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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,...


2

Just introduce the velocity as an additional variable and solve: $$\frac{d}{dt}(x,\dot{x})^t = (\dot{x}, k\sin(x))^t$$ You can then solve that with any ODE integrator, e.g. ode45 in Matlab, RK45 with Scipy... Note: I am quite confused as to why you would use a Newton's method to solve this problem... You can apply it to solve each time step of an implicit ...


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(...


1

The condition for finding a root that corresponds to positive definiteness of the Hessian in optimization, is that the function grows strictly monotonically. But this is not a useful condition. That is because even in optimization, positive definiteness of the Hessian does not actually guarantee convergence of the unmodified Newton method. What is important ...


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