matlab - 求解包含 Lambert-W 函数的非线性方程组

Question

我正在尝试使用 Newton-Raphson 方法求解非线性方程组。问题出在本文的第 2.1 节中。它只到第二次迭代，我得到了错误的答案。我将这些值用于iN和VN(N=1,2,3,4,5)：

V1 = 1.00400400e+00;
V2 = 1.02598500e+00;
V3 = 1.05325500e+00; 
V4 = 1.08812300e+00;
V5 = 1.13388700e+00;

i1 = -2.40036700e-02;
i2 = -3.59849700e-02;
i3 = -5.32552100e-02;
i4 = -7.81225600e-02;
i5 = -1.13887200e-01;

我用了：

a = Iph,
b = I0,
f = n,
c = Rs,
d = Rsh,
Vth = 0.0258

这是我的代码：

clear all; clc; 
N = 20;
tErr = 1e-6;
syms a b c d f
V1 = 1.00400400e+00; V2 = 1.02598500e+00; V3 = 1.05325500e+00; V4 = 1.08812300e+00; V5 = 1.13388700e+00; i1 = -2.40036700e-02; i2 = -3.59849700e-02; i3 = -5.32552100e-02; i4 = -7.81225600e-02 ; i5 = -1.13887200e-01; 
m1 = d*(-i1 + a + b)/(0.0258*f); m2 = d*(-i2 + a + b)/(0.0258*f); m3 = d*(-i3 + a + b)/(0.0258*f); m4 = d*(-i4 + a + b)/(0.0258*f);
m5 = d*(-i5 + a + b)/(0.0258*f); n = .0258*f; k = b*d/n;
f1 = -i1*c-i1*d+a*d-n*lambertw(b*d*exp(m1)/n)-V1;
f2 = -i2*c-i2*d+a*d-n*lambertw(b*d*exp(m2)/n)-V2;
f3 = -i3*c-i3*d+a*d-n*lambertw(b*d*exp(m3)/n)-V3;
f4 = -i4*c-i4*d+a*d-n*lambertw(b*d*exp(m4)/n)-V4;
f5 = -i5*c-i5*d+a*d-n*lambertw(b*d*exp(m5)/n)-V5;
fs = [f1; f2; f3; f4; f5];
xs = [a; b; f; c; d];
jac = [d/lambertw(b*d*exp(m1)/n), -(n*lambertw(b*d*exp(m1)/n)-b*d)/(b*(1+lambertw(b*d*exp(m1)/n))), lambertw(b*d*exp(m1)/n)*(-n*lambertw(b*d*exp(m1)/n)-i1*d+a*d+b*d)/(f*(1+lambertw(b*d*exp(m1)/n))), -i1, -(n*lambertw(b*d*exp(m1)/n)+i1*d-a*d-b*d)/((1+lambertw(b*d*exp(m1)/n))*d);
       d/lambertw(b*d*exp(m2)/n), -(n*lambertw(b*d*exp(m2)/n)-b*d)/(b*(1+lambertw(b*d*exp(m2)/n))), lambertw(b*d*exp(m2)/n)*(-n*lambertw(b*d*exp(m2)/n)-i2*d+a*d+b*d)/(f*(1+lambertw(b*d*exp(m2)/n))), -i2, -(n*lambertw(b*d*exp(m2)/n)+i2*d-a*d-b*d)/((1+lambertw(b*d*exp(m2)/n))*d);
       d/lambertw(b*d*exp(m3)/n), -(n*lambertw(b*d*exp(m3)/n)-b*d)/(b*(1+lambertw(b*d*exp(m3)/n))), lambertw(b*d*exp(m3)/n)*(-n*lambertw(b*d*exp(m3)/n)-i3*d+a*d+b*d)/(f*(1+lambertw(b*d*exp(m3)/n))), -i3, -(n*lambertw(b*d*exp(m3)/n)+i3*d-a*d-b*d)/((1+lambertw(b*d*exp(m3)/n))*d);
       d/lambertw(b*d*exp(m4)/n), -(n*lambertw(b*d*exp(m4)/n)-b*d)/(b*(1+lambertw(b*d*exp(m4)/n))), lambertw(b*d*exp(m4)/n)*(-n*lambertw(b*d*exp(m4)/n)-i4*d+a*d+b*d)/(f*(1+lambertw(b*d*exp(m4)/n))), -i4, -(n*lambertw(b*d*exp(m4)/n)+i4*d-a*d-b*d)/((1+lambertw(b*d*exp(m4)/n))*d);
       d/lambertw(b*d*exp(m5)/n), -(n*lambertw(b*d*exp(m5)/n)-b*d)/(b*(1+lambertw(b*d*exp(m5)/n))), lambertw(b*d*exp(m5)/n)*(-n*lambertw(b*d*exp(m5)/n)-i5*d+a*d+b*d)/(f*(1+lambertw(b*d*exp(m5)/n))), -i5, -(n*lambertw(b*d*exp(m5)/n)+i5*d-a*d-b*d)/((1+lambertw(b*d*exp(m5)/n))*d)];
x = [1.2; 0.001; 1.5; 1; 15];
for i=1:N
    f = double(subs(fs, xs, x));
    j = double(subs(jac, xs, x));
    x = x - inv(j)*f;
    
    err = abs(inv(j)*f);
    if(err<tErr)
        break;
    end
end
display(x);

标签： matlabmathdifferential-equations