最优化 二分法,Newton法,黄金分割法

二分法和newton法：x^2-2*x-1 >> syms x

>> f=x^2-2*x-1;

>> subplot(1,2,1)

>> [m,n]=erff(f,2,4,20,0.0001)

m =

2.4142

n =

14

>> subplot(1,2,2)

>> [o,p]=newton(f,4,20,0.0001)

o =

2.4142

p =

5

二分法函数：

function [root,n]=erff(f,a,b,N,eps)

k=0;

f1=subs(f,findsym(sym(f)),a);

f2=subs(f,findsym(sym(f)),b);

if(f1==0)

root=a;

n=k;

return;

end

if(f2==0)

root=b;

n=k;

return;

end

if(f1*f2>0)

disp('两端点点对应的函数值同号');

return;

end

while(1)

k=k+1;

r=subs(f,findsym(sym(f)),(a+b)/2); u(k)=abs(r);

if(r==0)

root=(a+b)/2;

n=k;

return;

end

if(f1*r<0)

b=(a+b)/2;

end

if(f2*r<0)

a=(a+b)/2;

end

if(abs(r)

root=(a+b)/2;

n=k;

plot(1:k,u(1:k),'r-');

xlabel('迭代次数');

ylabel('误差');

title('二分法');

return;

end

if(k>N)

disp('迭代N次不出结果');

return;

end

end

newton迭代法函数：

function [root n]=newton(f,x0,N,eps)

k=0;

df=diff(f);

if(subs(f,findsym(sym(f)),x0)==0)

root=x0;

n=k;

return;

end

while(1)

k=k+1;

x1=x0-subs(f,findsym(sym(f)),x0)/subs(df,findsym(sym(f)),x0); u(k)=abs(x1-x0);

if(abs(x1-x0)

root=x1;

n=k;

plot(1:k,u(1:k),'r');

xlabel('迭代次数');

ylabel('误差');

title('newton法');

return;

end

x0=x1;

if(k>N)

disp('不收敛');

return;

end

end

黄金分割法：x^2-sin(x)

>> syms x

>> phi=x^2-sin(x);

>> [s,phis,k,G,E]=golds(phi,0,1,1e-4,1e-5)

s =

0.4502

phis =

-0.2325

k =

21

G =

0 0.3820 0.6180 1.0000

0 0.2361 0.3820 0.6180

0.2361 0.3820 0.4721 0.6180

0.3820 0.4721 0.5279 0.6180

0.3820 0.4377 0.4721 0.5279

0.3820 0.4164 0.4377 0.4721

0.4164 0.4377 0.4508 0.4721

0.4377 0.4508 0.4590 0.4721

0.4377 0.4458 0.4508 0.4590

0.4458 0.4508 0.4540 0.4590

0.4458 0.4489 0.4508 0.4540

0.4489 0.4508 0.4520 0.4540

0.4489 0.4501 0.4508 0.4520

0.4489 0.4497 0.4501 0.4508

0.4497 0.4501 0.4504 0.4508

0.4497 0.4499 0.4501 0.4504

0.4499 0.4501 0.4502 0.4504

0.4501 0.4502 0.4503 0.4504

0.4501 0.4502 0.4502 0.4503

0.4501 0.4502 0.4502 0.4502

0.4502 0.4502 0.4502 0.4502

E =

1.0e-004 *

0.6611 0.0000

黄金分割法函数：

function [s,phis,k,G,E]=golds(phi,a,b,delta,epsilon)

t=(sqrt(5)-1)/2;h=b-a;

phia=subs(phi,findsym(sym(phi)),a);phib=subs(phi,findsym(sym(phi)),b) ;

p=a+(1-t)*h;q=a+t*h;

phip=subs(phi,findsym(sym(phi)),p);phiq=subs(phi,findsym(sym(phi)),q) ;

k=1;G(k,1:4)=[a,p,q,b];

while((abs(phib-phia)>epsilon)||(h>delta))

if(phip

b=q; phib=phiq; q=p; phiq=phip;

h=b-a; p=a+(1-t)*h; phip=subs(phi,findsym(sym(phi)),p);

else

a=p; phia=phip; p=q; phip=phiq;

h=b-a; q=a+t*h; phiq=subs(phi,findsym(sym(phi)),q);

end

k=k+1; G(k,1:4)=[a,p,q,b];

end

ds=abs(b-a); dphi=abs(phib-phia);

if(phip<=phiq)

s=p; phis=phip;

else

s=q; phis=phiq;

end

E=[ds,dphi];