Matlab非线性方程求根

 

                                                        非线性方程求根                     

 

二分法 :           

function root=MultiRoot(f,a,b,eps)

if(nargin==3)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

    tol=1;

    fun=diff(sym(f));

    ddf=diff(sym(fun));

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

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

    dfa=subs(sym(fun),findsym(sym(fun)),a);

    dfb=subs(sym(fun),findsym(sym(fun)),b);

    if(dfa>dfb)

        root=a;

    else

        root=b;

    end

    while(tol>eps)

        r1=root;

        fx=subs(sym(f),findsym(sym(f)),r1);

        dfx=subs(sym(fun),findsym(sym(fun)),r1);

        ddfx=subs(sym(ddf),findsym(sym(ddf)),r1);

        root=r1-fx*dfx/(dfx*dfx-fx*ddfx);

        tol=abs(root-r1);

    end

end

Example:

>> HalfInterval('x^3-3*x+1',0,1)

 

ans =

 

    0.3473

黄金分割法 :

function root=hj(f,a,b,eps)

if(nargin==3)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else   

    t1=a+(b-a)*0.382;

    t2=a+(b-a)*0.618;

    f_1=subs(sym(f),findsym(sym(f)),t1);

    f_2=subs(sym(f),findsym(sym(f)),t2);

    tol=abs(t1-t2);

    while(tol>eps)                           % 精度控制

        if(f_1*f_2<0)

            a=t1;

            b=t2;

        else

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

            if(f_1*fa>0)

                a=t2;

            else

                b=t1;

            end

        end

        t1=a+(b-a)*0.382;

        t2=a+(b-a)*0.618;

        f_1=subs(sym(f),findsym(sym(f)),t1);

        f_2=subs(sym(f),findsym(sym(f)),t2);

        tol=abs(t2-t1);

    end

    root=(t1+t2)/2;                             % 输出根

end

 

>> hj('x^3-3*x+1',0,1)

 

ans =

 

    0.3473

不动点迭代:

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

if(nargin==2)

    eps=1.0e-4;

end

 

tol=1;

root=x0;

n=0;

while(tol>eps)

    n=n+1;

    r1=root;

    root=subs(sym(f),findsym(sym(f)),r1)+r1;

    tol=abs(root-r1);

end

>> [r,n]=StablePoint('1/sqrt(x)+x-2',0.5)

 

r =

 

    0.3820

 

 

n =

 

     4

Atken 加速迭代法：

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

if(nargin==2)

    eps=1.0e-4;

end

 

tol=1;

root=x0;

x(1:2)=0;

n=0;

m=0;

a2=x0;

while(tol>eps)

    n=n+1;

    a1=a2;

    r1=root;

    root=subs(sym(f),findsym(sym(f)),r1)+r1;

    x(n)=root;

    if(n>2)

        m=m+1;

        a2=x(m)-(x(m+1)-x(m))^2/(x(m+2)-2*x(m+1)+x(m));

        tol=abs(a2-a1);

    end  

end

root=a2;

>> [r,n]=AtkenStablePoint('1/sqrt(x)+x-2',0.5)

 

r =

 

    0.3820

 

 

n =

 

     4

Steffensen 加速迭代：

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

if(nargin==2)

    eps=1.0e-4;

end

 

tol=1;

root=x0;

n=0;

while(tol>eps)

    n=n+1;

    r1=root;

    y=subs(sym(f),findsym(sym(f)),r1)+r1;

    z=subs(sym(f),findsym(sym(f)),y)+y;

    root=r1-(y-r1)^2/(z-2*y+r1);

    tol=abs(root-r1);

end

>> [r,n]=StevenStablePoint('1/sqrt(x)+x-2',0.5)

 

r =

 

    0.3820

 

 

n =

 

     4

弦截法：       

function root=Secant(f,a,b,eps)

if(nargin==3)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

    tol=1;

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

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

    root=a-(b-a)*fa/(fb-fa);

    while(tol>eps)

         r1=root;

        fx=subs(sym(f),findsym(sym(f)),r1);

        s=fx*fa;

        if(s==0)

            root=r1;

        else

            if(s>0)

                root=b-(r1-b)*fb/(fx-fb);

            else

                root=a-(r1-a)*fa/(fx-fa);

            end

        end

        tol=abs(root-r1);

    end

end

>> Secant('x^3-3*x+1',0,1)

 

ans =

 

    0.3473

Steffensen 弦截法：

function root=StevenSecant(f,a,b,eps)

if(nargin==3)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

    tol=1;

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

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

    faa=subs(sym(f),findsym(sym(f)),fa+a);

    root=a-fa*fa/(faa-fa);

    while(tol>eps)

        r1=root;

        fx=subs(sym(f),findsym(sym(f)),r1);

        v=fx+r1;

        fxx=subs(sym(f),findsym(sym(f)),v);

        root=r1-fx*fx/(fxx-fx);

        tol=abs(root-r1);

    end

end

>> StevenSecant('x^3-3*x+1',0,1)

 

ans =

 

    0.3473

抛物线法：

function root=Parabola(f,a,b,x,eps)

if(nargin==4)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

     tol=1;

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

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

    fx=subs(sym(f),findsym(sym(f)),x);

    d1=(fb-fa)/(b-a);

    d2=(fx-fb)/(x-b);

    d3=(d2-d1)/(x-a);

    B=d2+d3*(x-b);

    root=x-2*fx/(B+sign(B)*sqrt(B^2-4*fx*d3));

    t=zeros(3);

    t(1)=a;

    t(2)=b;

    t(3)=x;

    while(tol>eps)

        t(1)=t(2);

        t(2)=t(3);

        t(3)=root;

        f1=subs(sym(f),findsym(sym(f)),t(1));

        f2=subs(sym(f),findsym(sym(f)),t(2));

        f3=subs(sym(f),findsym(sym(f)),t(3));

        d1=(f2-f1)/(t(2)-t(1));

        d2=(f3-f2)/(t(3)-t(2));

        d3=(d2-d1)/(t(3)-t(1));

        B=d2+d3*(t(3)-t(2));

        root=t(3)-2*f3/(B+sign(B)*sqrt(B^2-4*f3*d3)); 

        tol=abs(root-t(3));

    end

end

>> Parabola('sqrt(x)+log(x)-2',1,4,2)

 

ans =

 

    1.8773

Newton 法：

function root=NewtonRoot(f,a,b,eps)

if(nargin==3)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

    tol=1;

    fun=diff(sym(f));

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

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

    dfa=subs(sym(fun),findsym(sym(fun)),a);

    dfb=subs(sym(fun),findsym(sym(fun)),b);

    if(dfa>dfb)

        root=a-fa/dfa;

    else

        root=b-fb/dfb;

    end

    while(tol>eps)

        r1=root;

        fx=subs(sym(f),findsym(sym(f)),r1);

        dfx=subs(sym(fun),findsym(sym(fun)),r1);

        root=r1-fx/dfx;

        tol=abs(root-r1);

    end

end

>> NewtonRoot('sqrt(x)-x^3+2',0.5,2)

 

ans =

 

    1.4759

简化Newton 法：

function root=SimpleNewton(f,a,b,eps)

if(nargin==3)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

    tol=1;

    fun=diff(sym(f));

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

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

    dfa=subs(sym(fun),findsym(sym(fun)),a);

    dfb=subs(sym(fun),findsym(sym(fun)),b);

 

    if(dfa>dfb)

        df0=1/dfa;

        root=a-df0*fa;

    else

        df0=1/dfb;

        root=b-df0*fb;

    end

    while(tol>eps)

        r1=root;  

        fx=subs(sym(f),findsym(sym(f)),r1);

        root=r1-df0*fx;

        tol=abs(root-r1);

    end

end

>> SimpleNewton('sqrt(x)-x^3+2',1.2,2)

 

ans =

 

    1.4759

Newton 下山法：

function root=NewtonDown(f,a,b,eps)

if(nargin==3)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

    tol=1;

    fun=diff(sym(f));

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

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

    dfa=subs(sym(fun),findsym(sym(fun)),a);

    dfb=subs(sym(fun),findsym(sym(fun)),b);

    if(dfa>dfb)

        root=a;

    else

        root=b;

    end

    while(tol>eps)

        r1=root;  

        fx=subs(sym(f),findsym(sym(f)),r1);   

        dfx=subs(sym(fun),findsym(sym(fun)),r1);

        toldf=1;

        alpha=2;

        while toldf>0

            alpha=alpha/2;

            root=r1-alpha*fx/dfx;

            fv=subs(sym(f),findsym(sym(f)),root);

            toldf=abs(fv)-abs(fx);

        end

        tol=abs(root-r1);

    end

end

>> NewtonDown('sqrt(x)-x^3+2',1.2,2)

 

ans =

 

1.4759

 

两步迭代法：

function root=TwoStep(f,a,b,type,eps)

if(nargin==4)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

    tol=1;

    fun=diff(sym(f));

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

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

    dfa=subs(sym(fun),findsym(sym(fun)),a);

    dfb=subs(sym(fun),findsym(sym(fun)),b);

    if(dfa>dfb)

        root=a;

    else

        root=b;

    end

    while(tol>eps)

        if(type==1)

            r1=root;

            fx1=subs(sym(f),findsym(sym(f)),r1);

            dfx=subs(sym(fun),findsym(sym(fun)),r1);

            r2=r1-fx1/dfx;

            fx2=subs(sym(f),findsym(sym(f)),r2);

            root=r2-fx2/dfx;

            tol=abs(root-r1);

        else

            r1=root;

            fx1=subs(sym(f),findsym(sym(f)),r1);

            dfx=subs(sym(fun),findsym(sym(fun)),r1);

            r2=r1-fx1/dfx;

            fx2=subs(sym(f),findsym(sym(f)),r2);

            root=r2-fx2*(r2-r1)/(2*fx2-fx1);

             tol=abs(root-r1);

        end

    end

end

>> Two_Step('log(x)-sin(x)+1',0.1,3,1)

 

ans =

 

    0.7013

重根的迭代法：

function root=MultiRoot(f,a,b,eps)

if(nargin==3)

    eps=1.0e-4;

end

 

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

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

if(f1==0)

    root=a;

end

if(f2==0)

    root=b;

end

 

if(f1*f2>0)

    disp(' 两端点函数值乘积大于0!');

    return;

else

    tol=1;

    fun=diff(sym(f));

    ddf=diff(sym(fun));

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

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

    dfa=subs(sym(fun),findsym(sym(fun)),a);

    dfb=subs(sym(fun),findsym(sym(fun)),b);

    if(dfa>dfb)

        root=a;

    else

        root=b;

    end

    while(tol>eps)

        r1=root;

        fx=subs(sym(f),findsym(sym(f)),r1);

        dfx=subs(sym(fun),findsym(sym(fun)),r1);

        ddfx=subs(sym(ddf),findsym(sym(ddf)),r1);

        root=r1-fx*dfx/(dfx*dfx-fx*ddfx);

        tol=abs(root-r1);

    end

end

>> MultiRoot('(sin(x)-x+2)*x*x',-2,3,1e-8)

 

ans =

 

     0

 

非线性方程组的数值解法

 

不动点迭代：

function [r,n]=mulStablePoint(myf,x0,eps)

if nargin==1

    eps=1.0e-4;

end

 

r=myf(x0);

n=1;

tol=1;

while tol>eps

    x0=r;

    r=myf(x0);

    tol=norm(r-x0);

    n=n+1;

    if(n>100000)

        disp(' 迭代步数太多，可能不收敛！');

        return;

    end

end

Newton 法 ：

function [r,n]=mulNewton(myf,dmyf,x0,eps)

if nargin==1

    eps=1.0e-4;

end

 

r=x0-myf(x0)/dmyf(x0);

n=1;

tol=1;

while tol>eps

    x0=r;

    r=x0-myf(x0)/dmyf(x0);

    tol=norm(r-x0);

    n=n+1;

    if(n>100000)

        disp(' 迭代步数太多，可能不收敛！');

        return;

    end

end

简化 Newton 法 ：

function [r,n]=mulSimNewton(myf,dmyf,x0,eps)

if nargin==1

    eps=1.0e-4;

end

 

r=x0-myf(x0)/dmyf(x0);

c=dmyf(x0);

n=1;

tol=1;

while tol>eps

    x0=r;

    r=x0-myf(x0)/c;

    tol=norm(r-x0);

    n=n+1;

    if(n>100000)

        disp(' 迭代步数太多，可能不收敛！');

        return;

    end

end

Newton 下山法：

function [r,n]=mulDNewton(myf,dmyf,x0,eps)

if nargin==1

    eps=1.0e-4;

end

 

r=x0-myf(x0)/dmyf(x0);

n=1;

tol=1;

while tol>eps

    x0=r;

    ttol=1;

    w=1;

    F1=norm(myf(x0));

    while ttol>=0

        r=x0-w*myf(x0)/dmyf(x0);

        ttol=norm(myf(r))-F1;

        w=w/2;

    end

    tol=norm(r-x0);

    n=n+1;

    if(n>100000)

        disp(' 迭代步数太多，可能不收敛！');

        return;

    end

end

拟 Newton 法 ：

function [r,n]=mulVNewton(myf,x0,A,eps)

if nargin==1 

    A=eye(length(x0));

else

    if nargin==2

        eps=1.0e-4;

    end

end

 

r=x0-myf(x0)/A;

n=1;

tol=1;

while tol>eps

    x0=r;

    r=x0-myf(x0)/A;

    y=r-x0;

    z=myf(r)-myf(x0);

    A1=A+(z-y*A)'*y/norm(y);

    A=A1;

    n=n+1;

    if(n>100000)

        disp(' 迭代步数太多，可能不收敛！');

        return;

    end

    tol=norm(r-x0);

end