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In this paper we extend complex homotopy methods to finding witness points on the irreducible components of real varieties. In particular we construct such witness points as the isolated real solutions of a constrained optimization problem.
First a random hyperplane characterized by its random normal vector is chosen. Witness points are computed by...
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Fractional order integrodifferential equations cannot be directly solved like ordinary differential equations. Numerical methods for such equations have additional algorithmic complexities. We present a particularly simple recipe for solving such equations using a Galerkin scheme developed in prior work. In particular, matrices needed for that meth...
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... For the real case, the methods in [56,57] yield real witness points as critical points of the distance from a random hyperplane to the real variety. Alternatively, the real witness points can be considered as critical points of the distance from a random point to the real variety [19]. ...
... When f satisfies the regularity assumptions in [56], all the real solutions of g = 0 can be obtained by the homotopy continuation method. These points are called real However, if f does not satisfy the regularity assumptions due to high multiplicity or a non-real radical ideal, then we apply a critical point approach [55] based on a penalty factor. ...
To find consistent initial data points for a system of differential-algebraic equations, requires the identification of its missing constraints. An efficient class of structural methods exploiting a dependency graph for this task was initiated by Pantiledes. More complete methods rely on differential-algebraic geometry but suffer from other issues (e.g. high complexity). In this paper we give a new class of efficient structural methods combined with new tools from numerical real algebraic geometry that has much improved completeness properties. Existing structural methods may fail for a system of differential-algebraic equations if its Jacobian matrix after differentiation is still singular due to symbolic cancellation or numerical degeneration. Existing structural methods can only handle degenerated cases caused by symbolic cancellation. However, if a system has parameters, then its parametric Jacobian matrix may be still singular after application of the structural method for certain values of the parameters. This case is called numerical degeneration. For polynomially nonlinear systems of differential-algebraic equations, numerical methods are given to solve both degenerated cases using numerical real algebraic geometry. First, we introduce a witness point method, which produces at least one witness point on every constraint component. This can help to ensure constant rank and detection of degeneration on all components of such systems. Secondly, we present a Constant Rank Embedding Lemma, and based on it propose an Index Reduction by Embedding (IRE) method which can construct an equivalent system with a full rank Jacobian matrix. Thirdly, IRE leads to a global structural differentiation method, to solve degenerated differential-algebraic equations on all components numerically. Application examples from circuits, mechanics, are used to demonstrate our method.
... In many applications, it suffices to computing an ϵ-approximation of the curve. In practice, an ϵ-approximation of the curve can be obtained by a sufficiently small perturbation of the constant term of P , denoted by P ϵ [49]. Moreover, this perturbed curve is smooth by Sard's theorem. ...
... There also exist robust homotopy continuation methods, like [6], to get certified points. For nonsmooth D, one can apply a symbolic approach [40,51,9] to compute the isolated singular points, or perturb D to obtain well-conditioned points [49,50], or employ a penalty function method to handle rankdeficiency [48], or compute "pseudo-singular points" instead [18]. Once the initial witness points are obtained, a polygonal chain approximation of the curve Γ ∩ B can be obtained by numerical continuation [18]. ...
We present an efficient geometric approach for computing the steady states of biparametric biological systems modeled by autonomous ordinary differential equations by taking advantage of their potential
block triangular structures. While the parametric steady states of a given system are initially described by an implicit real algebraic surface in high-dimensional space, this approach computes a border curve, approximated by polygonal chains, which separates a bounded box of the parametric space into finitely many open cells such that the parametric steady states are continuous functions of the parameters with disjoint graphs above each cell. A block triangular structure of the system is discovered by combining both Tarjan's algorithm for computing strongly connected components of a graph defined for the system and Gauss--Jordan elimination. This particular structure enables one to greatly reduce the size of polynomial systems defining border curves. The effectiveness of the approach is demonstrated by analyzing two biological systems with, respectively, 17 and 20
unknowns.
... Instead the method in [39,40] yields real witness points as critical points of the distance from a random hyperplane to the real variety. The reader can easily see this yield two real witness points for the circle example. ...
... An alternative numerical approach where the witness points are critical points of the distance from a random point to the real variety has been developed in [12]. The works [12,39,40] use Lagrange multipliers to set up the critical point problem in the following way. ...
Systems of differential-algebraic equations are routinely automatically produced by modeling enviroments such as Maplesim, System Modeler and Modelica. Structural methods are important for reducing the index and obtaining hidden constraints of such daes. This is especially the case for high index non-linear daes. Although such structural analysis is often successful for many dynamic systems, it may fail if the resulting Jacobian is still singular due to symbolic cancellation or numerical degeneration. Existing modified structural methods can handle some cases caused by symbolic cancellation, where assumes the determinant of a Jacobian matrix is identically zero. This paper removes such assumptions and provides numerical methods to analyze such degenerated cases using real algebraic geometry for polynomially nonlinear daes. Firstly, we provide a witness point method, which produces witness points on all components and can help to detect degeneration on all components of polynomially daes. Secondly, we propose an implicit index reduction method which can restore a full rank Jacobian matrix for degenerated dae. Thirdly, based on IIR, we introduce an improved structural method, which can numerically solve degenerated daes on all components. Examples are given to illustrate our methods and show their advantages for degenerated daes.
... Instead the method in [62,63] yields real witness points as critical points of the distance from a random hyperplane to the real variety. An alternative numerical approach where the witness points are critical points of the distance from a random point to the real variety has been developed in [64]. ...
... An alternative numerical approach where the witness points are critical points of the distance from a random point to the real variety has been developed in [64]. These works [62][63][64] use Lagrange multipliers to set up the critical point problem in the following way. ...
In this paper, we propose an iterative approach for estimating the domains of attraction for a class of discrete-time switched systems, where the state space is divided into several disjoint regions and each region is described by polynomial inequalities. At first, we introduce the basic concepts of Multi-step state subsequence, Multi-step state subspace, Multi-step basin of attraction and Multi-step multiple Lyapunov-like function. Secondly, beginning with an initial inner estimation, a theoretical framework is proposed for estimating the domain of attraction by iteratively calculating the Multi-step multiple Lyapunov-like functions. Thirdly, notice that the Multi-step state subspaces may be empty sets such that the corresponding constraints in the theoretical framework are redundant, we propose a numerical approach based on the homotopy continuation method to pre-check the non-emptiness of the Multi-step state subspaces, and then under-approximatively realize the framework by using S-procedure and sum of squares programming. At last, we implement our iterative approach and apply it to three discrete-time switched system examples with comparisons to existing methods in the literatures. These computation and comparison results show the advantages of our method.
... , f n−1 }, where J f is of full rank at any point of the curve, the main technical challenge would be to identify all branches of f −1 (0) and make sure that there is no jumping during curve tracing. Identifying all branches of f −1 (0) is the problem of computing witness points for every connected component of f −1 (0) [22,12,25]. Techniques for presenting or detecting curve jumping also exist [3,2,19,27,26]. ...
We propose a method for tracing implicit real algebraic curves defined by polynomials with rank-deficient Jacobians. For a given curve $f^{-1}(0)$, it first utilizes a regularization technique to compute at least one witness point per connected component of the curve. We improve this step by establishing a sufficient condition for testing the emptiness of $f^{-1}(0)$. We also analyze the convergence rate and carry out an error analysis for refining the witness points. The witness points are obtained by computing the minimum distance of a random point to a smooth manifold embedding the curve while at the same time penalizing the residual of $f$ at the local minima. To trace the curve starting from these witness points, we prove that if one drags the random point along a trajectory inside a tubular neighborhood of the embedded manifold of the curve, the projection of the trajectory on the manifold is unique and can be computed by numerical continuation. We then show how to choose such a trajectory to approximate the curve by computing eigenvectors of certain matrices. Effectiveness of the method is illustrated by examples.
... Others have continued to improve on these ideas using algebraic techniques such as triangular decompositions, Rational Univariate Representations, and geometric resolutions (e.g., [1,49,27]). A homotopy-based approach is presented in [32] and a similar approach uses critical points with respect to a generic line that can be translated rather than one point is found in [59]. ...
Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected compact component of a real (semi)-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the $n=4$ case. We also apply our method to design an efficient algorithm to compute the real dimension of (semi)-algebraic sets, the original motivation for this research.
... , λ n ) are introduced new variables. Let (x 0 , λ 0 ) be a point of V R (G) such that J F has full rank at x 0 , then by Proposition 2.2 of [36], for almost all choices of a in R n , (x 0 , λ 0 ) is an isolated point of V R (G). Therefore, we can apply homotopy continuation method to find such isolated points. ...
We present a new method for visualizing implicit real algebraic curves inside a bounding box in the $2$-D or $3$-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing space curve in higher-dimensional space. Since the topology of a curve near a singular point of it is not numerically stable, we trace only the curve outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is $\epsilon$-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the projection of the curve in $\mathbb{R}^2$, which is important for applications such as solving bi-parametric polynomial systems. The algorithm starts by computing three types of key points of the curve, namely the intersection of the curve with small spheres centered at singular points, regular critical points of every connected components of the curve, as well as intersection points of the curve with the given bounding box. It then traces the curve starting with and in the order of the above three types of points. This basic scheme is further enhanced by several optimizations, such as grouping singular points in natural clusters, tracing the curve by a try-and-resume strategy and handling "pseudo singular points". The effectiveness of the algorithm is illustrated by numerous examples. This manuscript extends our preliminary results that appeared in CASC 2018.
... Let RealWitnessPoint be the routine introduced in [27] for computing a set of witness points W of a real variety V R (P H ) satisfying Assumption (A 3 ). Recall that a set of witness points W of a real variety V is a finite subset of V such that W has non-empty intersection with every connected component of V . ...
... For solving systems of equations with multiple solutions, homotopy methods are extensively used, especially for systems of polynomial equations. The reader is referred to [10,11,77,95] for the latest developments. Mature and robust software implementations for solving polynomial systems are, for example, Bertini [11,12], and PHCpack [86,87]. ...
Tearing is a long-established decomposition technique, widely adapted across many engineering fields. It reduces the task of solving a large and sparse nonlinear system of equations to that of solving a sequence of low-dimensional ones. The most serious weakness of this approach is well-known: It may suffer from severe numerical instability. The present paper resolves this flaw for the first time. The new approach requires reasonable bound constraints on the variables. The worst-case time complexity of the algorithm is exponential in the size of the largest subproblem of the decomposed system. Although there is no theoretical guarantee that all solutions will be found in the general case, increasing the so-called sample size parameter of the method improves robustness. This is demonstrated on two particularly challenging problems. Our first example is the steady-state simulation a challenging distillation column, belonging to an infamous class of problems where tearing often fails due to numerical instability. This column has three solutions, one of which is missed using tearing, but even with problem-specific methods that are not based on tearing. The other example is the Stewart–Gough platform with 40 real solutions, an extensively studied benchmark in the field of numerical algebraic geometry. For both examples, all solutions are found with a fairly small amount of sampling.
... For example, in [23], a numerical homotopy method to find the extremum of Euclidean distance to a point as the objective function was presented. More recently, the Euclidean distance to a plane was proposed as a linear objective function in [25]. ...
... In this paper, we follow the work of [25] [26] to extend complex homotopy methods to finding witness points on the irreducible components of real varieties. To obtain such witness points, we first need to solve a special class of polynomial systems. ...
... In this section, we will combine the LPH algorithm in Section 4 and methods in [25] to compute a real witness set which has at least one point on each irreducible component of a real algebraic set, and give an illustrative example. ...
A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety.
