How can I solve multivariable Newton`s Method with implicit functions?

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Eric Bernard Dilger 2022-9-14

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https://ww2.mathworks.cn/matlabcentral/answers/1805500-how-can-i-solve-multivariable-newton-s-method-with-implicit-functions

评论： Torsten 2022-9-16

I am trying to solve a system of non linear equations using Newton`s Method. The function F(X), the unknown variables X and the initial guess X0 are presented below:

F(X)=[V1+V2+Vblk-Vstring;      I1(V1)-I2(V2);      I1(V1)-Iblk(V3)];
X = [V1;    V2;     Vblk;];
X0 = [Voc;  Voc;    0;];

Vstring is assigned from 91.1 to 0 so that Newtons`s Method must be applied for each value of Vstring to return the contribution of V1, V2 and Vblk across Vstring. However, in F(2) and F(3) the variables V1, V2 and Vblk are implicit functions of I1, I2 and Iblk. Before running Newton`s Method, the variables I1, I2 and Iblk are calculated by means of Lambert W function and then stored in arrays, so that variables I1, I2 and Iblk are known for any V1, V2 and Vblk values before Newton`s Method is applied. How could I solve the Newton`s Method considering these implicit functions for all Vstring values? I appreciate any help!

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Eric Bernard Dilger 2022-9-16

Hello @Torsten, thanks for support!

I replaced the code as you suggested and it worked. I also inclunded one more variable I3 raising the order of the system of equations and it`s working for most cases. However, when I set the input parameters which results in I1 = I2 = I3 equations are solved in a few iterations, but inaccurately. I tried setting TolFun and TolX equal to 1e-14 but it did not solve the problem. The purpose is to simulate a system of equations with order N = 33, but I don`t know if fsolve is suitable for this scale of application, it seems convergence problems may happen.