MATLAB lagrange插值 三次样条插值

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% Written by QQ137712826
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% 习题  对实验数据进行 lagrange 插值、三次样条插值
%          绘制出实验数据散点图，lagrange插值函数图，三次样条插值图
% ------------------------------------------------------------------------------------------------

% --------------------------------------------------------------------------------------------
% 实验数据 x y 初始赋值，画出原始数据折线图
% --------------------------------------------------------------------------------------------
clc,clear
x=[0.0 0.5 1.0 6.0 7.0 9.0];
y=[0.0 1.6 2.0 1.5 1.5 0.0];
scatter(x,y,'k');                                                      %画出原始数据散点图
hold on
% --------------------------------------------------------------------------------------------
% lagrange插值过程
% --------------------------------------------------------------------------------------------
syms t;                                                                 %系统变量
n = length(x);                                                       %x长度
f = 0.0;                                                                %lagrange插值多项式初始化
for i = 1:n                                                            %循环计算基函数
    l = y(i);
    for j = 1:i-1                                                       %计算基函数某个项前面一截，i之前的
        l = l*(t-x(j))/(x(i)-x(j));
    end
    for j = i+1:n                                                      %计算基函数某个项后面一截，i之后的
        l = l*(t-x(j))/(x(i)-x(j));
    end
    f = f + l;                                                           %得出最后的lagrange插值多项式
    simplify(f);                                                       %化简多项式
    if(i==n)
        f = collect(f);                                                %合并同类项
        f = vpa(f,6);                                                  %截断表达
    end
end
disp("lagrange插值多项式结果：");  
disp(f);                                                                %打印最后的lagrange插值多项式
% --------------------------------------------------------------------------------------------
% 画图
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x0=linspace(0,9,200);
f0 = subs(f,'t',x0);
plot(x0,f0,'r');                                                       %画图
hold on

% --------------------------------------------------------------------------------------------
% 实验数据 x y 初始赋值
% --------------------------------------------------------------------------------------------
x=[0.0 0.5 1.0 6.0 7.0 9.0];
y=[0.0 1.6 2.0 1.5 1.5 0.0];

% --------------------------------------------------------------------------------------------
% 三次样条 插值过程
% --------------------------------------------------------------------------------------------


n = length(x);                                                       %计算长度
a = y(1 : end - 1);                                                 %插入n-1个
b = zeros(n - 1, 1);                                               % 三次样条插值函数的系数
d = zeros(n - 1, 1);
dx = diff(x);                                                          %计算剖分差距h
dy = diff(y);                                                          %计算剖分差距
A = zeros(n);
B = zeros(n, 1);
A(1, 1) = 1;                                                          %边界条件
A(n, n) = 1;                                                          %边界条件
% --------------------------------------------------------------------------------------------
% 构建系数矩阵A*c=B
% --------------------------------------------------------------------------------------------
for i = 2 : n - 1
    A(i, i - 1) = dx(i - 1);
    A(i, i) = 2*(dx(i - 1) + dx(i));
    A(i, i + 1) = dx(i);
    B(i) = 3*(dy(i) / dx(i) - dy(i - 1) / dx(i - 1));
end
c = A \ B;                                                             %计算c
% --------------------------------------------------------------------------------------------
% 三次样条插值函数的系数
% --------------------------------------------------------------------------------------------
for i = 1 : n - 1
    d(i) = (c(i + 1) - c(i)) / (3 * dx(i));
    b(i) = dy(i) / dx(i) - dx(i)*(2*c(i) + c(i + 1)) / 3;
end
% --------------------------------------------------------------------------------------------
%  将点带入插值公式计算结果
% --------------------------------------------------------------------------------------------
xx=linspace(0,9,200);
XISHU=zeros(1,4);                                               %  三次样条系数

[mm, nn] = size(xx);
yy = zeros(mm, nn);
for i = 1 : mm*nn                                                  %  计算每一个元素
    for ii = 1 : n - 1                                                 %  选取分段函数系数
        if xx(i) >= x(ii) && xx(i) < x(ii + 1)
            j = ii;
            break;
        elseif xx(i) == x(n)
            j = n - 1;
        end
    end
    yy(i) = a(j) + b(j)*(xx(i) - x(j)) + c(j)*(xx(i) - x(j))^2 + d(j)*(xx(i) - x(j))^3;%计算插值结果
    XISHU=[XISHU;[a(j) , b(j),c(j),d(j)]];
end
XISHU=unique(XISHU,'rows');                              %  分段函数系数
XISHU=XISHU(2:6,:);
disp("三次样条分段函数系数矩阵：");
disp(XISHU);
% --------------------------------------------------------------------------------------------
%  画三次样条图
% --------------------------------------------------------------------------------------------
plot(xx,yy,'b');
hold on

legend("原始数据","lagrange插值","三次样条插值");
title("插值结果图")%标题

结果：