Finding more than one root with root finding algorithms

Question

What is the best way to find the roots of an equation with more than one root. I understand that no one method which can solve every equation, and that you have to use more than one, but I can't find a root finding algorithm that can solve for more than one root in even the simplest instance.

For example: y = x^2

Although a root solving algorithm to solve a basic equation like this is helpful, it would need to be something I could adapt to solve an equation with more than two roots.

One more thing to note is that the equations wouldn't be your typical polynomials, but could be something such as ln(x^2) + x - 15 = 0

What is a root finding algorithm that could solve for this, or how could you edit a root finding algorithm such as the Bisection/Newton/Brent method to solve this problem, (Assuming I'm correct in that Newton and Brent's method can only solve for one root).

Answer 1

I'd say that there's no general method to find all the roots of a general equation. However, one can try and devise methodologies once sufficient conditions have been specified. Even simple quadratic equations ax² + bx + c = 0 aren't completely trivial, because the existence of real roots depends on the sign of b²-4ac, which isn't immediately obvious. So there are lots of techniques to apply, e.g Newton-Raphson, but no general method for the general case, especially for equations like ln(x²)+x-15 = 0.

Answer 2

Bottom line: You need to isolate the roots yourself.

Details depend on the algorithm: If you're using bisection or Brent's method, you need to come up with a set of intervals each containing a unique root. If using the Newton's method, you need to come up with a set of starting estimates (since it converges to a root given a starting point, and with different starting points it may or may not converge to different roots).