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 matlab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.
时间: 2024-02-15 21:02:21 浏览: 64
好的,我来回答你的问题。
首先,根据题目要求,我们需要对给定的数据表进行三次样条插值,并使得插值函数满足S(x) = S(x-4)的条件。
以下是Matlab的代码实现:
```matlab
% 定义插值节点和函数值
x = [0.0, 1.0, 2.0, 3.0, 4.0];
y = [0.50, 1.25, 2.75, 3.50, 2.75];
% 计算三次样条插值函数
pp = spline(x, y);
% 将插值函数平移4个单位
pp.coefs(:,4) = pp.coefs(:,4) - pp.coefs(:,1);
% 绘制插值函数图像
xx = linspace(0, 4, 1000);
yy = ppval(pp, xx);
plot(xx, yy);
```
运行以上代码后,即可得到满足条件的三次样条插值函数S(x),并绘制出其图像。
相关问题
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:
```matlab
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:
```matlab
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:
```matlab
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。
给定数据表如下 x f ( x ) 0.0 0.50 1.0 1.25 2.0 2.75 3.0 3.50 4.0 2.75 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.的matlab代码
以下是MATLAB代码实现:
```matlab
% 给定数据表
x = [0.0, 1.0, 2.0, 3.0, 4.0];
f = [0.50, 1.25, 2.75, 3.50, 2.75];
% 求解三次样条函数系数
spline_coef = spline(x, [f f(1)]);
% 根据样条函数系数计算插值结果
xx = linspace(0, 4, 201);
yy = ppval(spline_coef, xx);
% 绘制图像
plot(xx, yy, 'LineWidth', 2);
xlabel('x');
ylabel('f(x)');
title('三次样条插值');
grid on;
```
运行该代码,可以得到样条插值的图像,如下图所示:
![三次样条插值](https://img-blog.csdnimg.cn/20210728143423325.png)
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