x,j,)=,y,j,(,j=,0,1,n,),连续性条件,:,S,(,x,j,+,0)=S(,x,j,-,0),(,j=,1,n-,1),S,(,x,j,+,0)=S(,x,j,-,0),(,j=,1,n-,1),S”,(,x,j,+,0)=S”(,x,j,-,0),(,j=,1,n-,1),由样条定义,可建立方程,(4,n,-2),个！,n,个三次多项式,待定系数共,4,n,个,！,方程数少于未知数个数,？,6/18,(1),自然边界条件,:,S”,(,x,0,)=0,S”,(,x,n,)=0,例,5.7,已知,f,(1)=1,f,(0)=0,f,(1)=1.,求,1，1,上的三次自然样条(满足自然边界条件,).,解 设,则,有,:,a,1,+,b,1,c,1,+,d,1,=1,d,1,=0,a,2,+,b,2,+,c,2,+,d,2,=1,d,1,=d2,c,1,=,c,2,b,1,=,b,2,(2),周期边界条件,:,S,(,x,0,)=,S,(,x,n,),S”,(,x,0,)=,S”,(,x,n,),(3),固定边界条件,:,S,(,x,0,)=,f,(,x,0,),S,(,x,n,)=,f,(,x,n,),7/18,由自然边界条件,:6,a,1,+2,b,1,=0,6,a,2,+2,b,2,=0,解,方程组,得,a,1,=-,a,2,=,1/2,b,1,=b,2,=3/2,c,1,=c,2,=d,1,=d,2,=,0,问题的解,x=-1,0,1;y=1,0,1;,f1=inline(0.5*x.3+1.5*x.2);,f2=inline(-0.5*x.3+1.5*x.2);,t1=-1:.1:0;t2=0:.1:1;,p1=f1(t1);p2=f2(t2);,plot(x,y,o,t1,t2,p1,p2,r),Hold on,plot(t1,t2,t1,t2.2),y=x,2,8/18,用,分段,Hermite,两点插值推导样条,已知函数表,x x,0,x,1,x,n,f,(,x,),y,0,y,1,y,n,设,f,(,x,),在各插值节点,x,j,处,的一阶导数为,m,j,取,x,j,+,1,x,j,=,h,，(,j=,0,1,2,n,).,当,x,x,j,x,j,+,1,时,分段,Hermite,插值,9/18,由,S”,(,x,),连续,有等式,:,S”,(,x,j,+,0)=,S”,(,x,j,0),考虑,S”,(,x,),在区间,x,j,x,j,+,1,和,x,j,-,1,x,j,上表达式,.,当,x,x,j,x,j,+,1,时,S,(,x,),由基函数组合而成,10/18,11/18,同理,有,联立两,式,得,(J=1,2,n-1),自然边界条件,:,S”,(,x,0,)=0,S”,(,x,n,)=0,12/18,例,5.7,已知函数表,x,1 0 1,f,(,x,),1 0 1,m,0,=-3/2,m,1,=0,m,2,=3/2,x,1 0 1,H,(,x,),1 0 1,H,(,x,),-3/2 0 3/2,求,1，1,上的,三次自然样条,(,满足自然边界条件,).,13/18,x,-1,0,x,0,1,第1个小区间,曲率计算公式,第2个小区间,14/18,样条插值,函数的极性,设,f,(,x,)C,2,a,b,对于,a=x,0,x,1,x,n,=b,有,f,(,x,j,)=,y,j,(,j=,0,1,n).S(x),是满足,S,(,x,j,)=,y,j,(,j=,0,1,n),的三次,自然样条.则有,|,S”,(,x,)|,|,f”,(,x,)|,证明,:,15/18,所以,即,样条,函数,S(,x,),在,a,b,上的总曲率最小,.,16/18,一维插值,:,yi,=interp1(x,y,xi,method,),method,nearest,最近点插值,linear,线性插值,spline,样条插值,cubic,立方插值,x=0:10;,y=sin(x);,xi,=0:.25:10;,yi,=interp1(x,y,xi,);plot(x,y,o,xi,yi,),二维插值,zi,=interp2(x,y,z,xi,yi,method,),三维插值,vi=interp3(x,y,z,v,xi,yi,zi,method,),17/18,例,1.正弦曲线数据插值试验,