三种插值方法的MATLAB实现

MATLAB 线性插值 二次插值 拉格朗日插值

插值方法介绍

(1)分段线性插值:

对于待求的x，选取[xi,xi+1]，使x属于[xi,xi+1]。再利用线性插值公式 即可求解。

(2)分段二次插值:

对于待求的x，选取与x距离最近的三个点xi-1,xi,xi+1。再利用二次插值公式 即可求解

(3)全区间上拉格朗日插值:

n=5，利用5次拉格朗日插值公式 即可求解

代码实现

分段线性插值：

function r = li( a )
%UNTITLED 此处显示有关此函数的摘要
%   此处显示详细说明
x=[0.0, 0.1, 0.195, 0.3, 0.401, 0.5];
y=[0.39894, 0.39695, 0.39142, 0.38138, 0.36812, 0.36206];
n = length(x);
m = length(y);
if n~=m
    disp('error program');
    return;
end
if a<x(1) || a>x(n)
    disp('error input');
    return;
end
for i = 2:n
    if a < x(i)
        z = i;
        break;
    end
end
b = (y(z) - y (z-1))/(x(z)-x(z-1));
r = y(z-1) + b*(a-x(z-1));
end

分段二次插值：

function r = twoint( a )
%UNTITLED 此处显示有关此函数的摘要
%   此处显示详细说明
x=[0.0, 0.1, 0.195, 0.3, 0.401, 0.5];
y=[0.39894, 0.39695, 0.39142, 0.38138, 0.36812, 0.36206];
n = length(x);
m = length(y);
if n~=m
    disp('error program');
    return;
end
if a<x(1) || a>x(n)
    disp('error input');
    return;
end
for i = 2:n
    if a < x(i)
        z = i;
        break;
    end
end
if z==2
    k =3;
else
    k =4;
end
if z~=2 && z~=6
  p = x(z+1)-a;
  q = x(z-2)-a;
   if abs(p)<abs(q)
    k = z+1;
   else
    k = z-2;
   end
end
b = (y(z) - y (z-1))/(x(z)-x(z-1));
c = (y(k) - y(z))/(x(k)-x(z));
d = (c-b)/(x(k)-x(z-1));
r = y(z-1) + b*(a-x(z-1)) + d*(a-x(z-1))*(a-x(z));
end

拉格朗日插值：

function [ z ] = wlangu(a)
%   此处显示详细说明
l =ones(1,6);
q = l;
x = [0,0.1,0.195,0.3,0.401,0.5];
y=[0.39894, 0.39695, 0.39142, 0.38138, 0.36812, 0.36206];
for  i = 1:6
    for j =1:6
       if i~=j
         l(i) = l(i)*(x(i) - x(j));
         q(i) = q(i)*(a-x(j));
       end
    end
end
z=0;
for i = 1:6
    z = z + y(i)*q(i)/l(i);
end
end

应用

x 0.0 0.1 0.195 0.3 0.401 0.5
f(x) 0.39894 0.39695 0.39142 0.38138 0.36812 0.35206

从函数表出发，计算f(0.15)，f(0.31)，f(0.47)的近似值

计算结果：

分段线性插值： f(0.15)=0.3944，f(0.31)=0.3802，f(0.47)=0.3619

分段二次插值： f(0.15)=0.3938，f(0.31)=0.3800，f(0.47)=0.3632

拉格朗日插值： f(0.15)=0.3944，f(0.31)=0.3802，f(0.47)=0.3619