Let's say there's an SDE $$dX_t=a(X_t)dt+b(X_t)dW_t$$ Then the semi-implicit Euler scheme is given by $$X_{n+1}=X_n+(\alpha a(X_{n+1})+(1-\alpha)a(X_n))\Delta t+b(X_n)\Delta W_n$$ where $0\leq\alpha\leq 1$. I need to derive an implementation for this scheme so that there won't be any need to use the Newton's method.
I think it means to get a formula for $X_{n+1}$ in terms of $X_n$ to get a recurrence relation of some sort. But I can't seem to figure out how to get there. I think if there was a clear representation of what the function (drift coefficient) $a(X_{n+1})$ looks like, then I'd be able to do that. But that's just my idea.