Welcome to our website! It is under construction.
Welcome to our website! It is under construction.
Prof. Eduard Danielian.
My wife Angela and me.
We introduce here Monotone Calculator. What it calculates and How?
The What. It allows to solve a list of very practical problems in Probability and Statistics, Optimal Control, Polynomials. To our knowledge majority of these problems were not calculable before. They are described in section Monotone Calculator. Particularly, for cumulative distribution functions on a finite interval, Monotone Calculator solves a set of minimization problems with up to 10 moments. At present Monotone Calculator solves a dozen of problems. Its number will reach a hundred in a year. We project a thousand quite practical problems solution in five year perspective. What unites all these at first sight quite unconnected or irrelevant problems? They have similar internal structure that is invisible at first sight. Therefore, to solve them we can apply the same method. The same How.
The How. Monotone Calculator uses an original method based exclusively on the notion of monotonicity. It does not use any of XVI-XXI centuries math. It could be used in Ancient Greece if they had calculators. Actually, even “plus”, “minus”, “multiplication” and “division” operations are not used. The method is using only “if”s and “<”s. Do we have something similar in computational mathematics? Sure. It is so called bisection method. Let us describe it. We should solve the equation
f(x) = 0,
where function f(x) is a monotone (increasing) function. What we do? We take any initial x . Then if f(x) < 0, we go right and if not we go left. Then we repeat. Thus, we reach a unique solution with the exactness we wish.
I remember, when I was a student and was introduced to bisection method during the course of Computational Methods I was disappointed that it does not have any continuation.
Here, by mere existence of Monotone Calculator we show that such extensions exist and we successfully apply them to solve a range of problems in Probability and Statistics, Optimal Control and Polynomials.
However, in spite of its “simplicity” and theoretical possibility to be discovered even in ancient Greece, to suspect that such sort of extension of bisection method exists a mathematician should have a very specific set of knowledge that takes its roots from Total Positivity up to … Algebraic Topology. I consider myself not an author but a discoverer of the method, or a group of methods that I called Monotone Programming. I still do not understand fully why does it work. I managed to write the code only because at some point I understood that it exists. Then, like an archeologist, I just unearthed it.
Monotone Calculator has a commercial value. We now in a process to evaluate it. It will take some time. That is why here we will not present the codes but will demonstrate the capacity of Monotone Calculator, the added value it brings to Statistics and Probability and some new results.