for j = 1:n+1 if j~=i c = (x-x1(j))/(x1(i)-x1(j)); ss = ss.*c; else continue end end y = y+ss.*y1(i); end % 插值多项式 x = -5:0.01:5; yy = eval(y); plot(x,yy) % 龙格现象图 End clc clear %% c题问题主程序 % Runge现象的呈现的程序 a = [5 10]; a1=-5;a2=5; for i = 1:length(a) Runge(a1,a2,a(i)); hold on end xlabel('Runge现象图') %% 分段线性插值 x1 = a1:(a2-a1)/a(2):a2; y1 = 1./(1+x1.^2); syms x s1 = []; for i = 1:a(2) b0 = (1-(x-x1(i))/(x1(i+1)-x1(i)))*y1(i); b1 = ((x-x1(i))/(x1(i+1)-x1(i)))*y1(i+1); s1{1,i} = b0+b1; end %% 分段三次插值 x1 = a1:(a2-a1)/a(2):a2; y1 = 1./(1+x1.^2); y2 = -2.*(x1)./((1+x1.^2).^2); s3 = []; for i = 1:a(2) h = x1(i+1)-x1(i); c0 = (((x-x1(i))/(x1(i+1)-x1(i))-1)^2)*(2*(x-x1(i))/(x1(i+1)-x1(i))+1)*y1(i); c1 = ((x-x1(i))/(x1(i+1)-x1(i)))^2*(-2*(x-x1(i))/(x1(i+1)-x1(i))+3)*y1(i+1); c2 = ((x-x1(i))/(x1(i+1)-x1(i)))*((x-x1(i))/(x1(i+1)-x1(i))-1)^2*(y2(i)); c3 = ((x-x1(i))/(x1(i+1)-x1(i)))^2*((x-x1(i))/(x1(i+1)-x1(i))-1)*y2(i+1); s3{1,i} = c0+c1+c2+c3; end %% 画图 figure x11 = a1:(a2-a1)/(a(2)*100):a2; y11 = 1./(1+x11.^2); % 分段线性插值图 for i = 1:a(2) x = x1(i):(a2-a1)/(10*a(2)):x1(i+1); yy = eval(s1{1,i}); plot(x,yy,'r'); hold on 11
end plot(x11,y11) % axis([a1,a2,-0.5,2]); xlabel('分段线性图 红色：插值 蓝色：原值') figure % 分段三次插值 for i = 1:a(2) x = x1(i):(a2-a1)/(10*a(2)):x1(i+1); yy = eval(s3{1,i}); plot(x,yy,''); hold on end plot(x11,y11) % axis([a1,a2,-0.5,2]); xlabel('分段线性图 红色：插值 蓝色：原值') D题实验程序 function [y0,N] = lagrange_eval(X,Y,x0) %% lagrange插值 % X，Y是一直的插值点的坐标点 % x0是插值点 % y0是lagrange多项式在x0处的值 % N是lagrange插值函数的权系数 m = length(X); N = zeros(m,1); y0 = 0; for i = 1:m N(i) = 1; for j = 1:m if j ~= i N(i) = N(i)*(x0-X(j))/(X(i)-X(j)); end end y0 = y0+Y(i)*N(i); end clc clear %% lagrange插值（D题） [v d] = xlsread('data.xls'); x = v(1,:); y = v(2,:); % 数据的读取 x1 = [0.5626 0.5635 0.5645]; %需插值的点 y1 = []; for i = 1:length(x1) [y0,N] = lagrange_eval(x,y,x1(i)); y1 = [y1 y0]; end fprintf('lagrange插值函数的结果 ') y1 % lagrange插值函数的结果
实验总结（由学生填写）：
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