function [min,solvec,stepnum]=variable_rotation(str,ini,err)
%  File name:variable_rotation.m      
%  Author:Yan Anxin
%  ID number:081810
%  Date:finished on 2008.11.18
%  Description:  
%       Here defines a function,to solve the multivariate unconstrained optimization problems by variable-rotation-method
%       In the syntax [min,solvec,stepnum]=variable_rotation(str,ini,err),
%       str----input,means the expression of objective function
%       ini----input,sets the starting point to ini 
%       err----input,sets admissible error
%       min=variable_rotation(str,ini,err) attempts to find min, i.e. the minimum of objective function   
%       [min,solvec]=variable_rotation(str,ini,err) returns solvec, i.e. the minimum point  
%       [min,solvec,stepnum]=variable_rotation(str,ini,err) returns stepnum, i.e. the times of iterative       

n=length(ini);       %gets the number of variables 
k=1;stepnum=0;x(:,k)=ini;
syms lam             %defines searching step----lam

while 1
    for t=1:n
        g=x(t,k)+lam;
        f=str;
        for p=1:n
            s=sym(['x' num2str(p)]);
            if t==p                
                f=subs(f,s,g);
            else
            f=subs(f,s,x(p,k));
            end
        end   
        f1=diff(f,lam);
        lamb=solve(f1,lam);     %gets the best step
        x(:,k+1)=x(:,k);x(t,k+1)=x(t,k+1)+lamb;  
        k=k+1;stepnum=stepnum+1;
    end
    if norm(x(:,k)-x(:,1))<=err,break,end       %halts cycle if meeting the requirements of error
    x(:,1)=x(:,k);
    k=1;
end

solvec=x(:,k);
min=minval(str,solvec);         %calls the function minval(str,solvec) to obtain the minimum value