Classes for fuzzy numbers and methods for computing real solutions of fuzzy polynomial systems

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This ticket implements generic classes for fuzzy numbers in their different representations, and methods to compute real solutions of a polynomial system with fuzzy coefficients and two indeterminates.

The function SolvingFuzzySystem finds real solutions of polynomial systems whose coefficients are L-R fuzzy numbers with bounded support and the same bijective spread functions. The real solutions of a given fuzzy system are deduced from solutions of some polynomial systems with real coefficients. This algorithm is based on new results that are universal because they are independent from the spread functions. These functions include the management of the fuzzy system’s solutions signs.

Up to now, given a fuzzy system (S) of s equations and k indeterminates (here k = 2), the existing algebraic methods have performed computations with the parametric representation of the coefficients to obtain the collected crisp form of (S) formed by 4s real equations. These computations are superfluous and we exhibit a formula that defines an equivalent system with 3s real equations. We call it the real transform T(S) of the system (S). T(S) is built with the function reconstruction_systems. As a main property, it has the same positive solutions as (S). Unlike the previous methods that were restricted to triangular fuzzy numbers, our results apply to any family F(L,R) where the spread functions L and R are bijective. Moreover there is no use to compute the inverse of the spread functions since the real transform is a universal formula independent from L and R.

For solving equations with fuzzy coefficients all in a same family F(L,R), one must face the issue of the sign of solutions. It is intrinsic to fuzzy numbers, since the product by a real scalar is expressed differently depending on the sign of this scalar. Our strategy has been to only focus on positive solutions by putting back the issue on the fuzzy coefficients. The real solutions of a system (S) are expressed from the positive solutions of at most 2 to the power k real induced systems. Our algorithm automatizes the research of solutions by avoiding the studies of signs needed in previous methods. Our approach is independent of the choice of the method to calculate the positive solutions of a system of polynomial with real coefficients. Here we use Wu's algorithm to compute the positive solutions of the system.

Among the 2 to the power k induced systems of (S), some of them are identical. It it is not rare to substantially reduce the number of induced systems to solve. We describe a strategy to avoid redundant branches of computations that leads to an optimized algorithm SolvingFuzzySystem.

Component: algebra

Keywords: real solving fuzzy polynomial systems

Author: Jeremy Marrez

Branch: github.com/JeremyMarrez/Fuzzy/blob/master/Real%20solving%20of%20fuzzy%20polynomial%20systems

Issue created by migration from https://trac.sagemath.org/ticket/28846

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Computing real solutions of polynomial fuzzy systems

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Attachment: AubryMarrezValibouze_Fuzzy_Ang_2017_12_18_HAL.pdf.gz

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