%%%做系统识别很重要，方法上完全符合系统识别最基础的理论

function [sun]=main(n)
fplot('1/(x+2)',[-1,1],'r');
x=ones(n+2,1);
for j=0:n+1
    x(j+1)=cos(pi*(n+1-j)/(n+1));
end
first=ones(n+2,1); 
f=1./(x+2);  %原函数
last=first;
for j=2:n+2
   last(j)=(-1)*last(j-1);
end
A=ones(n+2,n+2);
A(:,1)=first;
A(:,n+2)=last;
for j=2:n+1
    for t=2:j
        A(:,j)=x.*A(:,j);
    end
end
e=(1e-15)*first; %精度控制条件
sun=A\f;
while (1)    
    at=''; 
    for i=1:n
        for t=1:i
            if (t==1)
                at=strcat('(',num2str(sun(t+1)),')');
            elseif (t>1)
                xt='t';j=2; 
                while j<t
                    xt=strcat('t*',xt);j=j+1;
                end 
                at=strcat(num2str(t),'*(',num2str(sun(t+1)),')*',xt,'+',at);
            end
        end
    end
    %以下得到逼近函数
    ap1=sun(1:n+1,[1]);
    for i=1:n+1
        ap(i)=ap1(n+2-i);
    end
    yt=strcat('-1/(t+2)^2=',at);
    [y]=solve(yt,'t');
    y=numeric(y);
    %以下得到一组新的交错点组
    for i=1:n+1
        if y(i) < 1 & y(i)>-1
            for j=2:n+2
                if y(i)<x(j)&y(i)>x(j-1)
                    if (1/(x(j-1)+2)-polyval(ap,x(j-1)))*(1/(y(i)+2)-polyval(ap,y(i)))> 0
                        x(j-1)=y(i);
                    elseif (1/(x(j-1)+2)-polyval(ap,x(j-1)))*(1/(y(i)+2)-polyval(ap,y(i)))< 0
                        x(j)=y(i);
                    end
                end
            end
        end   
    end
    A=ones(n+2,n+2);
    A(:,1)=first;
    A(:,n+2)=last;
    for j=2:n+1
        for t=2:j
           A(:,j)=x.*A(:,j);
       end
    end
    f=1./(x+2);
    sun1=A\f;
    if(abs(sun1-sun)<e)
        break;
    end
    sun=sun1;
end
hold on;
funcion=poly2sym(ap);
ezplot(funcion,[-1,1]);
num=num2str(n);
legend('原函数曲线',strcat(num,'次逼近函数曲线'));
title('最佳逼近比较示意图');
xlabel('x的取值');
ylabel('f(x)的取值');
grid on;
end