I have the following equations that I need to solve. $$ \psi(\alpha)-\psi(\alpha+\beta)=X_0 \\ \psi(\beta)-\psi(\alpha+\beta)=Y_0 $$ $X_0$ and $Y_0$ are known constants. Is there a way to atleast approximately solve this equation?
Incase anyone wants to know I am moment matching $E(\ln \pi)$ and $E(\ln (1-\pi))$ for the beta distribution. I've already found out $X_0, Y_0$.
I need the calculation to be fairly quick (computationally) so hoping for something beyond newton's method. Something where I get the approximate answer in a few iterations.
One thing you may be aware that, if $X_0=Y_0$, this equation does not have unique solution (the Jacobian is singular). Therefore, you may instead consider choosing the initial start point carefully if the Newton method is used. If $X_0>Y_0$, then choose $\alpha(0) > \beta(0)$, and vice versa.
Another way to avoid simple Newton update is to consider proximal method ensuring the proximity between $(\alpha(t+1),\beta(t))$ and $(\alpha(t),\beta(t))$.