# How to choose an initial value for a multidimensional equation while using Newton-Raphson method to solve it?

I am trying to solve the following nonlinear equation for $$f \in \mathbb{R}^3$$ using the Newton-Raphson method, $$G(f) = {\sin{(||f||)} \over ||f|| }Jf + {1-\cos(||f||) \over ||f||^2} f \times Jf = g$$ where $$g$$ is a given vector in $$\mathbb{R}^3$$, $$J \in \mathbb{R}^{3 \times 3}$$ is a constant positive definite diagonal matrix and $$||.||:\mathbb{R}^3 \rightarrow \mathbb{R}-\{\mathbb{R}^-\}$$, represents Euclidean norm. The iterative equation obtained after applying this is the following, $$f_{i+1} = f_{i} + \nabla G (f_i)^{-1}(g-G(f_i))$$

Newton-Raphson method for an equation $$f(x)=0$$, where, $$f:\mathbb{R}\rightarrow \mathbb{R}$$, converges if the initial guess is very close to the root of the equation. We can roughly find a neighbourhood of the roots of such equations by plotting graphs or linearizing it but for equations like written above, I am not sure how would this work. I tried solving the above equation for different initial conditions, for some, it converges in at least $$9$$ iterations while for some other initial conditions it diverges badly. The measure of convergence is checked by computing the norm $$||g-G(f_i)||$$ in every iteration step.

Please find some iteration results below (performed in MATLAB):

Value of the matrix $$J = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$ and $$g = \begin{bmatrix} 6.13 \\ 1.73 \\ 3.20 \end{bmatrix}$$

1. For the initial condition $$f_0 = \begin{bmatrix} 1 \\ 1 \\ 1\end{bmatrix}$$, the value of the norm $$||g-G(f_i)||$$ converges to $$10^{-14}$$ in $$10$$ iterations. The solution of the equation comes out to be $$\begin{bmatrix} 1.84 \\ 1.16 \\ 1.42 \end{bmatrix}$$
2. For the initial condition $$f_0 = \begin{bmatrix} 2 \\ 2 \\ 1\end{bmatrix}$$, the equation diverges as the value of the norm $$||g-G(f_i)||$$ grows to $$10^{3}$$. The value of the determinant of the matrix $$\nabla G$$ becomes very small with subsequent iterations (i.e. the matrix $$\nabla G$$ loses its invertibility in the iterative equation)

It seems an issue of finding a proper initial value to solve it numerically using the Newton-Raphson method. Any other method for solving such equations would also really be appreciated. Looking forward to your suggestions!

Thank you, Dhananjay.