interpolation points x values
时间: 2024-05-29 10:12:31 浏览: 8
Interpolation points x values refer to the values of the independent variable (x) at which the function is known or given. These values are used to approximate the value of the dependent variable (y) at any point within the range of the given data. Interpolation is a method of estimating values that lie between the known data points. The accuracy of the interpolation depends on the number of interpolation points and the quality of the data. The x values should be evenly spaced for the most accurate interpolation.
相关问题
% 定义函数f(x)和区间[a, b] f = @(x) 1./(1 + x.^2); a = -5; b = 5; % 计算n+1个等距节点 n_values = [2, 4, 6, 8, 10, 20]; for i = 1:length(n_values) n = n_values(i); x = linspace(a, b, n+1); y = f(x); % 计算牛顿插值多项式和节点中点的近似值 p = @(t) newton_interpolation(x, y, t); midpoints = (x(1:end-1) + x(2:end))/2; % 绘制图像 subplot(2,3,i); fplot(f, [a, b], 'k'); hold on; fplot(p, [a, b], 'r'); plot(midpoints, p(midpoints), 'bo'); title(sprintf('n = %d', n)); legend('f(x)', 'p(x)'); hold off; end % 牛顿插值多项式的计算函数 function p = newton_interpolation(x, y, t) n = length(x) - 1; b = y; for j = 1:n for i = n+1:-1:j+1 b(i) = (b(i) - b(i-1))./(x(i) - x(i-j)); end end p = b(n+1); for j = n:-1:1 p = (t - x(j)).*p + b(j); end end给每行代码加注释
% 定义函数f(x)和区间[a, b]
f = @(x) 1./(1 + x.^2); % 定义函数f(x)为 1 / (1 + x^2)
a = -5; % 定义区间左端点为 -5
b = 5; % 定义区间右端点为 5
% 计算n+1个等距节点
n_values = [2, 4, 6, 8, 10, 20]; % 定义要计算的节点数
for i = 1:length(n_values) % 对每个节点数进行循环
n = n_values(i); % 取出当前节点数
x = linspace(a, b, n+1); % 生成n+1个等距节点
y = f(x); % 计算每个节点的函数值
% 计算牛顿插值多项式和节点中点的近似值
p = @(t) newton_interpolation(x, y, t); % 定义牛顿插值多项式
midpoints = (x(1:end-1) + x(2:end))/2; % 计算相邻节点的中点
% 绘制图像
subplot(2,3,i); % 将当前图像画在2x3的子图中的第i个位置
fplot(f, [a, b], 'k'); % 绘制原函数f(x)的图像
hold on; % 保持图像不变,方便后面继续绘制
fplot(p, [a, b], 'r'); % 绘制牛顿插值多项式的图像
plot(midpoints, p(midpoints), 'bo'); % 在相邻节点中点处绘制多项式的近似值
title(sprintf('n = %d', n)); % 在图像上添加标题,显示当前节点数
legend('f(x)', 'p(x)'); % 添加图例,方便区分f(x)和p(x)
hold off; % 结束绘制,释放图像
end
% 牛顿插值多项式的计算函数
function p = newton_interpolation(x, y, t)
n = length(x) - 1; % 计算节点数
b = y; % 初始化差商表,将y赋值给b
for j = 1:n % 对每列进行循环
for i = n+1:-1:j+1 % 对每行进行循环
b(i) = (b(i) - b(i-1))./(x(i) - x(i-j)); % 计算差商表的每个元素
end
end
p = b(n+1); % 取出差商表的最后一个元素,即多项式的最高次项系数
for j = n:-1:1 % 从最高次项开始对每项进行循环
p = (t - x(j)).*p + b(j); % 计算多项式的每一项
end
end
牛顿插值法matlab
以下是一个使用牛顿插值法的 MATLAB 代码示例:
function [p] = newton_interpolation(x, y, xi)
% x: known data points x values
% y: known data points y values
% xi: interpolation points x values
% p: interpolation points y values
n = length(x);
b = zeros(n, n);
% initialize the first column of b with y values
for i = 1:n
b(i, 1) = y(i);
end
% construct the rest of b using divided differences
for j = 2:n
for i = j:n
b(i, j) = (b(i, j-1) - b(i-1, j-1)) / (x(i) - x(i-j+1));
end
end
% evaluate the polynomial at xi
p = zeros(size(xi));
for i = 1:length(xi)
s = b(1, 1);
for j = 2:n
prod = 1;
for k = 1:j-1
prod = prod * (xi(i) - x(k));
end
s = s + b(j, j) * prod;
end
p(i) = s;
end
end
你可以将此代码保存为一个名为“newton_interpolation.m”的函数文件,并在 MATLAB 中调用它。下面是一个示例用法:
x = [0 1 2];
y = [1 4 3];
xi = 0:0.1:2;
p = newton_interpolation(x, y, xi);
plot(x, y, 'o', xi, p, '-')
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