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2

Newton's method can refer either to a method for solving $f(x)=0$ where $f: R^{n} \rightarrow R^{n}$, or to a method for minimizing/maximizing a function $g: R^{n} \rightarrow R$ by solving the system of equations $\nabla g(x)=0$. Your function $h$ maps $R^{2}$ to $R$ and you want to find a zero of the function. This is typically done by minimizing $\min h(...


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Although I do not know about the GSL package, I can give some tips for root-finding concerning domain issues. In the 1-D case, you can guarantee convergence with a bracketing method such as Newt-safe or bisection as you've suggested. This requires a bracketing interval given by two points where the function changes signs. In the general case, I recommend ...


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If you have a single vector equation $\vec{F}(\vec{x})=0$ then you solve it by representing that state vector $\vec{x}$ as a set of amplitudes $[x_0,x_1,...,x_{n-1}]$ after discretization by your favorite method (FD, FV, FEM, spectral); and we know how to solve it. If you also have a second equation $\vec{G}(\vec{y})=0$ then the full state vector is $[x_0,...


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