Methods to solve(root finding) the function with the restricted domain.
Suppose to solve the function
$sin^{-1}(\sqrt{E_n/V}) +a*\sqrt{2mE_n/h^2}=n*\pi$
where
$E_n,V,a,m,h n$ were all positive.
I tried to use Newton-Raphson Method to solve it where I realized that the input of those function must be positive thus from divergence
$iterate E_{n}(i+1)$
might be negative, thus produce a false result and stop the iteration.
The only solution I could found was thorough trust zone methods as
$E_n<=V$
was guaranteed by the function's physical meaning. However, it took much longer time.
Was there a way to use some fast computation like Newton-Raphson Method for this expression.
Summary from the comments to close the topic.
If the input of the funciton was restricted to, i.e.>0.
We could set m^2_n instead of x as squares of unconstrained variables – Ted Hopp
Secant method – meowgoesthedog
Ridders' method. - Shimuuar
Trust zone methods.