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1、1、 用 Gauss-Seidel 迭代法求解方程组。1.44491+0.79482+0.88013=10.69461+1.95682+0.17303= 00.62131+0.55262+1.97973=1解: function x,n=gauseidel(A,b,x0,eps,M)if nargin=3eps= 1.0e-6;M = 200;elseif nargin = 4M = 200;elseif nargin=epsx0=x;x=G*x0+f;n=n+1;if(n=M)disp(Warning: );return;endend A=1.446 0.7948 0.8801;0.6946
2、 1.9568 0.1730;0.6213 0.5526 1.9797; b=1 0 1; x0=zeros(3,1); x,n=gauseidel(A,b,x0)x = 0.5891-0.24340.3882n =112、 用 Jacobi 迭代法求解方程组。0.98891+0.000520.00023=10.00461+0.99462+0.0077= 00.00021+0.00922+0.99413=1解: function x,n=jacobi(A,b,x0,eps,varargin)if nargin=3eps= 1.0e-6;M = 200;elseif nargin=epsx0=x
3、;x=B*x0+f;n=n+1;if(n=M)disp(Warning: );return;endend A=0.9989 -0.0005 -0.0002;-0.0046 0.9946 0.0077;-0.0002 0.0092 0.9941; b=1 0 1; x0=zeros(3,1); x,n=jacobil(A,b,x0)x = 1.0114-0.00311.0062n =43、 求出下表所列各数据的拉格朗日插值多项式，并计算x=1.6 时 y 的值。X 1 1.2 1.8 2.5 4y 0.8415 0.9320 0.9738 0.5985 -0.7568解：function f =
4、 Language(x,y,x0)syms t;if(length(x) = length(y)n = length(x); elsedisp(xy);return;end %f = 0.0;for(i = 1:n)l = y(i); for(j = 1:i-1)l = l*(t-x(j)/(x(i)-x(j); end;for(j = i+1:n)l = l*(t-x(j)/(x(i)-x(j); %-end;f = f + l; %- simplify(f); %if(i=n)if(nargin = 3)f = subs(f,t,x0); %elsef = collect(f); %f =
5、 vpa(f,6); %6endendend x=1 1.2 1.8 2.5 4; y=0.8415 0.9320 0.9738 0.5985 -0.7568; f=Language(x,y)f =0.0328112*t4 - 0.204917*t3 - 0.0145485*t2 + 1.05427*t - 0.0261189 f=Language(x,y,1.6)f =0.99924、 求出下表所列各数据的均差形式的牛顿插值多项式，并计算 x=1.6 时 y 的值。X 1 1.2 1.8 2.5 4y 1 1.44 3.24 6.25 16解：function f = Newton(x,y,
6、x0)syms t;if(length(x) = length(y)n = length(x);c(1:n) = 0.0;elsedisp(xy);return;endf = y(1);y1 = 0;l = 1;for(i=1:n-1) for(j=i+1:n)y1(j) = (y(j)-y(i)/(x(j)-x(i);end c(i) = y1(i+1); l = l*(t-x(i); f = f + c(i)*l;simplify(f);y = y1;if(i=n-1)if(nargin = 3)f = subs(f,t,x0);elsef = collect(f); %f = vpa(f
7、, 6);endendend x=1 1.2 1.8 2.5 4; y=1 1.44 3.24 6.25 16; f=Newton(x,y)f =2.11471e-16*t4 - 1.69177e-15*t3 + 1.0*t2 - 4.82154e-15*t + 1.82711e-15 f=Newton(x,y,1.6)f =2.56005、 求出下表所列各数据的前向差分形式的牛顿插值多项式，并计算 x=1.55 时 y 的值。X 1 1.2 1.4 1.6 1.8y 0.8415 0.9320 0.9854 0.9996 0.9738解：function f = Newtonforward(
8、x,y,x0)syms t;if(length(x) = length(y)n = length(x);c(1:n) = 0.0;elsedisp(xy);return;endf = y(1);y1 = 0;xx =linspace(x(1),x(n),(x(2)-x(1);if(xx = x)disp();return;endfor(i=1:n-1)for(j=1:n-i)y1(j) = y(j+1)-y(j);endc(i) = y1(1); l = t;for(k=1:i-1)l = l*(t-k);end;f = f + c(i)*l/factorial(i);simplify(f);
9、y = y1;if(i=n-1)if(nargin = 3)f = subs(f,t,(x0-x(1)/(x(2)-x(1);elsef = collect(f);f = vpa(f, 6);endendend x=1:0.2:1.8; y=0.8415 0.9320 0.9854 0.9996 0.9738; f=Newtonforward(x,y) f=Newtonforward(x,y)f =0.0000541667*t4 - 0.000675*t3 - 0.0169042*t2 + 0.108025*t + 0.8415 f=Newtonforward(x,y,1.55)f =0.99
10、986、 求出下表所列各数据的高斯插值多项式，并计算 x=1.5时 y 的值。X 1 1.2 1.4 1.6 1.8y 0.8415 0.9320 0.9854 0.9996 0.9738解：function f = Gauss(x,y,x0)if(length(x) = length(y)n = length(x);elsedisp(xy);return;endxx =linspace(x(1),x(n),(x(2)-x(1);if( mod(n,2) =1)if(nargin = 2)f = GStirling(x,y,n);else if(nargin = 3)f = GStirling
11、(x,y,n,x0);endendelseif(nargin = 2)f = GBessel(x,y,n);else if(nargin = 3)f = GBessel(x,y,n,x0);endendendfunction f = GStirling(x,y,n,x0)syms t;nn = (n+1)/2;f = y(nn);for(i=1:n-1)for(j=i+1:n)y1(j) = y(j)-y(j-1);endif(mod(i,2)=1)c(i) = (y1(i+n)/2)+y1(i+n+2)/2)/2; elsec(i) = y1(i+n+1)/2)/2;endif(mod(i,
12、2)=1) l = t+(i-1)/2;for(k=1:i-1)l = l*(t+(i-1)/2-k);endelsel_1 = t+i/2-1;l_2 = t+i/2;for(k=1:i-1)l_1 = l_1*(t+i/2-1-k);l_2 = l_2*(t+i/2-k); endl = l_1 + l_2;endl = l/factorial(i);f = f + c(i)*l;simplify(f);f = vpa(f, 6);y = y1;if(i=n-1)if(nargin = 4)f = subs(f,t,(x0-x(nn)/(x(2)-x(1);endendendfunctio
13、n f = GBessel(x,y,n,x0)syms t;nn = n/2;f = (y(nn)+y(nn+1)/2;for(i=1:n-1)for(j=i+1:n)y1(j) = y(j)-y(j-1);endif(mod(i,2)=1)c(i) = y1(i+n+1)/2)/2; elsec(i) = (y1(i+n)/2)+y1(i+n+2)/2)/2;endif(mod(i,2)=0) l = t+i/2-1;for(k=1:i-1)l = l*(t+i/2-1-k);endelsel_1 = t+(i-1)/2;l_2 = t+(i-1)/2-1;for(k=1:i-1)l_1 =
14、 l_1*(t+(i-1)/2-k);l_2 = l_2*(t+(i-1)/2-1-k); endl = l_1 + l_2;endl = l/factorial(i);f = f + c(i)*l;simplify(f);f = vpa(f, 6);y = y1;if(i=n-1)if(nargin = 4)f = subs(f,t,(x0-x(nn)/(x(2)-x(1);endendend x=1:0.2:1.8; y=0 0.2630 0.4854 0.6781 0.8480; f=Gauss(x,y) f=Gauss(x,y)f =0.20755*t - 0.007425*t*(t - 1.0) - 0.007425*t*(t + 1.0) + 0.00148333*t*(t + 1.0)*(t - 1.0)
