An algorithm for checking whether a nonlinear function f is always positive

Question

Is there an algorithm for checking whether this (possibly non-linear) function f is always positive?

The idea I currently have is to find the roots of the function (using the Newton-raphson algorithm or similar methods, see http://en.wikipedia.org/wiki/Root-finding_algorithm ) and check the derivatives or find minimum f, but they do not seem to be the best solution to this problem, there are also many problems of convergence with root search algorithms.

For example, in Maple, a check function can do this, but I need to implement it in my own program. Maple help for verification: http://www.maplesoft.com/support/help/Maple/view.aspx?path=verify/function_shells Maple example: Assume (x, 'real'); check (x ^ 2 + 1.0, 'greater_than'); → returns true, since for each x we ​​have x ^ 2 + 1> 0

[edit] Some background to the question: The function $ f $ is a differential nonlinear model of the right side for the circuit. A nonlinear circuit can be modeled as a set of ordinary differential equations by applying modified nodal analysis (MNA), for simplicity we consider only systems with 1 dimension, therefore $ x '= f (x) $, where $ f $ describes a circuit, for example $ f $, can be $ f (x) = 10x - 100x ^ 2 + 200x ^ 3 - 300x ^ 4 + 100x ^ 5 $ (model for a nonlinear tunneling diode) or $ f = 10 - 2sin (4x) + 3x $ (model for josephson transition).

$ x $ is bounded, and $ f $ is defined only in the interval $ [a, b] \ in R $. $ f $ is continuous. I can also make the assumption that $ f $ is Lipschitz with the Lipschitz constant L> 0, but I don’t want it if I don’t have to.

Answer 1

If I understand your problem correctly, it comes down to counting the number of (real) roots in the interval, without necessarily identifying them. In fact, you don’t even need to get the exact number, just whether it is equal to zero or not.

If your function is a polynomial, I think the Sturm theorem may be applicable. The Wikipedia article claims that the other two procedures are preferred, so you can also check them out. I'm not sure if the Cartesian rule of signs works on an interval, but Budan’s theorem .