golden_section.m

无约束优化问题一维搜索的常用方法：黄金分割法（golden_section.m）、加步搜索法（plus_step_search.m）、牛顿法（newton.m）、抛物线法（parabola.m）

function [mini,point,times]=golden_section(str,ini,acc)
%  File name:golden_section.m      
%  Author:Yan Anxin
%  ID number:081810
%  Date:finished on 2008.11.24
%  Description:  
%       Here defines a function,to solve the one-dimensional searching problems by golden-section-method
%       In the syntax [mini,point,times]=golden_section(str,ini,acc),
%       str----input,means the expression of objective function
%       ini----input,sets the starting interval to ini 
%       err----input,sets admissible error
%       mini=golden_section(str,ini,acc) attempts to find mini, i.e. the minimum of objective function   
%       [mini,point]=golden_section(str,ini,acc) returns point, i.e. the independent variable value corresponding to minimum 
%       [mini,point,times]=golden_section(str,ini,acc) returns times, i.e. the times of iterative       
f=sym(str);
syms x
sca=(sqrt(5)-1)/2;        %golden ratio
k=1;times=0;              %setting initial value
a(k)=ini(1);b(k)=ini(2);
y(k+1)=a(k)+sca*(b(k)-a(k));
z(k+1)=a(k)+(1-sca)*(b(k)-a(k));

while 1                  %to narrow the search interval
if subs(f,x,z(k+1))<=subs(f,x,y(k+1))           
    a(k+1)=a(k);b(k+1)=y(k+1);
    if ((b(k+1)-a(k+1))/(b(1)-a(1)))<acc,break,end
    y(k+2)=z(k+1);z(k+2)=a(k+1)+(1-sca)*(b(k+1)-a(k+1));
    k=k+1;times=times+1;  
else                                           
    a(k+1)=z(k+1);b(k+1)=b(k);
    if ((b(k+1)-a(k+1))/(b(1)-a(1)))<acc,break,end
    z(k+2)=y(k+1);y(k+2)=a(k+1)+sca*(b(k+1)-a(k+1));
    k=k+1;times=times+1;      
end
end
point=0.5*(a(k+1)+b(k+1));
mini=subs(f,x,point);