Alizadeh, B.M., Rahmany, S., Basiri, A. Interval gröbner system and its applications (2018) Reliable Computing, 26, pp. 67-96.
ABSTRACT In this paper, the concept of the interval Gröbner system together with an algorithm for its computation are proposed to analyze algebraic polynomial systems with interval coefficients (interval polynomial systems). These systems appear in many computational problems arising from both the engineering and mathematical sciences. As opposed to linear interval polynomial systems, there is no method to solve and/or analyze a nonlinear interval polynomial system. Interval Gröbner systems enable us to determine whether an interval polynomial system has any solutions or not. If so, a finite decomposition of the solution set will be constructed by the elements of the computed interval Gröbner system. Furthermore, this concept allows us to verify whether two interval polynomial systems share a common solution or not. The concept of the interval Gröbner system is based on elimination tools on the set of interval polynomials. It is worth noting that this is not a trivial extension of usual techniques, since the set of interval polynomials does not satisfy the distributivity and additive inverse axioms of a ring with usual interval arithmetic. In doing so, we introduce the concept of the ideal family associated to an interval polynomial system which contains an infinite number of (non-interval) polynomial ideals. Then we analyze all of these ideals using an equivalence relation with a finite number of equivalence classes. This method is based on a novel computational algebraic tool, the concept of comprehensive Gröbner system, equipped with an interval-based criterion to omit unnecessary computations. We also provide some applications of interval Grobner systems to analyze interval polynomial systems, nding multiple roots and solving the divisibility problem of interval polynomials. Our algorithm for the computation of interval Grobner systems has been implemented in both Maple and Magma software packages. © 2018 Springer Netherlands. All rights reserved.
INDEX KEYWORDS: Equivalence classes; Inverse problems, Algebraic polynomials; Computational problem; Divisibility problem; Equivalence relations; Interval arithmetic; Interval coefficients; Interval polynomials; Mathematical science, Polynomials PUBLISHER: Springer Netherlands
Boroujeni, M., Basiri, A., Rahmany, S., Valibouze, A. Finding solutions of fuzzy polynomial equations systems by an Algebraic method (2016) Journal of Intelligent and Fuzzy Systems, 30 (2), pp. 791-800.
DOI: 10.3233/IFS-151801
ABSTRACT Finding solutions for fuzzy polynomial systems has recently received much attention and many efforts have been made to make the available algorithms for solving such problems more and more efficient. In the present paper, Wu's algorithm is introduced as a solution procedure to obtain fuzzy polynomial systems solutions. In this approach, the parametric form of the problem is first obtained from the computation of r-cuts of fuzzy polynomials. Wu's algorithm is then applied in order to convert the parametric form of a fuzzy polynomial system into a finite number of characteristic sets. We then have the right relation between solutions of these sets and those of the polynomial system. The most outstanding advantage of the proposed method lies in the fact that it leads to solve triangular sets amenable to easy solution. © 2016 - IOS Press and the authors. All rights reserved.
AUTHOR KEYWORDS: characteristic sets; fuzzy polynomial systems; Polynomial systems; Wu's algorithm INDEX KEYWORDS: Algebra; Algorithms; Parameter estimation, Algebraic method; Characteristic set; Finding solutions; Parametric forms; Polynomial equation; Polynomial systems; Solution procedure; Triangular sets, Polynomials PUBLISHER: IOS Press
Boroujeni, M., Basiri, A., Rahmany, S., Valibouze, A. Solving Fuzzy Systems in Dual Form Using Wu's Method (2015) International Journal of Fuzzy Systems, 17 (2), pp. 170-180.
DOI: 10.1007/s40815-015-0033-4
ABSTRACT In this paper, fuzzy polynomial systems in dual form are considered and an algebraic approach for finding their solutions is presented. A dual fuzzy polynomial system in the form AX + B = CX + D, where A, B, C, and D are fuzzy matrices, is converted to a system with real coefficients and variables first. Then, Wu's algorithm is used as a solution procedure for solving this system. This algorithm leads to solving characteristic sets that are amenable to easy solution. Finally, the accuracy of the presented algorithm is shown via some examples. © 2015 Taiwan Fuzzy Systems Association and Springer-Verlag.
AUTHOR KEYWORDS: Characteristic sets; Fuzzy polynomial systems; Polynomial systems; Wu's algorithm INDEX KEYWORDS: Algorithms, Algebraic approaches; Characteristic set; Dual form; Fuzzy matrix; Polynomial systems; Real coefficients; Solution procedure; Wu's method, Polynomials PUBLISHER: Springer Berlin Heidelberg
Boroujeni, M., Basiri, A., Rahmany, S., Valibouze, A. F4-invariant algorithm for computing SAGBI-Gröbner bases (2015) Theoretical Computer Science, 573, pp. 54-62.
DOI: 10.1016/j.tcs.2015.01.045
ABSTRACT This paper introduces a new algorithm for computing SAGBI-Gröbner bases for ideals of invariant rings of permutation groups. This algorithm is based on F4 algorithm. A first implementation of this algorithm has been made in SAGE and MAPLE computer algebra systems and has been successfully tried on a number of examples. © 2015 Elsevier B.V.
AUTHOR KEYWORDS: F4 algorithm; F5 algorithm; Invariant ring; SAGBI-Gröbner basis INDEX KEYWORDS: Algebra, Computer algebra systems; Invariant algorithms; Invariant ring; Permutation group, Algorithms PUBLISHER: Elsevier
Farahani, H., Rahmany, S., Basiri, A. Determining of level sets for a fuzzy surface using Gröbner basis (2015) International Journal of Fuzzy System Applications, 4 (2), pp. 1-14.
DOI: 10.4018/IJFSA.2015040101
ABSTRACT In this paper, a manner to determine the level sets of a fuzzy surface using the benefits of Gröbner basis is presented. Fuzzy surfaces are constructed from incomplete datasets or from data that contain uncertainty which has not statistical nature. The authors firstly define the concept of level sets for the fuzzy surfaces. Then, employing Gröbner bases benefits a criterion is proposed for when the level sets of the fuzzy surface are nonempty sets. Moreover, a new algorithm is designed to determine the level sets. The big advantage of the proposed method lies in the fact that it attains all members of the level sets of the fuzzy surface at a time. Finally, some applied numerical examples are illustrated to demonstrate the proficiency of the given approach. Copyright © 2015, IGI Global.
AUTHOR KEYWORDS: Fuzzy numbers; Fuzzy surfaces; Gröbner bases; Level sets PUBLISHER: IGI Global
Farahani, H., Rahmany, S., Basiri, A., Abbasi Molai, A. Resolution of a system of fuzzy polynomial equations using eigenvalue method (2014) Soft Computing, 19 (2), pp. 283-291.
DOI: 10.1007/s00500-014-1249-1
ABSTRACT In this paper, a system of fuzzy polynomial equations is studied. Two solution types are defined for this system, called solution and (Formula presented.)-cut solution. Then sufficient and necessary conditions are proposed for existence of solution and (Formula presented.)-cut solution of the system, respectively. The solution set of the system is also determined. Moreover, a new algorithm is designed to find all the solutions and all the (Formula presented.)-cut solutions of the system based on the eigenvalue method. Finally, some examples are given to illustrate the concepts and the algorithm. © 2014, Springer-Verlag Berlin Heidelberg.
AUTHOR KEYWORDS: Eigenvalue method; Fuzzy numbers; Gröbner basis; Real solution; System of fuzzy polynomial equations INDEX KEYWORDS: Eigenvalues and eigenfunctions; Fuzzy sets, Eigenvalue methods; Existence of Solutions; Fuzzy numbers; Polynomial equation; Real solutions; Solution set; Solution types; S