Newton's method returns NaN when solving

Question

Editor 2022-12-21

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1882802-newton-s-method-returns-nan-when-solving

评论： Editor 2022-12-22

I am trying to solve a 7D system of equations using the Newton's method. However, my code returns NaN after running. I have checked through my equations and I do not have the case of 0/0. I have tried to change initial guesses but to no success. I cannot figure out why I get such values. Can someone please give me a third eye or suggest a better method that I can use? Thanks in anticipation. Below is my code:

clc; close all; clear
syms x1 x2 x3 x4 x5 x6 x7 x8
f1(x1, x2, x3, x4, x5, x6, x7) = (1.*x2.*(x3+x6) + x1.*x7.*x4)./(2.*x1.*x7 + 2.*x2) + x5./2;
f2(x1, x2, x3, x4, x5, x6, x7) = (2.*x2.*(x3+x6) + x1.*x7.*x4)./(2.*x1.*x7 + 4.*x2) + x5./2;
f3(x1, x2, x3, x4, x5, x6, x7) = (3.*x2.*(x3+x6) + x1.*x7.*x4)./(2.*x1.*x7 + 6.*x2) + x5./2;
f4(x1, x2, x3, x4, x5, x6, x7) = (4.*x2.*(x3+x6) + x1.*x7.*x4)./(2.*x1.*x7 + 8.*x2) + x5./2;
f5(x1, x2, x3, x4, x5, x6, x7) = sqrt( (1.*x2.*(x3+x6).*x5)./(x1.*x7 + 1.*x2) );
f6(x1, x2, x3, x4, x5, x6, x7) = sqrt( (2.*x2.*(x3+x6).*x5)./(x1.*x7 + 2.*x2) );
f7(x1, x2, x3, x4, x5, x6, x7) = sqrt( (3.*x2.*(x3+x6).*x5)./(x1.*x7 + 3.*x2) );
x=[8.0; 8.0; 4.0; 4.0; 3.0; 2.5; 1.6];
e=10^(-8);
n=20;
f1x1(x1,x2,x3,x4,x5,x6,x7) = diff(f1,x1);
f1x2(x1,x2,x3,x4,x5,x6,x7) = diff(f1,x2);
f1x3(x1,x2,x3,x4,x5,x6,x7) = diff(f1,x3);
f1x4(x1,x2,x3,x4,x5,x6,x7) = diff(f1,x4);
f1x5(x1,x2,x3,x4,x5,x6,x7) = diff(f1,x5);
f1x6(x1,x2,x3,x4,x5,x6,x7) = diff(f1,x6);
f1x7(x1,x2,x3,x4,x5,x6,x7) = diff(f1,x7);
f2x1(x1,x2,x3,x4,x5,x6,x7) = diff(f2,x1);
f2x2(x1,x2,x3,x4,x5,x6,x7) = diff(f2,x2);
f2x3(x1,x2,x3,x4,x5,x6,x7) = diff(f2,x3);
f2x4(x1,x2,x3,x4,x5,x6,x7) = diff(f2,x4);
f2x5(x1,x2,x3,x4,x5,x6,x7) = diff(f2,x5);
f2x6(x1,x2,x3,x4,x5,x6,x7) = diff(f2,x6);
f2x7(x1,x2,x3,x4,x5,x6,x7) = diff(f2,x7);
f3x1(x1,x2,x3,x4,x5,x6,x7) = diff(f3,x1);
f3x2(x1,x2,x3,x4,x5,x6,x7) = diff(f3,x2);
f3x3(x1,x2,x3,x4,x5,x6,x7) = diff(f3,x3);
f3x4(x1,x2,x3,x4,x5,x6,x7) = diff(f3,x4);
f3x5(x1,x2,x3,x4,x5,x6,x7) = diff(f3,x5);
f3x6(x1,x2,x3,x4,x5,x6,x7) = diff(f3,x6);
f3x7(x1,x2,x3,x4,x5,x6,x7) = diff(f3,x7);
f4x1(x1,x2,x3,x4,x5,x6,x7) = diff(f4,x1);
f4x2(x1,x2,x3,x4,x5,x6,x7) = diff(f4,x2);
f4x3(x1,x2,x3,x4,x5,x6,x7) = diff(f4,x3);
f4x4(x1,x2,x3,x4,x5,x6,x7) = diff(f4,x4);
f4x5(x1,x2,x3,x4,x5,x6,x7) = diff(f4,x5);
f4x6(x1,x2,x3,x4,x5,x6,x7) = diff(f4,x6);
f4x7(x1,x2,x3,x4,x5,x6,x7) = diff(f4,x7);
f5x1(x1,x2,x3,x4,x5,x6,x7) = diff(f5,x1);
f5x2(x1,x2,x3,x4,x5,x6,x7) = diff(f5,x2);
f5x3(x1,x2,x3,x4,x5,x6,x7) = diff(f5,x3);
f5x4(x1,x2,x3,x4,x5,x6,x7) = diff(f5,x4);
f5x5(x1,x2,x3,x4,x5,x6,x7) = diff(f5,x5);
f5x6(x1,x2,x3,x4,x5,x6,x7) = diff(f5,x6);
f5x7(x1,x2,x3,x4,x5,x6,x7) = diff(f5,x7);
f6x1(x1,x2,x3,x4,x5,x6,x7) = diff(f6,x1);
f6x2(x1,x2,x3,x4,x5,x6,x7) = diff(f6,x2);
f6x3(x1,x2,x3,x4,x5,x6,x7) = diff(f6,x3);
f6x4(x1,x2,x3,x4,x5,x6,x7) = diff(f6,x4);
f6x5(x1,x2,x3,x4,x5,x6,x7) = diff(f6,x5);
f6x6(x1,x2,x3,x4,x5,x6,x7) = diff(f6,x6);
f6x7(x1,x2,x3,x4,x5,x6,x7) = diff(f6,x7);
f7x1(x1,x2,x3,x4,x5,x6,x7) = diff(f7,x1);
f7x2(x1,x2,x3,x4,x5,x6,x7) = diff(f7,x2);
f7x3(x1,x2,x3,x4,x5,x6,x7) = diff(f7,x3);
f7x4(x1,x2,x3,x4,x5,x6,x7) = diff(f7,x4);
f7x5(x1,x2,x3,x4,x5,x6,x7) = diff(f7,x5);
f7x6(x1,x2,x3,x4,x5,x6,x7) = diff(f7,x6);
f7x7(x1,x2,x3,x4,x5,x6,x7) = diff(f7,x7);
f11=matlabFunction(f1);
f22=matlabFunction(f2);
f33=matlabFunction(f3);
f44=matlabFunction(f4);
f55=matlabFunction(f5);
f66=matlabFunction(f6);
f77=matlabFunction(f7);
f1x11=matlabFunction(f1x1);
f1x22=matlabFunction(f1x2);
f1x33=matlabFunction(f1x3);
f1x44=matlabFunction(f1x4);
f1x55=matlabFunction(f1x5);
f1x66=matlabFunction(f1x6);
f1x77=matlabFunction(f1x7);
f2x11=matlabFunction(f2x1);
f2x22=matlabFunction(f2x2);
f2x33=matlabFunction(f2x3);
f2x44=matlabFunction(f2x4);
f2x55=matlabFunction(f2x5);
f2x66=matlabFunction(f2x6);
f2x77=matlabFunction(f2x7);
f3x11=matlabFunction(f3x1);
f3x22=matlabFunction(f3x2);
f3x33=matlabFunction(f3x3);
f3x44=matlabFunction(f3x4);
f3x55=matlabFunction(f3x5);
f3x66=matlabFunction(f3x6);
f3x77=matlabFunction(f3x7);
f4x11=matlabFunction(f4x1);
f4x22=matlabFunction(f4x2);
f4x33=matlabFunction(f4x3);
f4x44=matlabFunction(f4x4);
f4x55=matlabFunction(f4x5);
f4x66=matlabFunction(f4x6);
f4x77=matlabFunction(f4x7);
f5x11=matlabFunction(f5x1);
f5x22=matlabFunction(f5x2);
f5x33=matlabFunction(f5x3);
f5x44=matlabFunction(f5x4);
f5x55=matlabFunction(f5x5);
f5x66=matlabFunction(f5x6);
f5x77=matlabFunction(f5x7);
f6x11=matlabFunction(f6x1);
f6x22=matlabFunction(f6x2);
f6x33=matlabFunction(f6x3);
f6x44=matlabFunction(f6x4);
f6x55=matlabFunction(f6x5);
f6x66=matlabFunction(f6x6);
f6x77=matlabFunction(f6x7);
f7x11=matlabFunction(f7x1);
f7x22=matlabFunction(f7x2);
f7x33=matlabFunction(f7x3);
f7x44=matlabFunction(f7x4);
f7x55=matlabFunction(f7x5);
f7x66=matlabFunction(f7x6);
f7x77=matlabFunction(f7x7);
for i=1:n
    F=[f11(x(1),x(2),x(3),x(4),x(5),x(6),x(7));f22(x(1),x(2),x(3),x(4),x(5),x(6),x(7));...
        f33(x(1),x(2),x(3),x(4),x(5),x(6),x(7)); f44(x(1),x(2),x(3),x(4),x(5),x(6),x(7));...
        f55(x(1),x(2),x(3),x(4),x(5),x(6),x(7));f66(x(1),x(2),x(3),x(4),x(5),x(6),x(7));...
        f77(x(1),x(2),x(3),x(4),x(5),x(6),x(7))]; 
    J=[f1x11(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f1x22(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f1x33(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f1x44(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f1x55(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f1x66(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f1x77(x(1),x(2),x(3),x(4),x(5),x(6),x(7)); 
       f2x11(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f2x22(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f2x33(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f2x44(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f2x55(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f2x66(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f2x77(x(1),x(2),x(3),x(4),x(5),x(6),x(7)); 
       f3x11(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f3x22(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f3x33(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f3x44(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f3x55(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f3x66(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f3x77(x(1),x(2),x(3),x(4),x(5),x(6),x(7));
       f4x11(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f4x22(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f4x33(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f4x44(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f4x55(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f4x66(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f4x77(x(1),x(2),x(3),x(4),x(5),x(6),x(7));
       f5x11(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f5x22(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f5x33(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f5x44(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f5x55(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f5x66(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f5x77(x(1),x(2),x(3),x(4),x(5),x(6),x(7)); 
       f6x11(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f6x22(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f6x33(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f6x44(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f6x55(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f6x66(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f6x77(x(1),x(2),x(3),x(4),x(5),x(6),x(7)); 
       f7x11(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f7x22(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f7x33(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f7x44(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f7x55(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),f7x66(x(1),x(2),x(3),x(4),x(5),x(6),x(7)),...
       f7x77(x(1),x(2),x(3),x(4),x(5),x(6),x(7))]; 
    y=-J\F;
    x=x+y
end

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Answer 1

Torsten 2022-12-21

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1882802-newton-s-method-returns-nan-when-solving#answer_1133802

编辑：Torsten 2022-12-22

If you rename y(1)= x(3)+x(6) and y(2)=x(1)*x(7) in your equations for f1,...,f7, you will see that you have 7 equations for 5 unknowns (x(2),x(4),x(5),y(1),y(2)). This doesn't sound good for solvability of the system, especially for the regularity of the Jacobian.

So remove two of the equations f1,...,f7 and replace x(3)+x(6) by y(1) and x(1)*x(7) by y(2).

One possible solution for your equations is

x(2) = 0, y(1) = const1, y(2) = const2 (~=0), x(4) = const3, x(5) = -const3

where const1, const2 ~=0 and const3 are arbitrary constants.