Groebner Basis Methods Research Articles

Solve the Polynomial Functions Conditional Extreme by Applying the Groebner Basis Method

In this paper, an algebraic method which is based on the groebner bases theory is proposed to solve the polynomial functions conditional extreme. Firstly, we describe how to solve conditional extreme value problems by establishing Lagrange functions and calculating the differential equations derived from the Lagrange functions. Then, by solving the single variable polynomials in the groebner basis, the solution of polynomial equations could be derived successively. We overcome the high number of variables and constraints in the extreme value problem. Finally, this paper illustrates the calculation process of this method through the general procedures and examples in solving questions of conditional extremum of polynomial function.

Solution region synthesis methodology of spatial 1CS-4SS linkages for six given positions

This paper proposes a 1CS-4SS spatial linkage and presents the solution region synthesis methodology for six specified positions. Firstly, synthesis equations of the SS dyads and the CS dyad are proposed, the discrete solutions of the CS dyad are obtained by Bertini numerical method. The 1CS-4SS spatial linkage are obtained by selecting four SS dyads on the solution curves of the SS dyad and one CS dyad on the discrete solutions of the CS dyad. We display all solutions of linkages in a plane, which is called the solution region. Secondly, The Jacobian matrix is deduced to determinate the defective linkages. The Bertini numerical method and Groebner basis method are used for position analysis of the linkages, and the analysis results from the two methods are compared and mutually verified. The feasible solution region can be obtained after eliminating defective linkages and linkages that fail to satisfy the other requirements from the solution region plane. Finally, the validity of the formulas and applicability of the proposed approach are illustrated by an example.

Satisfying positivity requirement in the Beyond Complex Langevin approach

The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case.

A degree-increasing [N to N+1] homotopy for Chebyshev and Fourier spectral methods

Hitherto, as a tool for tracing all branches of nonlinear differential equations, resolution-increasing homotopy methods have been applied only to finite difference discretizations. However, spectral Galerkin algorithms typically match the error of fourth order differences with one-half to one-fifth the number of degrees of freedom N in one dimension, and a factor of eight to a hundred and twenty-five in three dimensions. Let u → N be the vector of spectral coefficients and R → N the vector of N Galerkin constraints. A common two-part procedure is to first find all roots of R → N ( u → N ) = 0 → using resultants, Groebner basis methods or block matrix companion matrices. (These methods are slow and ill-conditioned, practical only for small N .) The second part is to then apply resolution-increasing continuation. Because the number of solutions is an exponential function of N , spectral methods are exponentially superior to finite differences in this context. Unfortunately, u → N is all too often outside the domain of convergence of Newton’s iteration when N is increased to ( N + 1 ) . We show that a good option is the artificial parameter homotopy H → ( u → ; τ ) ≡ R → N + 1 ( u → ) − ( 1 − τ ) R → N + 1 ( u → N ) , τ ∈ [ 0 , 1 ] . Marching in small steps in τ , we proceed smoothly from the N -term to the N + 1 -term approximations.

Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares

In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of $H_\infty$ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.

Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases

This paper is devoted to the study of vibration of mechanical systems with geometric nonlinearities. The harmonic balance method is used to derive systems of polynomial equations whose solutions give the frequency component of the possible steady states. Groebner basis methods are used for computing all solutions of polynomial systems. This approach allows to reduce the complete system to an unique polynomial equation in one variable driving all solutions of the problem. In addition, in order to decrease the number of variables, we propose to first work on the undamped system, and recover solution of the damped system using a continuation on the damping parameter. The search for multiple solutions is illustrated on a simple system, where the influence of the retained number of harmonic is studied. Finally, the procedure is applied on a simple cyclic system and we give a representation of the multiple states versus frequency.

Groebner Basis Methods for Stationary Solutions of a Low-Dimensional Model for a Shear Flow

We use Groebner basis methods to extract all stationary solutions for the nine-mode shear flow model described in Moehlis et al. (New J Phys 6:56, 2004). Using rational approximations to irrational wave numbers and algebraic manipulation techniques we reduce the problem of determining all stationary states to finding roots of a polynomial of order 30. The coefficients differ by 30 powers of 10, so that algorithms for extended precision are needed to extract the roots reliably. We find that there are eight stationary solutions consisting of two distinct states, each of which appears in four symmetry-related phases. We discuss extensions of these results for other flows.

Evaluation of Groebner Basis Methodology as an Aid to Harmonic Balance

A preliminary exploration is carried out of the utility of the Groebner basis method for implementing the harmonic balance analysis of nonlinear steady state vibrations. The method is found to be worthy of further investigation.

On a Finiteness Result for the Number of Critical Points of the H2 Approximation Criterion

The long-standing open problem about whether the number of critical points in the H2 SISO real model order reduction problem is finite is answered in the positive in the case the transfer function of the to-be-reduced model has distinct poles (ie. only has poles of algebraic multiplicity one). This has important implications for various search methods for finding critical points or local minima of the criterion function for this model reduction problem. In fact more is shown namely that in a particular parametrization the number of complex solutions of the first order conditions of the H2 real model order reduction problem is finite and lies in a bounded set of which the bound can be determined from information about the to-be-reduced model. This implies that the H2 model order reduction problem can be solved in principle by constructive algebra methods (such as Groebner basis methods) in case the to-be-reduced model has distinct poles. It simplifies the methods for co-order three reduction as described in a companion paper.

Dynamics of Coherent Structures in the Coupled Complex Ginzburg-Landau Equations

Problem statement: In this study, we study the analytical construction of some exact solutions of a system of coupled physical differential equations, namely, the Complex Ginzburg-Landau Equations (CGLEs). CGLEs are intensively studied models of pattern formation in nonlinear dissipative media, with applications to biology, hydrodynamics, nonlinear optics, plasma physics, reaction-diffusion systems and many other fields. Approach: A system of two coupled CGLEs modeling the propagation of pulses under the combined influence of dispersion, self and cross phase modulations, linear and nonlinear gain and loss will be discussed. A Solitary Pulse (SP) is a localized wave form and a front (also termed as shock) refers to a transition connecting two constant, but unequal, asymptotic states. A SP-front pair solution can be analytically obtained by the modified Hirota bilinear method. Results: These wave solutions are deduced by a system of six nonlinear algebraic equations, allowing the amplitudes, wave-numbers, frequency and velocities to be determined. Conclusion: The final exact solution can then be computed by applying the Groebner basis method with a large amount of algebraic simplifications done by the computer software Maple.

A geometric view of cryptographic equation solving

This paper considers the geometric properties of the Relinearisation algorithm and of the XL algorithm used in cryptology for equation solving. We give a formal description of each algorithm in terms of projective geometry, making particular use of the Veronese variety. We establish the fundamental geometrical connection between the two algorithms and show how both algorithms can be viewed as being equivalent to the problem of finding a matrix of low rank in the linear span of a collection of matrices, a problem sometimes known as the MinRank problem. Furthermore, we generalise the XL algorithm to a geometrically invariant algorithm, which we term the GeometricXL algorithm. The GeometricXL algorithm is a technique which can solve certain equation systems that are not easily soluble by the XL algorithm or by Groebner basis methods.

Groebner basis methods for multichannel sampling with unknown offsets

In multichannel sampling, several sets of sub-Nyquist sampled signal values are acquired. The offsets between the sets are unknown, and have to be resolved, just like the parameters of the signal itself. This problem is nonlinear in the offsets, but linear in the signal parameters. We show that when the basis functions for the signal space are related to polynomials, we can express the joint offset and signal parameter estimation as a set of polynomial equations. This is the case for example with polynomial signals or Fourier series. The unknown offsets and signal parameters can be computed exactly from such a set of polynomials using Gröbner bases and Buchberger's algorithm. This solution method is developed in detail after a short and tutorial overview of Gröbner basis methods. We then address the case of noisy samples, and consider the computational complexity, exploring simplifications due to the special structure of the problem.

Normal Forms for Coupled Takens-Bogdanov Systems

The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and is best described by giving a Stanley decomposition of that module. In this paper Groebner basis methods are used to determine a Groebner basis for the ideal of relations and a Stanley decomposition for the ring of invariants that arise in normal forms for Takens-Bogdanov systems. An algorithm developed by Murdock, is then used to produce a Stanley decomposition for the (normal form module) module of the equivariants from the Stanley decomposition for the ring of invariants.

Recursive Computation of Free Resolutions and a Generalized Koszul Complex

This article describes a new algorithm to compute a free resolution of an ideal of (or module over) a commutative ring R combining the Koszul complex with Groebner basis methods. The algorithm computes the resolution of an ideal I via the resolutions of a sequence of subideals \({{(0)=I_0 \subset\ldots\subset I_r = I}}\) differing by one generator each time. The article discusses special orderings and criteria applicable to this algorithm and gives some timings based on an implementation within the computer algebra system Singular.

Discrete versions of continuous isoperimetric problems

Discrete isoperimetric variational problems which model single and double integral isoperimetric problems are formulated and some multiplier rules are derived. For quardratic functionals, the Euler-Lagrange equation is linear and can be analyzed bydeterminant methods. difference equations methods. or numerically bythealgebraic eigenvalue problem. A specific example is given wherethe eigenfuntions and eigenvalue of this discrete problem converge. as the step size goes to zero, to the eigenfunctionsand eigenvalues of the corresponding, continuous problem Recent developments in algebraic geometry offer the hopeof using Groebner (=Gröbner or Grobner) basis methods to numerically solve some systems of equationsgenerated by Lagrange multiplier methods. These methods mayapply to nonlinear systems of equations whenever they can be reduced to systems of polynomial equations. The Groebner basis algorithm of Buchberger is an extension of Gaussian elimination to polynomial systems.

Constructive Algebra Methods for the L2-Problem For Stable Linear Systems

We investigate the feasibility of using computer algebra for solving L2 system approximation problems. The first-order optimally conditions yield a set of polynomial equations which can, in principle, be solved using Groebner basis methods. A general solution along these lines would be tremendously useful, although it does not appear to be feasible at present except for rather low McMillan degrees. We demonstrate that it can be feasible for specific examples; in such cases global optima can be found reliably. We show that the use of a Schwarz-like canonical form simplifies the structure of the problem, and the required computations.