1. Algorithm principle
Set the curve F (x) and find its extreme value interval [x1, x2], so that it satisfies f (x1)> f ((x1 + x2) / 2), f ((x1 + x2) / 2) <f (x2), use the values of these three points to fit a parabolic equation
f (x) = ax ^ 2 + bx + c, abc is the coefficient.
| ax1^2+bx1+c=f(x1) |
| ax2^2+bx2+c=f(x2) |
| ax3^2+bx3+c=f(x3) Written in matrix form x1^2 x1 1 a f(x1) x2^2 x2 1 * b = f(x2) x3^2 x3 1 c f(x3) Simplified to vander (x1 x2 x3) * [abc] '= [f (x1) f (x2) f (x3)]' After using Matlab to find the expression, find the minimum point xp of f (x). Determine the value of f (xp) and f ((x1 + x2) / 2) If f (xp)> f ((x1 + x2) / 2) 1, xp <(x1 + x2) / 2 <x2, the new interval is [xp, x2]; 2. (x1 + x2) / 2 <xp <x2, then the new interval [x1, xp] If f (xp) <f ((x1 + x2) / 2) 1, x1 <xp <(x1 + x2) / 2, then the new interval is [x1, (x1 + x2) / 2]; 2. (x1 + x2) / 2 <xp <x2, then the new interval [(x1 + x2) / 2, x2]; The principle is to ensure that the value of the function is distributed in a high degree, and iterate continuously.
|
Second, the matlab program
clc
clear
f=@(x) x.^3-6*x+9;
ezplot(f,[-100 100])
[x,fx]=Min_erci(f,[0 5],100) % a 函数值 b横坐标
function [x,result]=Min_erci(f,x0,k) %x0为初始区间端点，k为迭代次数
x1=x0(1);
x3=x0(2);
x2=(x1+x3)/2;
n=1;
while n < k
% 确定抛物线的系数
f1=f(x1);
f2=f(x2);
f3=f(x3);
A=[x1^2 x1 1;
x2^2 x2 1;
x3^2 x3 1;];
b=[f1;f2;f3];
XS=A\b; %求出抛物线系数a b c 存放在xs中
xp=-XS(2)/(2*XS(1)); %二次多项式的极值点在x=-b/2a
fp=f(xp); %求出该点函数值
if abs(xp-x2) < 1e-8 %该点满足极值点条件
x=xp; %输出极值点
result=f(x); %输出函数值
return;
end
if fp > f2 %判断新的迭代区间
if xp < x2
x1=xp;
else
x3=xp;
end
else
if xp < x2
x3=x2;
x2=xp;
else
x1=x2;
x2=xp;
end
end
n=n+1; %迭代次数+1
end
if n == k %如果超出迭代次数
x=[]; %输出空
result=[];
disp('超过迭代次数');
end
end
