I have 3 non linear equations where the variables are $a,b,\phi$. Equations are $$ \frac{x}{\sqrt{x^{2} + y^{2}}} = -1 + \frac{1}{1.2}\times \cos(\phi)$$ $$ \frac{y}{\sqrt{x^{2} + y^{2}}} = \frac{1}{1.2}\times \sin(\phi)$$ $$ \arg((\frac{x- jy}{\sqrt{x^{2}+y^{2}}})-a) = \frac{\pi}{2} - \phi$$
Where $x = \cos(a) - ab\sin(a) - 1$ and $y = ab\cos(a) + \sin(a)$ $x$ & $y$ are functions of $a$.
I tried to solve it but the problem is whatever initial value I choose, I couldn't get inverse of Jacobian Matrix.
Reference: https://www.dropbox.com/s/3sv4hs5z5jn3mfr/xyz.pdf?dl=0
Make sure you do not have terms in the denominator because of numerical reasons. If possible rearrange the square root terms (bring them maybe to the other side). That should work then.
Furthermore, there are ways to solve it without the Jacobian. Find the right algorithm that does not require it.