基于MATLAB的常用一维搜索方法

一维搜索一般步骤

1、选取初始点x0，令k=0；
2、选取方向pk，使f（x）沿plk方向函数值下降（pk为搜索方向）；
3、从xk出发以pk为方向作射线xk+lamdapk，在此射线上求f（x）的最小值；即：
min f（xk+lamdapk）=f（x[k+1]） 其中，lamda>=0；
4、判别x[k+1]是否为最优解x*，若|f’（x[k+1]）|

成功-失败法

%成功-失败法函数
function [xo,fo]=succeed_fail(x,f,h,e)
h = abs(h);
e = abs(e);
phi1=subs(f,x)
while 1
    while 1
        phi2=double(subs(f,x+h))
        if phi1 > phi2
            x=x+h
            phi1=phi2
            h=2*h
            break;
        end
        if abs(h) > e
            h=-h/4
            break;
        end
    end
    if abs(h) <= e
        break;
    end
end
xo = x;
fo = double(subs(f,x));

%成功-失败法函数测试
clear all;clc;
syms x
f = x^4 - 4*x^3 - 6*x^2 - 16*x + 4
[x0,f0] = succeed_fail(0,f,1,0.001)

0.618法（黄金分割法）

%0.618法函数
function [xo,fo]=zero_point_618(a,b,f,e)
if a>b
    t=a;a=b;b=t;
end
e=abs(e)
x1=a+0.382*(b-a)
x2=a+b-x1
L=double(subs(f,x1))
R=double(subs(f,x2))
while 1
    if L

%0.618法函数测试
clear all;clc;
syms x
f = x^4 - 4*x^3 - 6*x^2 - 16*x + 4
[x0,f0] = zero_point_618(0,5,f,0.01)

二分法

%二分法函数
function [xo,fo]=twohalf(a,b,f,df,e)
if a>b
    t=a;a=b;b=t;
end
e=abs(e)
while 1
    m = (a+b)/2;
    if abs(subs(df,m))

%二分法函数测试
clear all;clc;
syms x
f = x^4 - 4*x^3 - 6*x^2 - 16*x + 4
df = diff(f)
[x0,f0] = twohalf(5,2,f,df,0.01)

牛顿法

%牛顿法函数
function [xo,fo]=Newton(x,f,df,ddf,e)
e=abs(e);
while 1
    xo=x-double(subs(df,x)/subs(ddf,x))
    if double(subs(df,xo))

%牛顿法函数测试
clear all;clc;
t1=clock;
syms x
f = x^4 - 4*x^3 - 6*x^2 - 16*x + 4
df = diff(f)
ddf =diff(df)
[x0,f0] = Newton(6,f,df,ddf,0.01)
t2=clock;
etime(t2,t1)

二次插值法

%二次插值法函数
function [xo,fo]=interpolation2(x1,x2,x3,f,e)
while 1
    c1=(subs(f,x3)-subs(f,x1))/(x3-x1);
    c2=((subs(f,x2)-subs(f,x1))/(x2-x1)-c1)/(x2-x3);
    xo=double(0.5*(x1+x3-c1/c2));
    if abs(xo-x2)0
        x1=x2;
        x2=xo;
    else
        x3=x2;
        x2=xo;
    end
end
fo=double(subs(f,xo));
        

%二次插值法函数测试
clear all;clc;
t1=clock;
syms x
f = x^4 - 4*x^3 - 6*x^2 - 16*x + 4;
[x0,f0] = interpolation2(3,3.5,5,f,0.01)
t2=clock;
etime(t2,t1)

三次插值法

%三次插值法函数
function [xo,fo]=interpolation3(f,df,a,h,e)
e=abs(e);
v=subs(df,a);
u = v;
if abs(v) < e
    xo = a
else
    while abs(v) >= e
        if v < 0
            h = abs(h)
        else
            h = -abs(h)
        end
        while u * v >= 0
            b = a + h
            u = double(subs(df,b))
            if abs(u) < e
                xo = b;break;
            else
                if u * v <0
                    break;
                else
                    h = 2 * h
                    v = u
                    a = b
                end
            end
        end
        if abs(u) < e
            break; 
        end
        s = double(3*(subs(f,a)-subs(f,b))/(b-a))
        z = double(s-u-v)
        w = double((z^2-u*v)^0.5)
        a = double(a-(b-a)*v/(z-w*sign(v)-v))
        v = double(subs(df,a))
        if abs(v) < e
            xo = a;break;
        else
            h = h/10
        end
    end
end
fo=double(subs(f,xo));    

%三次插值法函数测试
clear all;clc;
t1=clock;
syms x
f = x^4 - 4*x^3 - 6*x^2 - 16*x + 4;
df = diff(f);
[x0,f0] = interpolation3(f,df,0,1,0.5)
t2=clock;
etime(t2,t1)