function [s,phis,k,ds,dphi,S]=alpha_min(phi,a,b,delta,epsilon)
%功能: 精确线搜索之抛物线法
%输入: phi 是目标函数, a和b是搜索区间的端点
% delta,epsilon是容许误差
%输出: s是近似极小点, phis是对应的近似极小值;
% ds, dphi分别是s, phis的误差界; S是迭代向量
s0=a; maxj=20; maxk=30; big=1e6; err=1; k=1;
S(k)=s0; cond=0; h=1; ds=0.00001;
if (abs(s0)>1e4), h=abs(s0)*(1e-4); end
while (k<maxk & err>epsilon &cond~=5)
f1=(feval(phi,s0+ds)-feval(phi,s0-ds))/(2*ds);
if(f1>0), h=-abs(h); end
s1=s0+h; s2=s0+2*h; bars=s0;
phi0=feval(phi,s0); phi1=feval(phi,s1); phi2=feval(phi,s2);
barphi=phi0; cond=0; j=0;
%确定h使得phi1<phi0且phi1<phi2
while(j<maxj&abs(h)>delta&cond==0)
if (phi0<=phi1),
s2=s1; phi2=phi1; h=0.5*h;
s1=s0+h; phi1=feval(phi,s1);
else if (phi2<phi1),
s1=s2; phi1=phi2; h=2*h;
s2=s0+2*h; phi2=feval(phi,s2);
else
cond=-1;
end
end
j=j+1;
if(abs(h)>big|abs(s0)>big), cond=5; end
end
if(cond==5)
bars=s1; barphi=feval(phi,s1);
else
%二次插值求phis
d=2*(2*phi1-phi0-phi2);
if(d<0),
barh=h*(4*phi1-3*phi0-phi2)/d;
else
barh=h/3; cond=4;
end
bars=s0+barh; barphi=feval(phi,bars);
h=abs(h); h0=abs(barh);
h1=abs(barh-h); h2=abs(barh-2*h);
%确定下一次迭代的h值
if(h0<h), h=h0; end
if(h1<h), h=h1; end
if(h2<h), h=h2; end
if(h==0), h=barh; end
if(h<delta), cond=1; end
if(abs(h)>big|abs(bars)>big), cond=5; end
%迭代终止判别
err=abs(phi1-barphi);
%e0=abs(phi0-barphi); e1=abs(phi1-barphi); e2=abs(phi2-barphi);
%if(e0~=0 & e0<err), err=e0; end
%if(e1~=0 & e1<err), err=e1; end
%if(e2~=0 & e2<err), err=e2; end
%if(e0~=0 & e1==0 & e2==0), err=0; end
%if(err<epsilon), cond=2; end
s0=bars; k=k+1; S(k)=s0;
end
if(cond==2&h<delta), cond=3; end
end
s=s0; phis=feval(phi,s);
ds=h; dphi=err;
