newton.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]=newton(str,ini,err)
%  File name:newton.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 newton-method
%       In the syntax [mini,point,times]=newton(str,ini,err),
%       str----input,means the expression of objective function
%       ini----input,sets the starting point to ini 
%       err----input,sets admissible error
%       mini=newton(str,ini,err) attempts to find mini, i.e. the minimum of objective function   
%       [mini,point]=newton(str,ini,err) 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
k=1;times=0;y(k)=ini;           %setting initial value
while 1                         %applying Newton's iterative method
    df=diff(f);ddf=diff(df);
    if abs(subs(df,x,y(k)))<err,break,end
    y(k+1)=y(k)-subs(df,x,y(k))/subs(ddf,x,y(k));
    k=k+1;times=times+1;
end
point=y(k);
mini=subs(f,x,point);