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计算物理作业1 浦正国 2013202020010 已知函数节点 、计算和拉格朗日插值多项式的值， 、求带有自然边界条件的三次样条函数表达式中的参数计算 、画出的图形（各描41点） 运行程序（Matlab） x0=-1:.2:1;y0=1./(1+25.*(x0.^2));%列出已知点x0.y0 syms x l %求拉格朗日插值基 for j=1:11 ll=1; pp=1; for i=1:11 if i==j continue; end ll=(x-x0(i))*ll; pp=(x0(j)-x0(i))*pp; end l(j)=ll/pp; end l(11) y=0; for i=1:11 y=y0(i)*l(i)+y; %拉格朗日差值函数 end disp(y); x1=-1:.05:1; y1=subs(y,x,x1); figure(1) subplot(1,2,1) x2=-1:.05:1; y2=1./(1+25.*(x2.^2));y3=subs(l(11),x,x1); %用x2,y2表示被插函数 plot(x1,y1,-.r,x2,y2,x1,y3); legend(拉格朗日插值,原函数); a=1/2;b=ones(1,11); %自然边界条件的三次样条函数 b(1)=15*(y0(2)-y0(1))-0.1*(2*2500-50*26)/(26^3); for i=2:10 b(i)=15/2*(y0(i+1)-y0(i-1)); end b(11)=15*(y0(11)-y0(10))+0.1*(2*2500-50*26)/(26^3); A=ones(1,11);B=ones(1,11); A(1)=-1/2;B(1)=b(1)/2; for i=2:11 A(i)=-1*a/(2+(1-a)*A(i-1)); B(i)=(b(i)-(1-a)*B(i-1))/(2+(1-a)*A(i-1));%求得A（i）,B(i) end m=ones(1,11); m(11)=B(11); for i=10:-1:1 m(i)=A(i)*m(i+1)+B(i); %求得m(i) end for i=1:10 x3(1+(i-1)*5:i*5)=[-1+(i-1)*0.2:.05:-1+i*0.2]; y3(1+(i-1)*5:i*5)=[(1+2.*(x3(1+(i-1)*5:i*5)-x0(i))./(x0(i+1)-x0(i))).*(( x3(1+(i-1)*5:i*5)-x0(i+1))./(x0(i)-x0(i+1))).^2.*y0(i)+(1+2.*(x3(1+(i-1)*5:i*5)-x0(i+1))./(x0(i)-x0(i+1))).*(( x3(1+(i-1)*5:i*5)-x0(i))./(x0(i+1)-x0(i))).^2.*y0(i+1)+... (x3(1+(i-1)*5:i*5)-x0(i)).*((x3(1+(i-1)*5:i*5)-x0(i+1))./(x0(i)-x0(i+1))).^2.*m(i)+(x3(1+(i-1)*5:i*5)-x0(i+1)).*((x3(1+(i-1)*5:i*5)-x0(i))./(x0(i+1)-x0(i))).^2.*m(i+1)]; end subplot(1,2,2) plot(x3,y3,-.r,x2,y2); legend(三次样条插值,原函数); title(自然边界条件的三次样条插值函数); 运行结果 ans = (18014398509481984*x*(x + 1)*(x - 1/5)*(x + 1/5)*(x - 2/5)*(x + 2/5)*(x - 3/5)*(x + 3/5)*(x - 4/5)*(x + 4/5))/6693954489467723 (72057594037927936*x*(x - 1)*(x + 1)*(x - 1/5)*(x + 1/5)*(x - 2/5)*(x + 2/5)*(x - 3/5)*(x - 4/5)*(x + 4/5))/5950181768415755 - (144115188075855872*x*(x - 1)*(x + 1)*(x -