Find intersection(s) of any two functions - solving simultaneous equations

Question

Imagine having any two functions. You need to find intersections of that functions. You definitely don't want to try all x values to check for f(x)==g(x). Normally in math, you create simultaneous equations derived from f(x)==g(x). But I see no way how to implement equations in any programing language.
So once more, what am I looking for:

1. Conceptual algorithm to solve equations.
2. The same for simultaneous and quadratic equations.

I believe there should be some workaround using function derivations, but I've recently learned derivation concept at school and I have no idea how to use it in this case.

Answer 1

That is a much harder problem than you would imagine. A good place to start for learning about these things is the Newton-Raphson method, which gives numerical approximations to equations of the form h(x) = 0. (When you set h(x) = g(x) - f(x), this provides solutions for the problem you are asking about.)

Exact, algebraic solving of equations (as implemented in Mathematica, for example) are even more difficult, you basically have to recreate everything you would do in your head when solving an equation on a piece of paper.

Answer 2

Obviously this problem is not solvable in the general case because you can construct a "function" which is arbitrarily complex. For example, if you have a "function" with 5 trillion terms in it including various transcendental and complex transformations in it, the computer could take years just to compute a single value, much less intersect it with another similar function.

So, first of all you need to define what you mean by a "function". If you mean a polynomial of degree less than 4 then the problem becomes much more straightforward. In such cases you combine the terms of the polynomial and find the roots of the equation, which will be the intersections.

If the polynomial has more than 5 terms (a quintic or greater) then there is no easy symbolic solution. In this case the terms are combined and you find the roots by iterative approximation. See Root Finding Algorithms.

If the function involves transcendentals such sin/cos/log/e^x, etc, you can potentially find the intersection by representing the functions as a series or a continued fraction. You then subtract one series from the other and set the value to zero. The solution of the continuous equation yields an approximation of the root(s).