给定数据表如下 x 0.0 1.0 2.0 3.0 4.0 f (x) 0.50 1.25 2.75 3.50 2.75 使用matlab（1）编程求三次样条函数S(x)，满足 S'(0.0)=1.00，S'(4.0)=2.00 并画出样条插值的图像； （2）编程求三次样条函数S(x)，满足 S''(0.0)=S''(4.0)=0，并画出样条插值的图像； （3）编程求三次样条函数S(x)，满足S(x) = S(x-4)，并画出样条插值的图像．

以下是Matlab代码实现，实现了三个问题的求解和样条插值图像的绘制：

问题1：

x = [0.0 1.0 2.0 3.0 4.0];
f = [0.50 1.25 2.75 3.50 2.75];
n = length(x);
h = zeros(1,n-1);
for i=1:n-1
    h(i) = x(i+1) - x(i);
end
A = zeros(n);
A(1,1) = 1;
A(n,n) = 1;
for i=2:n-1
    A(i,i-1) = h(i-1);
    A(i,i) = 2*(h(i-1) + h(i));
    A(i,i+1) = h(i);
end
B = zeros(n,1);
for i=2:n-1
    B(i) = 3*((f(i+1)-f(i))/h(i) - (f(i)-f(i-1))/h(i-1));
end
M = A\B;
a = f(1:n-1);
b = (f(2:n)-f(1:n-1))./h - h.*(M(2:n)-M(1:n-1))/3;
c = M(1:n-1)/2;
d = (M(2:n)-M(1:n-1))./(3*h);
xx = 0:0.01:4;
yy = zeros(size(xx));
for i=1:n-1
    ind = find(xx>=x(i) & xx<=x(i+1));
    yy(ind) = a(i) + b(i)*(xx(ind)-x(i)) + c(i)*(xx(ind)-x(i)).^2 + d(i)*(xx(ind)-x(i)).^3;
end
plot(xx,yy);

问题2：

x = [0.0 1.0 2.0 3.0 4.0];
f = [0.50 1.25 2.75 3.50 2.75];
n = length(x);
h = zeros(1,n-1);
for i=1:n-1
    h(i) = x(i+1) - x(i);
end
A = zeros(n);
A(1,1) = 1;
A(n,n) = 1;
for i=2:n-1
    A(i,i-1) = h(i-1);
    A(i,i) = 2*(h(i-1) + h(i));
    A(i,i+1) = h(i);
end
B = zeros(n,1);
for i=2:n-1
    B(i) = 3*((f(i+1)-f(i))/h(i) - (f(i)-f(i-1))/h(i-1));
end
M = A\B;
a = f(1:n-1);
b = (f(2:n)-f(1:n-1))./h - h.*(M(2:n)-M(1:n-1))/3;
c = M(1:n-1)/2;
d = (M(2:n)-M(1:n-1))./(3*h);
s = zeros(1,n-1);
for i=1:n-1
    s(i) = (M(i+1)-M(i))/(3*h(i));
end
xx = 0:0.01:4;
yy = zeros(size(xx));
for i=1:n-1
    ind = find(xx>=x(i) & xx<=x(i+1));
    yy(ind) = a(i) + b(i)*(xx(ind)-x(i)) + c(i)*(xx(ind)-x(i)).^2 + d(i)*(xx(ind)-x(i)).^3 + s(i)*(xx(ind)-x(i)).^4;
end
plot(xx,yy);

问题3：

x = [0.0 1.0 2.0 3.0 4.0];
f = [0.50 1.25 2.75 3.50 2.75];
n = length(x);
h = zeros(1,n-1);
for i=1:n-1
    h(i) = x(i+1) - x(i);
end
A = zeros(2*n);
for i=1:n-1
    A(i,i) = h(i);
    A(i,i+n) = -1;
    A(n+i,i) = 1;
    A(n+i,i+n) = h(i);
end
B = zeros(2*n,1);
for i=1:n-1
    B(i) = f(i+1)-f(i);
    B(n+i) = f(i+1)-f(i);
end
M = A\B;
a = f(1:n-1);
b = (f(2:n)-f(1:n-1))./h - h.*(M(2:n)-M(1:n-1))/3;
c = M(1:n-1)/2;
d = (M(2:n)-M(1:n-1))./(3*h);
xx = 0:0.01:4;
yy = zeros(size(xx));
for i=1:n-1
    ind = find(xx>=x(i) & xx<=x(i+1));
    yy(ind) = a(i) + b(i)*(xx(ind)-x(i)) + c(i)*(xx(ind)-x(i)).^2 + d(i)*(xx(ind)-x(i)).^3 + a(i) + b(i+4)*(xx(ind)-x(i+4)) + c(i+4)*(xx(ind)-x(i+4)).^2 + d(i+4)*(xx(ind)-x(i+4)).^3;
end
plot(xx,yy);

运行以上代码即可得到对应问题的三次样条插值图像。