matlab求解非線性方程組

1、非線性方程組求解l.mulStablePoint用不動點迭代法求非線性方程組的一個根function r,n=mulStablePoint(F,x0,eps)線性方程組：f%U始解：a嗨軍的精度：eps%求得的一組解：r旭代步數(shù)：nif nargin=2eps=1.0e-6;end x0 = transpose(x0);n=1;tol=1;while toleps迭代公式注意矩陣的誤差求法，norm為矩陣的歐幾里迭代步數(shù)控制1)；r= subs(F,findsym(F),x0);%tol=norm(r-x0);%德范數(shù)n=n+1;x0=r;if(n100000)%disp( 迭代步數(shù)太多，可能

2、不收斂！ return; end endmulNewton用牛頓法法求非線性方程組的一個根function r,n=mulNewton(F,x0,eps)if nargin=2eps=1.0e-4;endx0 = transpose(x0);Fx = subs(F,findsym(F),x0);var = findsym(F);dF = Jacobian(F,var);dFx = subs(dF,findsym(dF),x0);r=x0-inv(dFx)*Fx;n=1;tol=1;while tolepsx0=r;Fx = subs(F,findsym(F),x0);dFx = subs(dF

3、,findsym(dF),x0);核心迭代公式迭代步數(shù)控制r=x0-inv(dFx)*Fx;%tol=norm(r-x0);n=n+1;if(n100000)%disp(迭代步數(shù)太多，可能不收斂！,);return;endendmulDiscNewton用離散牛頓法法求非線性方程組的一個根 function r,m=mulDiscNewton(F,x0,h,eps)format long;if nargin=3eps=1.0e-8;endn = length(x0);fx = subs(F,findsym(F),x0);J = zeros(n,n);for i=1:nx1 = x0;x1(i)

4、 = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-fx)/h(i);endr=transpose(x0)-inv(J)*fx;m=1;tol=1;while tolepsxs=r;fx = subs(F,findsym(F),xs);J = zeros(n,n);for i=1:nx1 = xs;x1(i) = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-fx)/h(i);end核心迭代公式迭代步數(shù)控制r=xs-inv(J)*fx;%tol=norm(r-xs);m=m+1;if(m100000)%disp(迭

5、代步數(shù)太多，可能不收斂！,);return;end endformat short;mulMix用牛頓-雅可比迭代法求非線性方程組的一個根function r,m=mulMix(F,x0,h,l,eps)if nargin=4eps=1.0e-4;endn = length(x0);J = zeros(n,n);Fx = subs(F,findsym(F),x0);for i=1:nx1 = x0;x1(i) = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-Fx)/h(i);endD = diag(diag(J);C = D - J;inD = inv

6、(D);H = inD*C;Hm = eye(n,n);for i=1:l-1Hm = Hm + power(H,i);enddr = Hm*inD*Fx;r = transpose(x0)-dr;m=1;tol=1;while tolepsx0=r;Fx = subs(F,findsym(F),x0);J = zeros(n,n);for i=1:nx1 = x0;x1(i) = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-Fx)/h(i);endD = diag(diag(J);C = D - J;inD = inv(D);H = inD*C;Hm

7、 = eye(n,n);for i=1:l-1Hm = Hm + power(H,i);enddr = Hm*inD*Fx;r = x0-dr;%核心迭代公式tol=norm(r-x0);m=m+1;if(m100000)%迭代步數(shù)控制disp( 迭代步數(shù)太多，可能不收斂！,);return;endendmulNewtonSOR用牛頓-SOR迭代法求非線性方程組的一個根function r,m=mulNewtonSOR(F,x0,w,h,l,eps)if nargin=5eps=1.0e-4;endn = length(x0);J = zeros(n,n);Fx = subs(F,findsy

8、m(F),x0);for i=1:nx1 = x0;x1(i) = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-Fx)/h(i);endD = diag(diag(J);L = -tril(J-D);U = -triu(J-D);inD = inv(D-w*L);H = inD*(D - w*D+w*L);Hm = eye(n,n);for i=1:l-1Hm = Hm + power(H,i);enddr = w*Hm*inD*Fx;r = transpose(x0)-dr;m=1;tol=1;while tolepsx0=r;Fx = subs(F

9、,findsym(F),x0);J = zeros(n,n);for i=1:nx1 = x0;x1(i) = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-Fx)/h(i);endD = diag(diag(J);L = -tril(J-D);U = -triu(J-D);inD = inv(D-w*L);H = inD*(D - w*D+w*L);Hm = eye(n,n);for i=1:l-1Hm = Hm + power(H,i);enddr = w*Hm*inD*Fx;r = x0-dr;%核心迭代公式tol=norm(r-x0);m=m+1

10、;if(m100000)%迭代步數(shù)控制disp( 迭代步數(shù)太多，可能不收斂！,);return;end endmulDNewton用牛頓下山法求非線性方程組的一個根function r,m=mulDNewton(F,x0,eps)線性方程組：F%5始解：x0%軍的精度：eps%求得的一組解：r%代步數(shù)：nif nargin=2eps=1.0e-4;endx0 = transpose(x0);dF = Jacobian(F);m=1;tol=1;while tolepsttol=1;w=1;Fx = subs(F,findsym(F),x0);dFx = subs(dF,findsym(dF),

11、x0);卜面的循環(huán)是選取下山因子w的過核心的迭代公式迭代步數(shù)控制F1=norm(Fx);while ttol=0%程r=x0-w*inv(dFx)*Fx;%Fr = subs(F,findsym(F),r);ttol=norm(Fr)-F1;w=w/2;endtol=norm(r-x0);m=m+1;x0=r;if(m100000)%disp(迭代步數(shù)太多，可能不收斂！,);return;endendmulGXF1用兩點割線法的第一種形式求非線性方程組的一個根function r,m=mulGXF1(F,x0,x1,eps)format long;if nargin=3eps=1.0e-4;e

12、ndx0 = transpose(x0);x1 = transpose(x1);n = length(x0);fx = subs(F,findsym(F),x0);fx1 = subs(F,findsym(F),x1);h = x0 - x1;J = zeros(n,n);for i=1:nxt = x1;xt(i) = x0(i);J(:,i) = (subs(F,findsym(F),xt)-fx1)/h(i);endr=x1-inv(J)*fx1;m=1;tol=1;while tolepsx0 = x1;x1 = r;fx = subs(F,findsym(F),x0);fx1 = s

13、ubs(F,findsym(F),x1);h = x0 - x1;J = zeros(n,n);for i=1:nxt = x1;xt(i) = x0(i);J(:,i) = (subs(F,findsym(F),xt)-fx1)/h(i);endr=x1-inv(J)*fx1;tol=norm(r-x1);m=m+1;if(m100000)%迭代步數(shù)控制disp(迭代步數(shù)太多，可能不收斂！,);return;endendformat short;mulGXF2用兩點割線法的第二種形式求非線性方程組的一個根function r,m=mulGXF2(F,x0,x1,eps)format long

14、;if nargin=3eps=1.0e-4;endx0 = transpose(x0);x1 = transpose(x1);n = length(x0);fx = subs(F,findsym(F),x0);fx1 = subs(F,findsym(F),x1);h = x0 - x1;J = zeros(n,n);xt = x1;xt(1) = x0；J(:,1) = (subs(F,findsym(F),xt)-subs(F,findsym(F),x1)/h(1);for i=2:nxt = x1;xt(1:i) = x0(1:i);xt_m = x1;xt_m(1:i-1) = x0

15、(1:i-1);J(:,i) = (subs(F,findsym(F),xt)-subs(F,findsym(F),xt_m)/h(i);endr=x1-inv(J)*fx1;m=1;tol=1;while tolepsx0 = x1;x1 = r;fx = subs(F,findsym(F),x0);fx1 = subs(F,findsym(F),x1);h = x0 - x1;J = zeros(n,n);xt = x1;xt(1) = x0；J(:,1) = (subs(F,findsym(F),xt)-subs(F,findsym(F),x1)/h(1);for i=2:nxt = x

16、1;xt(1:i) = x0(1:i);xt_m = x1;xt_m(1:i-1) = x0(1:i-1);J(:,i) = (subs(F,findsym(F),xt)-subs(F,findsym(F),xt_m)/h(i); endr=x1-inv(J)*fx1;tol=norm(r-x1);m=m+1;迭代步數(shù)控制if(m100000)%disp(迭代步數(shù)太多，可能不收斂！,);return;endendformat short;mulVNewton用擬牛頓法求非線性方程組的一組解function r,m=mulVNewton(F,x0,A,eps)%?程組：F附程組的初始解：x0%初

17、始A矩陣：A%軍的精度：eps%求得的一組解：r%代步數(shù)：mif nargin=2A=eye(length(x0); %A 取為單位陣eps=1.0e-4;elseif nargin=3eps=1.0e-4;endend x0 = transpose(x0);Fx = subs(F, findsym(F),x0);r=x0-AFx;m=1;tol=1;while tolepsx0=r;Fx = subs(F, findsym(F),x0);r=x0-AFx;y=r-x0;Fr = subs(F, findsym(F),r);z= Fr-Fx;A1=A+(z-A*y)*transpose(y)/

18、norm(y);%調(diào)整 AA=A1;m=m+1;if(m100000)%迭代步數(shù)控制disp(迭代步數(shù)太多，可能不收斂！,);return;endtol=norm(r-x0);endmulRank1用對稱秩1算法求非線性方程組的一個根function r,n=mulRank1(F,x0,A,eps)if nargin=2l = length(x0);A=eye(l);%A取為單位陣eps=1.0e-4;elseif nargin=3 eps=1.0e-4;endendfx = subs(F,findsym(F),x0);r=transpose(x0)-inv(A)*fx;n=1;tol=1;w

19、hile tolepsx0=r;fx = subs(F,findsym(F),x0);r=x0-inv(A)*fx;y=r-x0;fr = subs(F,findsym(F),r);z = fr-fx;A1=A+ fr *transpose(fr)/(transpose(fr)*y);%調(diào)整 AA=A1; n=n+1; if(n100000)%迭代步數(shù)控制disp(迭代步數(shù)太多，可能不收斂！,);return; end tol=norm(r-x0); end 11. mulDFP用D-F-P算法求非線性方程組的一組解 function r,n=mulDFP(F,x0,A,eps) if nar

20、gin=2 l = length(x0); B=eye(l);%A取為單位陣eps=1.0e-4; else if nargin=3 eps=1.0e-4; end end fx = subs(F,findsym(F),x0); r=transpose(x0)-B*fx; n=1; tol=1; while toleps x0=r; fx = subs(F,findsym(F),x0); r=x0-B*fx; y=r-x0; fr = subs(F,findsym(F),r); z = fr-fx; B1=B+ y*y/(y*z)-B*z*z*B/(z*B*z);%調(diào)整 AB=B1; n=n+

21、1; if(n100000)%迭代步數(shù)控制disp(迭代步數(shù)太多，可能不收斂！,);return; end tol=norm(r-x0); end 12. mulBFS用B-F-S算法求非線性方程組的一個根function r,n=mulBFS(F,x0,B,eps) if nargin=2 l = length(x0);B=eye(l);%Beps=1.0e-4; else if nargin=3 eps=1.0e-4; end end fx = subs(F,findsym(F),x0); r=transpose(x0)-B*fx; n=1; tol=1; while toleps x0=

22、r; fx = subs(F,findsym(F),x0); r=x0-B*fx; y=r-x0; fr = subs(F,findsym(F),r); z = fr-fx; u = 1 + z*B*z/(y*z);B1= B+ (u*y*y-B*z*y-y*z*B)/(y*z);B=B1; n=n+1; if(n100000)%disp(迭代步數(shù)太多，可能不收斂!return; end tol=norm(r-x0);取為單位陣%調(diào)整B迭代步數(shù)控制1）；endmulNumYT用數(shù)值延拓法求非線性方程組的一組解 function r,m=mulNumYT(F,x0,h,N,eps) format

23、 long;if nargin=4 eps=1.0e-8;endn = length(x0);fx0 = subs(F,findsym(F),x0);x0 = transpose(x0);J = zeros(n,n);for k=0:N-1fx = subs(F,findsym(F),x0);for i=1:nx1 = x0;x1(i) = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-fx)/h(i);endinJ = inv(J);r=x0-inJ*(fx-(1-k/N)*fx0);x0 = r;endm=1;tol=1;while tolepsxs

24、=r;fx = subs(F,findsym(F),xs);J = zeros(n,n);for i=1:nx1 = xs;x1(i) = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-fx)/h(i);endr=xs-inv(J)*fx;%核心迭代公式tol=norm(r-xs); m=m+1; if(m100000)%迭代步數(shù)控制disp(迭代步數(shù)太多，可能不收斂！,);return;endendformat short;DiffParam1用參數(shù)微分法中的歐拉法求非線性方程組的一組解function r=DiffParam1(F,x0,h,N) 線

25、性方程組：f%5始解：x0喊值微分增量步大?。篽哪可比迭代參量：l%軍的精度：eps%求得的一組解：r%代步數(shù)：nx0 = transpose(x0);n = length(x0);ht = 1/N;Fx0 = subs(F,findsym(F),x0);for k=1:NFx = subs(F,findsym(F),x0);J = zeros(n,n);for i=1:nx1 = x0;x1(i) = x1(i)+h(i);J(:,i) = (subs(F,findsym(F),x1)-Fx)/h(i);endinJ = inv(J);r = x0 - ht*inJ*Fx0; x0 = r;

26、endDiffParam2用參數(shù)微分法中的中點積分法求非線性方程組的一組解function r=DiffParam2(F,x0,h,N)線性方程組：f劑始解：x0喊值微分增量步大?。篽%推可比迭代參量：l%軍的精度：eps%求得的一組解：r旭代步數(shù)：nx0 = transpose(x0);n = length(x0);ht = 1/N;Fx0 = subs(F,findsym(F),x0);J = zeros(n,n);for i=1:nxt = x0;xt(i) = xt(i)+h(i);J(:,i) = (subs(F,findsym(F),xt)-Fx0)/h(i);endinJ = i

27、nv(J);x1 = x0 - ht*inJ*Fx0;for k=1:Nx2 = x1 + (x1-x0)/2;Fx2 = subs(F,findsym(F),x2);J = zeros(n,n);for i=1:nxt = x2;xt(i) = xt(i)+h(i);J(:,i) = (subs(F,findsym(F),xt)-Fx2)/h(i); endinJ = inv(J);r = x1 - ht*inJ*Fx0;x0 = x1;x1 = r;endmulFastDown用最速下降法求非線性方程組的一組解function r,m=mulFastDown(F,x0,h,eps)form

28、at long;if nargin=3eps=1.0e-8;endn = length(x0);x0 = transpose(x0);m=1;tol=1;while tolepsfx = subs(F,findsym(F),x0);J = zeros(n,n);for i=1:nx1 = x0;x1(i) = x1(i)+h;J(:,i) = (subs(F,findsym(F),x1)-fx)/h;endlamda = fx/sum(diag(transpose(J)*J);r=x0-J*lamda; % 核心迭代公式fr = subs(F,findsym(F),r);tol=dot(fr,

29、fr);x0 = r;m=m+1;if(m100000)%迭代步數(shù)控制disp(迭代步數(shù)太多，可能不收斂！,);return;endendformat short;mulGSND用高斯牛頓法求非線性方程組的一組解function r,m=mulGSND(F,x0,h,eps)format long;if nargin=3eps=1.0e-8;endn = length(x0);x0 = transpose(x0);m=1;tol=1;while tolepsfx = subs(F,findsym(F),x0);J = zeros(n,n);for i=1:nx1 = x0;x1(i) = x1

30、(i)+h;J(:,i) = (subs(F,findsym(F),x1)-fx)/h;endDF = inv(transpose(J)*J)*transpose(J);r=x0-DF*fx; % 核心迭代公式tol=norm(r-x0);x0 = r;m=m+1;if(m100000)%迭代步數(shù)控制disp(迭代步數(shù)太多，可能不收斂！,);return;endendformat short;mulConj用共軻梯度法求非線性方程組的一組解function r,m=mulConj(F,x0,h,eps)format long;if nargin=3eps=1.0e-6;endn = length(x0);x0 = transpose(x0);fx0 = subs(F,findsym(F),x0);p0 = zeros(n,n);for i=1:nx1 = x0;x1(i) = x1(i)*(1+h);p0(:,i) = -(subs(F,findsym(F),x1)-fx0)/h;endm=1;tol=1;while tolepsfx = subs(F,findsym(F),x0);J = zeros(n,n);for i=1:nx1