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Is there a numerical method to solve this equation, considering that we cannot write out the specific expressions for F or G, but can numerically compute the gradient or function value information at a certain point x? Can anyone provide some suggestions? In regard to my specific problem, $F(x) = \frac{||f(x)||^2_2}{2}$. f(x) is a known function from $R^n$ to $R^n$. $G(x)=\frac{a}{x}+b$.

$a$ and $b$ are constant vectors, and the division operation is element-wise division.

I can also compute the function value of $f^{-1}(x)$， I'm not sure if this would be helpful.

yllgl
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梁港林
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    What do you mean by “solve”? If $F$ and $G$ are both known, what is there to solve? – kieransquared Apr 14 '23 at 03:17
  • I want to find a solution for some x that makes this equation equal. – 梁港林 Apr 14 '23 at 05:05
  • This equation is not valid for any arbitrary x; I want to find the numerical solution for x that makes the equation hold true. "Known" means that the function value and derivative can be calculated at a specific point, but the explicit expression is unknown. – 梁港林 Apr 14 '23 at 05:12
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    Ah, your differential equations tags led me to believe you were solving for a function. To solve any kind of nonlinear equation you can use a root-finding algorithm like Newton’s method. – kieransquared Apr 14 '23 at 14:17

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