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Journal articles on the topic 'Solving polynomial systems of equations'

Author: Grafiati

Published: 3 June 2025

Last updated: 13 July 2025

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1

Manocha, D. "Solving systems of polynomial equations." IEEE Computer Graphics and Applications 14, no. 2 (1994): 46–55. http://dx.doi.org/10.1109/38.267470.

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2

Moszyński, Krzysztof. "Remarks on polynomial methods for solving systems of linear algebraic equations." Applications of Mathematics 37, no. 6 (1992): 419–36. http://dx.doi.org/10.21136/am.1992.104521.

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3

LLIBRE, JAUME, and CLAUDIA VALLS. "POLYNOMIAL FIRST INTEGRALS FOR THE CHEN AND LÜ SYSTEMS." International Journal of Bifurcation and Chaos 22, no. 11 (2012): 1250262. http://dx.doi.org/10.1142/s0218127412502628.

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We characterize all the values of the parameters for which the Chen and Lü systems have polynomial first integrals by using weight homogeneous polynomials and the method of characteristics for solving partial differential equations. We improve previous results which were not complete.

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4

Wang, Dongming. "Solving polynomial equations: Characteristic sets and triangular systems." Mathematics and Computers in Simulation 42, no. 4-6 (1996): 339–51. http://dx.doi.org/10.1016/s0378-4754(96)00008-0.

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5

Serdyukova, S. I. "Solving large systems of polynomial equations using REDUCE." Programming and Computer Software 26, no. 1 (2000): 28–29. http://dx.doi.org/10.1007/bf02759175.

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6

Chukanov, Sergei Nikolaevich, and Ilya Stanislavovich Chukanov. "The Investigation of Nonlinear Polynomial Control Systems." Modeling and Analysis of Information Systems 28, no. 3 (2021): 238–49. http://dx.doi.org/10.18255/1818-1015-2021-3-238-249.

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The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

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7

Raghavan, M., and B. Roth. "Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators." Journal of Mechanical Design 117, B (1995): 71–79. http://dx.doi.org/10.1115/1.2836473.

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Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

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8

Raghavan, M., and B. Roth. "Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators." Journal of Vibration and Acoustics 117, B (1995): 71–79. http://dx.doi.org/10.1115/1.2838679.

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Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

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9

Barotov, Dostonjon Numonjonovich, and Ruziboy Numonjonovich Barotov. "Polylinear Transformation Method for Solving Systems of Logical Equations." Mathematics 10, no. 6 (2022): 918. http://dx.doi.org/10.3390/math10060918.

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In connection with applications, the solution of a system of logical equations plays an important role in computational mathematics and in many other areas. As a result, many new directions and algorithms for solving systems of logical equations are being developed. One of these directions is transformation into the real continuous domain. The real continuous domain is a richer domain to work with because it features many algorithms, which are well designed. In this study, firstly, we transformed any system of logical equations in the unit n-dimensional cube Kn into a system of polylinear–polynomial equations in a mathematically constructive way. Secondly, we proved that if we slightly modify the system of logical equations, namely, add no more than one special equation to the system, then the resulting system of logical equations and the corresponding system of polylinear–polynomial equations in Kn+1 is equivalent. The paper proposes an algorithm and proves its correctness. Based on these results, further research plans are developed to adapt the proposed method.

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10

Rojas, J. Maurice, and Yuyu Zhu. "A complexity chasm for solving sparse polynomial equations over p -adic fields." ACM Communications in Computer Algebra 54, no. 3 (2020): 86–90. http://dx.doi.org/10.1145/3457341.3457343.

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The applications of solving systems of polynomial equations are legion: The real case permeates all of non-linear optimization as well as numerous problems in engineering. The p -adic case leads to many classical questions in number theory, and is close to many applications in cryptography, coding theory, and computational number theory. As such, it is important to understand the complexity of solving systems of polynomial equations over local fields. Furthermore, the complexity of solving structured systems --- such as those with a fixed number of monomial terms or invariance with respect to a group action --- arises naturally in many computational geometric applications and is closely related to a deeper understanding of circuit complexity (see, e.g., [8]). Clearly, if we are to fully understand the complexity of solving sparse polynomial systems, then we should at least be able to settle the univariate case, e.g., classify when it is possible to separate and approximate roots in deterministic time polynomial in the input size.

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11

Ivanyos, Gábor, and Miklos Santha. "Solving systems of diagonal polynomial equations over finite fields." Theoretical Computer Science 657 (January 2017): 73–85. http://dx.doi.org/10.1016/j.tcs.2016.04.045.

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12

Farahani, Hamed, and Hossein Jafari. "SOLVING FULLY FUZZY POLYNOMIAL EQUATIONS SYSTEMS USING EIGENVALUE METHOD." Advances in Fuzzy Sets and Systems 24, no. 1 (2019): 29–54. http://dx.doi.org/10.17654/fs024010029.

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13

Barotov, Dostonjon, Aleksey Osipov, Sergey Korchagin, et al. "Transformation Method for Solving System of Boolean Algebraic Equations." Mathematics 9, no. 24 (2021): 3299. http://dx.doi.org/10.3390/math9243299.

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In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube Kn with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in Kn is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reliable evidence. Based on this work, plans for further research to improve the proposed method are outlined.

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14

Jiang, Zhaolin. "Fast Algorithms for Solving FLSR-Factor Block Circulant Linear Systems and Inverse Problem ofAX=b." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/340803.

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Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLSR-factor block circulant (retrocirculant) matrix over fieldF. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a fieldFis nonsingular. Fast algorithms for solving the unique solution of the inverse problem ofAX=bin the class of the level-2 FLS(R,r)-circulant(retrocirculant) matrix of type(m,n)over fieldFare given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.

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15

Alshabanat, Amal, and Bessem Samet. "A numerical study of a coupled system of fractional differential equations." Filomat 34, no. 8 (2020): 2585–600. http://dx.doi.org/10.2298/fil2008585a.

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We consider a certain class of coupled systems of fractional differential equations involving ?-Caputo fractional derivatives. A numerical approach is provided for solving this class of systems. The method is based on operational matrix of fractional integration of an arbitrary ?-polynomial basis. A theoretical study related to the numerical scheme and the convergence of the method is presented. Next, several numerical examples are given using different types of polynomials aiming to confirm the efficiency of our approach.

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16

Nedashkovska, Anastasiya. "Solving systems of matrix equations of the second degree." Physico-mathematical modelling and informational technologies, no. 33 (September 3, 2021): 52–56. http://dx.doi.org/10.15407/fmmit2021.33.052.

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Matrix equations and systems of matrix equations are widely used in control system optimization problems. However, the methods for their solving are developed only for the most popular matrix equations – Riccati and Lyapunov equations, and there is no universal approach for solving problems of this class. This paper summarizes the previously considered method of solving systems of algebraic equations over a field of real numbers [1] and proposes a scheme for systems of polynomial matrix equations of the second degree with many unknowns. A recurrent formula for fractionalization a solution into a continued matrix fraction is also given. The convergence of the proposed method is investigated. The results of numerical experiments that confirm the validity of theoretical calculations and the effectiveness of the proposed scheme are presented.

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17

Wampler, C. W., A. P. Morgan, and A. J. Sommese. "Numerical Continuation Methods for Solving Polynomial Systems Arising in Kinematics." Journal of Mechanical Design 112, no. 1 (1990): 59–68. http://dx.doi.org/10.1115/1.2912579.

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Many problems in mechanism design and theoretical kinematics can be formulated as systems of polynomial equations. Recent developments in numerical continuation have led to algorithms that compute all solutions to polynomial systems of moderate size. Despite the immediate relevance of these methods, they are unfamiliar to most kinematicians. This paper attempts to bridge that gap by presenting a tutorial on the main ideas of polynomial continuation along with a section surveying advanced techniques. A seven position Burmester problem serves to illustrate the basic material and the inverse position problem for general six-axis manipulators shows the usefulness of the advanced techniques.

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18

Woba, Moumouni Djassibo. "Solving Some Problems and Elimination in Systems of Polynomial Equations." American Journal of Computational Mathematics 14, no. 03 (2024): 333–45. http://dx.doi.org/10.4236/ajcm.2024.143016.

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19

Bates, Daniel J., Andrew J. Newell, and Matthew Niemerg. "BertiniLab: A MATLAB interface for solving systems of polynomial equations." Numerical Algorithms 71, no. 1 (2015): 229–44. http://dx.doi.org/10.1007/s11075-015-0014-6.

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20

Anil Kumar, Sachin Kumar. "New Operational Matrix Via Gnocchi Polynomial for Solving Non-Linear Fractional Differential Equations." Communications on Applied Nonlinear Analysis 32, no. 9s (2025): 2969–81. https://doi.org/10.52783/cana.v32.4595.

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Fractional differential equations (FDEs) have emerged as essential tools in modeling complex dynamical systems exhibiting memory and hereditary properties. Traditional operational matrices arising from Legendre, Chebyshev, and Jacobi polynomials are generally known to be numerically unstable, computationally expensive, and inefficient in approximating fractional operators. In this study an operational matrix based on Gnocchi polynomial is introduced for solving non linear fractional differential equations (NFDE) with better sparsity, stability and computational efficiency. The proposed method transforms NFDEs into tractable algebraic systems by constructing a fractional differentiation operational matrix using Gnocchi polynomials. The method is validated by theoretical formulations, spectral convergence analysis, error estimation proofs. It is also compared with existing polynomial based approaches to demonstrate better performance in function approximation and numerical stability. The Gnocchi operational matrix is based on Gnocchi, and it achieves exponential convergence, reduced computational complexity and increased numerical robustness compared to classical techniques. It is effective in fractional modeling because it can accurately approximate non-linear fractional operators. The author develops a mathematically rigorous, computationally efficient framework to solve NFDEs. Further improvements will be done by other researchers in the future for higher dimension applications, for adaptive techniques in the spectral method and for hybrid AI assisted optimization.

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21

Yannakoudakis, Aristotle G. "Full Static Output Feedback Equivalence." Journal of Control Science and Engineering 2013 (2013): 1–17. http://dx.doi.org/10.1155/2013/491709.

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We present a constructive solution to the problem of full output feedback equivalence, of linear, minimal, time-invariant systems. The equivalence relation on the set of systems is transformed to another on the set of invertible block Bezout/Hankel matrices using the isotropy subgroups of the full state feedback group and the full output injection group. The transformation achieving equivalence is calculated solving linear systems of equations. We give a polynomial version of the results proving that two systems are full output feedback equivalent, if and only if they have the same family of generalized Bezoutians. We present a new set of output feedback invariant polynomials that generalize the breakaway polynomial of scalar systems.

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22

Khakimova, Zilya Nailevna, Larisa Nikolaevna Timofeeva, and Ajkanush Ashotovna Atojan. "Applying a Power Transformation to the Orbit of the 2nd Painleve Equation and Solving Differential Equations with Polynomial Right-hand Sides Via the 2nd Painleve Transcendent and in Polynomials." Differential Equations and Control Processes, no. 4 (2023): 142–54. http://dx.doi.org/10.21638/11701/spbu35.2023.407.

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The 2nd Painleve equation is considered as a representative of the second-order class of ordinary differential equations (ODEs) with polynomial right-hand sides, as well as of the more general second-order class of equations with fractional polynomial right-hand sides. The second PainlevГ© equation with three terms on the right side has an orbit in the class of fractional polynomial equations with respect to the pseudogroup of the 36th order, and in the absence of the 3rd term вЂ“ the 60th order. This paper presents a power transformation with an arbitrary parameter that preserves the polynomial or fractional polynomial form of the equations. This power-law transformation is applied to the orbital equations of the 2nd PainlevГ© equation with three and two terms on the right-hand sides of the equations. Pseudogroups of transformations induced by the above-mentioned pseudogroups of the 36th and 60th orders are constructed. All equations with one-constant arbitrariness corresponding to the vertices of the graphs of induced pseudogroups are found. General solutions of all found equations are obtained through the 2nd PainlevГ© transcendental or in polynomials. A theorem is presented that allows, using the scaling operation, to find general solutions to all the above equations with arbitrary coefficients.

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23

Barotov, Dostonjon N., and Ruziboy N. Barotov. "Construction of smooth convex extensions of Boolean functions." Russian Universities Reports. Mathematics, no. 145 (2024): 20–28. http://dx.doi.org/10.20310/2686-9667-2024-29-145-20-28.

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Systems of Boolean equations are widely used in mathematics, computer science, and applied sciences. In this regard, on the one hand, new research methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of these methods is that, firstly, the system of Boolean equations given over the ring of Boolean polynomials is transformed into a system of equations over the field of real numbers, and secondly, the transformed system is reduced either to the problem of numerical minimization of the corresponding objective function, to a MILP or QUBO problem, to a system of polynomial equations solved on the set of integers, or to an equivalent system of polynomial equations solved by symbolic methods. There are many ways to transform a system of Boolean equations into a continuous minimization problem, since the fundamental difference between such methods and “brute force” local search algorithms is that at each iteration of the algorithm, the shift along the antigradient is performed on all variables simultaneously. But one of the main problems that arise when applying these methods is that the objective function to be minimized in the desired area can have many local minima, which greatly complicates their practical use. In this paper, a non-negative convex and continuously differentiable extension of any Boolean function is constructed, which is applied to solving an arbitrary system of Boolean equations. It is argued that the problem of solving an arbitrary system of Boolean equations can be constructively reduced to the problem of minimizing a function, any local minimum of which in the desired domain is a global minimum.

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24

Zajac, Pavol. "MRHS Equation Systems that can be Solved in Polynomial Time." Tatra Mountains Mathematical Publications 67, no. 1 (2016): 205–19. http://dx.doi.org/10.1515/tmmp-2016-0040.

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Abstract In this article we study the difficulty of solving Multiple Right-Hand Side (MRHS) equation systems. In the first part we show that, in general, solving MRHS systems is NP-hard. In the next part we focus on special (large) families of MRHS systems that can be solved in polynomial time with two algorithms: one based on linearisation of MRHS equations, and the second one based on decoding problems that can be solved in polynomial time.

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25

Il’ev, A. V., and V. P. Il’ev. "ALGORITHMS FOR SOLVING SYSTEMS OF EQUATIONS OVER VARIOUS CLASSES OF FINITE GRAPHS." Prikladnaya Diskretnaya Matematika, no. 53 (2021): 89–102. http://dx.doi.org/10.17223/20710410/53/6.

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The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language L consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations S with k variables over an arbitrary simple n-vertex graph Γ is N P-complete. The computational complexity of the procedure for checking compatibility of a system of equations S over a simple graph Γ and the procedure for finding a general solution of this system is calculated. The computational complexity of the algorithm for solving a system of equations S with k variables over an arbitrary simple n-vertex graph Γ involving these procedures is O(k 2n k/2+1(k + n) 2 ) for n > 3. Polynomially solvable cases are distinguished: systems of equations over trees, forests, bipartite and complete bipartite graphs. Polynomial time algorithms for solving these systems with running time O(k 2n(k + n) 2 ) are proposed.

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26

GERDT, VLADIMIR P. "COMPUTER ALGEBRA, SYMMETRY ANALYSIS AND INTEGRABILITY OF NONLINEAR EVOLUTION EQUATIONS." International Journal of Modern Physics C 04, no. 02 (1993): 279–86. http://dx.doi.org/10.1142/s012918319300029x.

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A computer algebra-aided symmetry approach to investigating integrability of polynomial-nonlinear evolution equations in one-temporal and one-spatial dimensions is presented. The approach is based on verifying the existence of higher conservation laws and symmetries. If the equations contain arbitrary numerical parameters, the problem of selection of all the integrable cases is reduced to the solving polynomial equations in those parameters. The Gröbner basis technique is used in order to simplify and to solve such polynomial systems which typically have infinitely many solutions.

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27

Nigim, K. A., M. M. A. Salama, and M. Kazerani. "Solving Polynomial Algebraic Equations of the Stand Alone Induction Generator." International Journal of Electrical Engineering & Education 40, no. 1 (2003): 45–54. http://dx.doi.org/10.7227/ijeee.40.1.5.

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This paper describes the use of MathCAD's solving block which uses ‘Given’ and ‘Find’ built-in functions to solve nth order nonlinear algebraic equations. Introducing complex energy systems to electrical engineering students in their undergraduate studies is essential to complement many energy conversion courses. Various electric energy-capturing schemes use electric equivalent circuit models that incorporate nonlinear elements with complex mathematical formulas requiring numerical computation. Without incorporating programming tools, the taught material could be vague and a burden for both the student and the lecturer, hindering comprehension of the complexity of the system during the limited lecture hours. This paper introduces a ‘ready to use’ computational and mathematical tool that can be used to solve the non-linear equations quickly. The performance of an energy scheme with non-linear, interrelated variables such as the self-excited induction generator (SEIG) under variable excitation and loading conditions is used as an example. SEIG systems have been intensively proposed for energy capturing to supply power to remote areas from renewable energy resources such as wind and hydro prime mover systems.

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28

Malaschonok, Natasha. "Solving Differential Equations by Parallel Laplace Method with Assured Accuracy." Serdica Journal of Computing 1, no. 4 (2007): 387–402. http://dx.doi.org/10.55630/sjc.2007.1.387-402.

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We produce a parallel algorithm realizing the Laplace transform method for the symbolic solving of differential equations. In this paper we consider systems of ordinary linear differential equations with constant coefficients, nonzero initial conditions and right-hand parts reduced to sums of exponents with polynomial coefficients.

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29

Pichuev, K. D., and A. N. Rybalov. "On the complexity of solving of equations in the bicyclic monoid." Herald of Omsk University 29, no. 1 (2024): 8–17. http://dx.doi.org/10.24147/1812-3996.2024.1.8-17.

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In this article we prove that the problem of solvability of systems of equations over the bicyclic a monoid is NP-hard. On the other hand, we prove polynomial decidability of this problem for some natural class of equations in one variable.

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30

Lichtblau, Daniel. "Approximate Gröbner Bases, Overdetermined Polynomial Systems, and Approximate GCDs." ISRN Computational Mathematics 2013 (March 21, 2013): 1–12. http://dx.doi.org/10.1155/2013/352806.

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We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to absolute and relative error from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we show handling of approximate polynomial GCD computations, using benchmarks from the literature.

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31

Parkinson, Suzanna, Hayden Ringer, Kate Wall, et al. "Analysis of normal-form algorithms for solving systems of polynomial equations." Journal of Computational and Applied Mathematics 411 (September 2022): 114235. http://dx.doi.org/10.1016/j.cam.2022.114235.

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32

Chen, Tianran, and Tien-Yien Li. "Homotopy continuation method for solving systems of nonlinear and polynomial equations." Communications in Information and Systems 15, no. 2 (2015): 119–307. http://dx.doi.org/10.4310/cis.2015.v15.n2.a1.

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33

Romeuf, Jean-François. "A polynomial algorithm for solving systems of two linear diophantine equations." Theoretical Computer Science 74, no. 3 (1990): 329–40. http://dx.doi.org/10.1016/0304-3975(90)90082-s.

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34

Hakk, K. "TWO-GRID ITERATION METHOD FOR WEAKLY SINGULAR INTEGRAL EQUATIONS." Mathematical Modelling and Analysis 5, no. 1 (2000): 76–85. http://dx.doi.org/10.3846/13926292.2000.9637130.

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For the solution of weakly singular integral equations by the piecewise polynomial collocation method it is necessary to solve large linear systems. In the present paper a two‐grid iteration method for solving such systems is constructed and the convergence of this method is investigated.

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35

Rybalov, A. N. "On the generic complexity of solving equations over natural numbers with addition." Prikladnaya Diskretnaya Matematika, no. 64 (2024): 72–78. http://dx.doi.org/10.17223/20710410/64/6.

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We study the general complexity of the problem of determining the solvability of equations systems over natural numbers with the addition. The NP-completeness of this problem is proved. A polynomial generic algorithm for solving this problem is proposed. It is proved that if P ̸= NP and P = BPP, then for the problem of checking the solvability of systems of equations over natural numbers with zero there is no strongly generic polynomial algorithm. For a strongly generic polynomial algorithm, there is no efficient method for random generation of inputs on which the algorithm cannot solve the problem. To prove this theorem, we use the method of generic amplification, which allows us to construct generically hard problems from problems that are hard in the classical sense. The main feature of this method is the cloning technique, which combines the input data of a problem into sufficiently large sets of equivalent input data. Equivalence is understood in the sense that the problem is solved similarly for them.

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36

Bataineh, Ahmad Sami, Osman Rasit Isik, Moa’ath Oqielat, and Ishak Hashim. "An Enhanced Adaptive Bernstein Collocation Method for Solving Systems of ODEs." Mathematics 9, no. 4 (2021): 425. http://dx.doi.org/10.3390/math9040425.

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In this paper, we introduce two new methods to solve systems of ordinary differential equations. The first method is constituted of the generalized Bernstein functions, which are obtained by Bernstein polynomials, and operational matrix of differentiation with collocation method. The second method depends on tau method, the generalized Bernstein functions and operational matrix of differentiation. These methods produce a series which is obtained by non-polynomial functions set. We give the standard Bernstein polynomials to explain the generalizations for both methods. By applying the residual correction procedure to the methods, one can estimate the absolute errors for both methods and may obtain more accurate results. We apply the methods to some test examples including linear system, non-homogeneous linear system, nonlinear stiff systems, non-homogeneous nonlinear system and chaotic Genesio system. The numerical shows that the methods are efficient and work well. Increasing m yields a decrease on the errors for all methods. One can estimate the errors by using the residual correction procedure.

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37

Vorontsov, Oleg, Valeriy Usenko, and Iryna Vorontsova. "DETERMINATION OF SUPERPOSITION COEFFICIENTS FOR DISCRETE FORMATION OF POLYNOMIAL FUNCTIONAL DEPENDENCIES." APPLIED GEOMETRY AND ENGINEERING GRAPHICS, no. 107 (February 26, 2025): 42–53. https://doi.org/10.32347/0131-579x.2024.107.42-53.

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The shape control of a discretely represented curve (DRC) in the static-geometric method can be achieved not only by varying the functional external load but also through the coefficients in computational templates. These templates form the basis for constructing systems of finite-difference equations for DRC formation and indicate the proportional contribution of adjacent nodes to the desired formation. This article proposes a general approach to creating computational templates for modeling geometric objects (GOs) using superpositions of point sets. This aims to further study the influence of superposition coefficients, both arbitrary and of adjacent nodes of numerical sequences, on the formation of discrete analogs of elementary functional dependencies. One of the objectives of this study is to continue exploring the modeling of discrete geometric objects (DGOs) based on the classical finite difference method, the static-geometric method, and the geometric apparatus of superpositions. Since any polynomial of degree n is defined by n+1 points, determining the ordinate of any point given its abscissa requires substituting the coordinates of n+1 points into the polynomial function equation. This results in a system of algebraic equations containing n+1 equations and n+1 variables. Solving this system yields the polynomial coefficients a0 , a1 , a2 , a3 , … , an are found. In contrast to this approach, the recursive formula and the formula for determining the superposition coefficients proposed in this study allow for calculating the ordinate of any point of a polynomial of degree n given its abscissa without constructing and solving a system of n+1 equations. The ordinate of any curve point is determined as a superposition of the ordinates of n+1 points. In the proposed method of geometric curve modeling, the superposition coefficients are derived from systems of equations that contain one equation fewer than those used to calculate polynomial coefficients . Computational templates have been developed for the discrete formation of polynomial functional dependencies using superpositions of adjacent points’ coordinates. The approach presented in the article can be used to obtain expressions similar to formula (3) for calculating superposition coefficients for adjacent points of polynomials with two variables. Varying the superposition coefficients in the developed computational templates allows for studying the impact of these coefficients, both arbitrary and for adjacent nodes of numerical sequences, on the formation of discrete analogs of elementary functional dependencies. The shape control of a discretely represented curve (DRC) in the static-geometric method can be achieved not only by varying the functional external load but also through the coefficients in computational templates. These templates form the basis for constructing systems of finite-difference equations for DRC formation and indicate the proportional contribution of adjacent nodes to the desired formation. This article proposes a general approach to creating computational templates for modeling geometric objects (GOs) using superpositions of point sets. This aims to further study the influence of superposition coefficients, both arbitrary and of adjacent nodes of numerical sequences, on the formation of discrete analogs of elementary functional dependencies. One of the objectives of this study is to continue exploring the modeling of discrete geometric objects (DGOs) based on the classical finite difference method, the static-geometric method, and the geometric apparatus of superpositions. Since any polynomial of degree n is defined by n+1 points, determining the ordinate of any point given its abscissa requires substituting the coordinates of n+1 points into the polynomial function equation. This results in a system of algebraic equations containing n+1 equations and n+1 variables. Solving this system yields the polynomial coefficients a0 , a1 , a2 , a3 , … , an are found. In contrast to this approach, the recursive formula and the formula for determining the superposition coefficients proposed in this study allow for calculating the ordinate of any point of a polynomial of degree n given its abscissa without constructing and solving a system of n+1 equations. The ordinate of any curve point is determined as a superposition of the ordinates of n+1 points. In the proposed method of geometric curve modeling, the superposition coefficients are derived from systems of equations that contain one equation fewer than those used to calculate polynomial coefficients . Computational templates have been developed for the discrete formation of polynomial functional dependencies using superpositions of adjacent points’ coordinates. The approach presented in the article can be used to obtain expressions similar to formula (3) for calculating superposition coefficients for adjacent points of polynomials with two variables. Varying the superposition coefficients in the developed computational templates allows for studying the impact of these coefficients, both arbitrary and for adjacent nodes of numerical sequences, on the formation of discrete analogs of elementary functional dependencies.

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38

Nigay, Ruslan, Evgeny Nigay, and Lubov Mironova. "Investigation of rigid dynamic systems on the example of modelling a tape drive mechanism." MATEC Web of Conferences 329 (2020): 03003. http://dx.doi.org/10.1051/matecconf/202032903003.

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The features of modeling the dynamics of mechanical systems on the example of the operation of the tape drive mechanism related to real technological processes are stated. An approach to solving stiff systems of differential equations by the numerical-analytical method is noted. The approach is based on solving systems of higher-order differential equations using elementary functions using procedures for precision search for the roots of the characteristic polynomial of the system. A mathematical model of the tape drive mechanism of a VCR is given as an example of a precision electromechanical object.

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39

Ajileye Ganiyu, Richard Taparki, Ojo Olamiposi Aduroja, and Rahimat Oziohu Onsachi. "Volterra Integral Equations: A Numerical Solution Method Using Shifted Chebyshev Polynomial." International Journal of Latest Technology in Engineering Management & Applied Science 14, no. 4 (2025): 940–44. https://doi.org/10.51583/ijltemas.2025.140400114.

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Abstract: This study presents a numerical method for solving Volterra integral equations of the second kind using shifted Chebyshev polynomials. Volterra integral equations arise in various scientific and engineering applications, including population dynamics, physics, and control systems. Due to their complexity, obtaining analytical solutions is often challenging, making numerical techniques crucial. We employ shifted Chebyshev polynomials as basis functions to approximate the solution, transforming the integral equation into a system of algebraic equations. The shifted Chebyshev polynomials offer excellent approximation properties, improving convergence rates and accuracy. The proposed method is analyzed for stability and efficiency, and numerical experiments demonstrate its effectiveness in solving different classes of Volterra integral equations. The results highlight the advantages of the approach compared to traditional numerical methods.

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40

Kashpur, O. F. "Conditions for the solvability of nonlinear equations systems in Euclidean spaces." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1 (2021): 74–80. http://dx.doi.org/10.17721/1812-5409.2021/1.9.

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The solution of many applied problems is to ﬁnd a solution of nonlinear equations systems in ﬁnite- dimensional Euclidean spaces. The problem of ﬁnding the solution of a nonlinear system is divided into two problems: 1. The existence of a solution of a nonlinear equations system; in the case of nonunique of the solution, it is necessary to ﬁnd the number of these solutions and their surroundings. 2. Finding the solution of a system of nonlinear equations with a given accuracy. Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence and estimates of the solution accuracy. In contrast to problem 2, for problem 1 there is no general algorithm for solving this task, there are no constructive conditions for the existence of a solution of a nonlinear equations system in Euclidean spaces. In this article, in ﬁnite-dimensional Euclidean spaces, the constructive conditions for the existence of a solution of nonlinear systems of polynomial form are found. The connection of these conditions with the linear polynomial interpolant of the minimum norm, generated by a scalar product with Gaussian measure and the conditions of its existence, is given.

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Salim, S. H., R. K. Saeed, and K. H. F. Jwamer. "Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods." Bulletin of the Karaganda University-Mathematics 117, no. 1 (2025): 155–69. https://doi.org/10.31489/2025m1/155-169.

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In this paper, a method that utilizes a non-polynomial spline function based on the weighted residual technique to approximate solutions for linear Volterra-Fredholm integral equations is presented. The approach begins with the selection of a series of knots along the integration interval. We then create a set of basis functions, defined as non-polynomial spline functions, between each pair of adjacent knots. The unknown function is expressed as a linear combination of these basis functions to approximate the solution of integral equations. The coefficients of the spline function are calculated by solving a system of linear equations derived from substituting the spline approximation into the integral equation while maintaining continuity and smoothness at the knots. Non-polynomial splines are beneficial for approximating functions with complex shapes and for solving integral equations with non-smooth kernels. However, the solution’s accuracy significantly relies on the selection of knots, and the method may require extensive computational resources for large systems. To illustrate the effectiveness of the method, three examples are presented, implemented using Python version 3.9. The paper also addresses the error analysis theorem relevant to the proposed non-polynomial spline function.

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42

GOULIANAS, K., A. MARGARIS, I. REFANIDIS, and K. DIAMANTARAS. "Solving polynomial systems using a fast adaptive back propagation-type neural network algorithm." European Journal of Applied Mathematics 29, no. 2 (2017): 301–37. http://dx.doi.org/10.1017/s0956792517000146.

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This paper proposes a neural network architecture for solving systems of non-linear equations. A back propagation algorithm is applied to solve the problem, using an adaptive learning rate procedure, based on the minimization of the mean squared error function defined by the system, as well as the network activation function, which can be linear or non-linear. The results obtained are compared with some of the standard global optimization techniques that are used for solving non-linear equations systems. The method was tested with some well-known and difficult applications (such as Gauss–Legendre 2-point formula for numerical integration, chemical equilibrium application, kinematic application, neuropsychology application, combustion application and interval arithmetic benchmark) in order to evaluate the performance of the new approach. Empirical results reveal that the proposed method is characterized by fast convergence and is able to deal with high-dimensional equations systems.

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43

Sommese, Andrew J., Jan Verschelde, and Charles W. Wampler. "Advances in Polynomial Continuation for Solving Problems in Kinematics." Journal of Mechanical Design 126, no. 2 (2004): 262–68. http://dx.doi.org/10.1115/1.1649965.

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For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higher-dimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.

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44

Chistov, A. L. "Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations: 33 Years Later. III." Journal of Mathematical Sciences 247, no. 5 (2020): 738–57. http://dx.doi.org/10.1007/s10958-020-04836-8.

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45

Chistov, A. L. "Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations: 33 Years Later. I." Journal of Mathematical Sciences 232, no. 2 (2018): 177–203. http://dx.doi.org/10.1007/s10958-018-3868-z.

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Chistov, A. L. "Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations 33 Years Later. II." Journal of Mathematical Sciences 240, no. 5 (2019): 594–616. http://dx.doi.org/10.1007/s10958-019-04378-8.

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47

Le, Huu Phuoc, and Mohab Safey El Din. "Solving parametric systems of polynomial equations over the reals through Hermite matrices." Journal of Symbolic Computation 112 (September 2022): 25–61. http://dx.doi.org/10.1016/j.jsc.2021.12.002.

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Ayad, Ali, Ali Fares, and Youssef Ayyad. "An algorithm for solving zero-dimensional parametric systems of polynomial homogeneous equations." Journal of Nonlinear Sciences and Applications 05, no. 06 (2012): 426–38. http://dx.doi.org/10.22436/jnsa.005.06.03.

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Wolf, T. "On solving large systems of polynomial equations appearing in discrete differential geometry." Programming and Computer Software 34, no. 2 (2008): 75–83. http://dx.doi.org/10.1134/s0361768808020047.

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50

Buchanan, S. Alasdair. "Some theoretical problems when solving systems of polynomial equations using Gröbner bases." ACM SIGSAM Bulletin 25, no. 2 (1991): 24–27. http://dx.doi.org/10.1145/122520.122523.

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