Solving equalities

Question

I need to write efficient code for solving the following kinds of equalities:
12 / b = b + 1
and yes I know the answer is 3.
Anybody know a good algorithim or a another online resource where I might find one?

Mykonine · Answer

You can try using the Newton-Rhapson approximation.
Just convert the expression f(x) = g(x) to
g(x)-f(x) = h(x) = 0
and then use the guessing algorithm:
x = x - h(x)/h'(x) where h'(x) is the derivative of h(x). The derivative can also be approximated by using a very small delta-x (something close to epsilon, the smallest positive value expressible by a floating point number) using the following expression:
h'(x) = [ h(x+delta) - h(x) ]/delta
The math involved is complicated (stopping just short of calculus), but that should be a very fast algorithm for the generic case of f(x) = g(x).
Most importantly, you need to be able to parse and evaluate an algebraic expression consisting of "^", "+", "-", "*", "/", "(", ")", and "x" (x being plugged in during evaluation) before you can do anything else. This is best done recursively.
All in all, it's complicated because programs generally can't do algebraic expressions that aren't hardcoded; solved and reduced to a simple "answer = f(x)".