text stringlengths 128 2.05k |
|---|
An analysis of the manipulators’ behavior, briefly mentioned at the beginning of this section, directing towards game theory seems promising as well. (Even more so since we identified polynomial-time algorithms for a few variants of coalitional manipulation.) One very interesting question about coalitions is, for examp... |
Acknowledgments We thank the anonymous IJCAI ’17 reviewers for their constructive and valuable feedback. Robert Bredereck was from mid-September 2016 to mid-September 2017 on postdoctoral leave at the University of Oxford, supported by the DFG fellowship BR 5207/2. Andrzej Kaczmarczyk was supported by the DFG project A... |
# Source: arxiv 1806.10584 # Title: Implementation of a Near-Optimal Complex Root Clustering Algorithm # Sections: all # Downloaded: 2026-03-03T02:19:56.627813+00:00 |
Implementation of a Near-Optimal Complex Root Clustering Algorithm Abstract We describe Ccluster a software for computing natural [MATH] -clusters of complex roots in a given box of the complex plane. This algorithm from Becker et al. (2016) is near-optimal when applied to the benchmark problem of isolating all complex... |
of the first implementations of a near-optimal algorithm for complex roots. We describe some low level techniques for speeding up the algorithm. Its performance is compared with the well-known MPSolve library and Maple |
Introduction The problem of root finding for a polynomial [MATH] is a classical problem from antiquity, but remains the subject of active research to the present |
We consider a classic version of root finding: Local root isolation problem Given: a polynomial [MATH] a box [MATH] [MATH] Output: a set [MATH] |
of pairwise-disjoint discs of radius [MATH] each containing a unique root of [MATH] in [MATH] It is local because we only look for roots in a locality, as specified by [MATH] The local problem is useful in applications (especially in geometric computation) where we know where to look for the roots of interest. There ar... |
[MATH] that contains all roots of [MATH] If we omit [MATH] , it amounts to setting [MATH] representing the pure isolation problem. |
Our main interest is a generalization of root isolation, to the lesser-studied problem of root clustering It is convenient to introduce two definitions: for any set [MATH] , let [MATH] denote the set of roots of [MATH] in [MATH] , and let [MATH] |
count the total multiplicity of the roots in [MATH] Typically, [MATH] is a disc or a box. For boxes and discs, we may write [MATH] (for any [MATH] ) to denote the dilation of [MATH] by factor [MATH] , keeping the same center. The following problem was introduced in |
Local root clustering problem Given: a polynomial [MATH] , a box [MATH] [MATH] Output: a set of pairs [MATH] where [MATH] ’s are pairwise-disjoint discs of radius |
[MATH] [MATH] for all [MATH] , and [MATH] This generalization of root isolation is necessary when we consider polynomials whose coefficients are non-algebraic (or when [MATH] is an analytic function, as in |
). The requirement that [MATH] ensures that our output clusters are natural a polynomial of degree [MATH] has at most [MATH] natural clusters (see 16 , Lemma 1] ). The local root clustering algorithm for analytic functions of |
has termination proof, but no complexity analysis. By restricting [MATH] to a polymomial, Becker et al. succeeded in giving an algorithm and also its complexity analysis based on the geometry of the roots. When applied to the benchmark problem , where [MATH] is an integer polynomial of degree [MATH] with [MATH] -bit co... |
calls such bounds near-optimal (at least when [MATH] ). The clustering algorithm studied in this paper comes from , which in turn is based on |
Previously, the Pan-Schönhage algorithm has achieved near-optimal bounds with divide-and-conquer methods but was the first subdivision algorithm to achieve the near-optimal bound for complex roots. For real roots, Sagraloff-Mehlhorn |
had earlier achieved near-optimal bound via subdivision. Why the emphasis on “subdivision”? It is because such algorithms are implementable and quite practical (e.g., |
). Thus the near-optimal real subdivision algorithm of was implemented shortly after its discovery, and reported in with excellent results. In contrast, all the asymptotically efficient root algorithms (not necessarily near-optimal) based on divide-and-conquer methods of the last 30 years have never been implemented; a... |
). Computer algebra systems mainly rely on algorithms with a priori guarantees of correctness. But in practice, algorithms without such guarantees are widely used. For complex root isolation, one of the most highly regarded multiprecision software is MPSolve |
The original algorithm in MPSolve was based on Erhlich-Aberth (EA) iteration; but since 2014, a “hybrid” algorithm was introduced. It is based on the secular equation, and combines ideas from EA and eigensolve |
These algorithms are inherently global solvers (they must approximate all roots of a polynomial simultaneously). Another theoretical limitation is that the global convergence of these methods is not proven. |
In this paper, we give a preliminary report about Ccluster , our implementation of the root clustering algorithm from To illustrate the performance for the local versus global problem, consider the Bernoulli polynomials |
[MATH] where [MATH] ’s are the Bernoulli numbers. Figure (Left) shows the graphical output of Ccluster for [MATH] Table has four timings [MATH] |
(for [MATH] in seconds: [MATH] is the time for solving the local problem over a box [MATH] [MATH] is the time for the global problem over the box [MATH] (which contains all the roots). The other two timings from MPSolve [MATH] for unisolve, [MATH] for secsolve) will be explained later. For each instance, we also indica... |
shows that MPSolve remains unchallenged for the global problem. But in applications where locality can be exploited, local methods may win, as seen in the last two rows of the table. The corresponding time for Maple ’s fsolve is also given; fsolve is not a guaranteed algorithm and may fail. |
Overview of Paper In Section 2, we describe the experimental setup for Ccluster Sections 3-5 describe some techniques for speeding up the basic algorithm. We conclude with Section 6. |
Implementation and Experiments The main implementation of Ccluster is in language. We have an interface for Julia We based our big number computation on the |
arb library. The arb library implements ball arithmetic for real numbers, complex numbers and polynomials with complex coefficients. Each arithmetic operation is carried out with error bounds. |
Test Suite We consider 7 families of polynomials, classic ones as well as some new ones constructed to have interesting clustering or multiple root structure. |
(F1) The Bernoulli polynomial [MATH] of degree [MATH] is described in Section (F2) The Mignotte polynomial [MATH] for a positive integer [MATH] has two roots whose separation is near the theoretical minimum separation bound. |
(F3) The Wilkinson polynomials [MATH] (F4) The Spiral Polynomial [MATH] See Figure (Right) for [MATH] (F5) Wilkinson Multiple: [MATH] |
[MATH] has degree [MATH] where the root [MATH] has multiplicity [MATH] (for [MATH] ). (F6) Mignotte Cluster: [MATH] This polynomial has degree [MATH] (assuming [MATH] and has a cluster of [MATH] roots near |
[MATH] and a cluster of [MATH] roots near [MATH] (F7) Nested Cluster: [MATH] has degree [MATH] and is defined by induction on [MATH] |
[MATH] with roots [MATH] where [MATH] Inductively, if the roots of [MATH] are [MATH] , then we define [MATH] See Figure for the natural |
[MATH] -clusters of [MATH] Timing Running times are sequential times on a Intel(R) Core(TM) i3 CPU 530 @ 2.93GHz machine with linux. |
Ccluster implements the algorithm described in with differences coming from the improvements described in Sections below. Unless explicitly specified, the value of [MATH] for Ccluster is set to [MATH] ; roughly speaking, it falls back to asking for 15 guaranteed decimal digits. |
MPSolve For external comparison, we use MPSolve . It was shown to be superior to major software such as Maple or Mathematica There are two root solvers in MPSolve the original unisolve |
which is based on the Ehrlich-Aberth iteration and the new hybrid algorithm called secsolve These are called with the commands mpsolve -au -Gi -o [MATH] -j1 and |
mpsolve -as -Gi -o [MATH] -j1 (respectively). -Gi means that MPSolve tries to find for each root a unique complex disc containing it, such that Newton iteration is guaranteed to converge quadratically toward the root starting from the center of the disc. |
-o [MATH] means that [MATH] is used as an escape bound, i.e., the algorithm stops when the complex disc containing the root has radius less that [MATH] , regardless of whether it is isolating or not. Unless explicitly specified, we set [MATH] |
-j1 means that the process is not parallelized. Although MPSolve does not do general local search, it has an option to search only within the unit disc. This option does not seem to lead to much improvement. |
Improved Soft Pellet Test The key predicate in is a form of Pellet test denoted [MATH] (with implicit [MATH] ). This is modified in Figure by adding an outer while-loop to control the number of Graeffe-Dandelin iterations. We try to get a definite decision (i.e., anything other than unresolved ) from the soft compariso... |
in the innermost while-loop. Thus we have two versions of our algorithm: (V1) uses the original [MATH] in and (V2) uses the modified form in Figure Let [MATH] V1 and [MATH] V2 be timings for the 2 versions. Table |
shows the time [MATH] V1 (in seconds) and the ratio [MATH] V1/ [MATH] V2. We see that (V2) achieves a consistent 2.3 to 3-fold speed up. |
In (V2), as in , we use [MATH] (defined as [MATH] to prove that a box [MATH] has no root. We propose a new version (V3) that uses [MATH] |
(defined as [MATH] , where [MATH] is the degree of [MATH] instead of [MATH] to achieve this goal: instead of just showing that [MATH] has no root, it upper bounds [MATH] Although counter-intuitive, this yields a substantial improvement because it led to fewer Graeffe iterations overall. The timing for (V3) is [MATH] V3... |
Filtering A technique for speeding up the evaluation of predicates is the idea of filters (e.g., ). The various Pellet tests can be viewed as a box predicate [MATH] that maps a box [MATH] to a value |
in [MATH] If [MATH] is another box predicate with property that [MATH] implies [MATH] , we call [MATH] falsehood filter If [MATH] is efficient relatively to [MATH] and “efficacious” (informally, [MATH] is likely to yield [MATH] ), then it is useful to first compute |
[MATH] . If [MATH] , we do not need to compute [MATH] The predicate [MATH] used in Ccluster is defined as follows: [MATH] is true if [MATH] returns [MATH] |
(then [MATH] contains no root of [MATH] and is false if [MATH] returns [MATH] or [MATH] (then [MATH] may contain some roots of [MATH] ). We next present the falsehood filter [MATH] for [MATH] |
Let [MATH] denote the Taylor shift of [MATH] in [MATH] [MATH] its [MATH] -th Graeffe iterate, [MATH] the [MATH] -th coefficient of [MATH] and [MATH] the absolute value of the [MATH] -th coefficient. Let [MATH] be the degree of [MATH] The assertion below is a direct consequence of the classical test of Pellet (see |
[p. 12]) and justify the correctness of our filters: [MATH] if [MATH] then [MATH] returns [MATH] or [MATH] Our [MATH] filter computes [MATH] |
[MATH] and [MATH] and checks hypothesis of [MATH] using IntCompare [MATH] and [MATH] can respectively be computed as [MATH] and [MATH] |
[MATH] can be computed with the following well known formula: [EQUATION] Obtaining [MATH] with eq. ( ) requires to know [MATH] coefficients of [MATH] |
[MATH] coefficients of [MATH] and finally [MATH] coefficients of [MATH] In particular, it requires to compute entirely the iterations |
[MATH] such that [MATH] , and it is possible to do it more efficiently that with eq. ( (for instance with the formula given in definition 2 of |
). Our [MATH] filter takes as input a precision [MATH] , the Taylor shift [MATH] of the [MATH] bit approximation of [MATH] and its [MATH] -th Graeffe iteration [MATH] |
such that [MATH] and [MATH] It computes [MATH] [MATH] and the [MATH] first coefficients of [MATH] for [MATH] with eq. ( ). Then it checks the hypothesis of [MATH] using IntCompare and returns false if it is verified, and true otherwise. In practice, it is implemented within the procedure implementing |
[MATH] Incorporating [MATH] into Version (V3), we obtain (V4) and the speed up can be seen in Table Filtering with [MATH] becomes more effective as degree grows and this is because one has [MATH] for smaller |
[MATH] (recall that [MATH] ). Escape Bound The [MATH] parameter is usually understood as the precision desired for roots. But we can also view it as an escape bound for multiple roots as follows: we do not refine a disc that contains a simple root, even if its radius is [MATH] But for clusters of size greater than one,... |
MPSolve has a similar option. This variant of (V4) is denoted (V4’). We see from Table that (V4’) gives a modest improvement (up to 25% speedup) over (V4) when [MATH] This improvement generally grows with [MATH] |
(but [MATH] shows no difference). Conclusion Implementing subdivision algorithms is relatively easy but achieving state-of-art performance requires much optimization and low-level development. This paper explores several such techniques. We do well compared to fsolve in Maple , but the performance of MPSolve is superio... |
# Source: arxiv 1806.10640 # Title: A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems # Sections: all # Downloaded: 2026-03-03T01:48:15.703428+00:00 |
A DIAGRAMMATIC REPRESENTATION OF PHASE PORTRAITS AND BIFURCATION DIAGRAMS OF TWO-DIMENSIONAL DYNAMICAL SYSTEMS Abstract We treat the problem of characterizing in a systematic way the qualitative features of two-dimensional dynamical systems. To that end, we construct a representation of the topological features of phas... |
keywords: two-dimensional systems, bifurcations, nonlinear dynamics Introduction By understanding the dynamics displayed by a nonlinear system we typically refer to the capacity to list all the qualitatively different phase space portraits that the system can display for different values of its control parameters. Even... |
. In that case, one starts with some sets of attractors, obtained for different parameter values, which a priori are not “close” in any way. Actually, a similar situation is faced when a system (whose equations are known) is explored numerically. Is it possible to algorithmically list and classify the dynamical possibi... |
In this work we explore this question for planar systems. These systems are near and dear to the hearts of dynamicists, since two is the minimal dimensionality in which we can embed nontrivial, recurrent dynamics. Moreover, it is typical to study the different behaviors that these bidimensional models can display when ... |
The work is organized as follows. In Section we introduce a way of representing phase portraits by using diagrams that capture important qualitative information of the system’s dynamics. Specifically, they encode what the limit sets of the system are, their stability and their distribution in phase space. |
Section discusses how smooth modifications of these diagrams give rise naturally to bifurcations, in which phase portraits change qualitatively. All codimension 1 bifurcations are obtained in this way. We introduce a representation of bifurcation curves by means of “dressed” lines, that encode the direction in which ne... |
In Section we show, as an instructive example, how the theoretical framework developed here can be applied to the Wilson-Cowan oscillator. We present our conclusions in Section |
Diagram Representation of Phase Portraits We now describe a representation of phase portraits by diagrams that discard all the quantitative information that portraits convey (i.e., the specific trajectories in phase space), while preserving the qualitative features. Unlike the phase portrait, the resulting diagram is r... |
We will restrict ourselves to two-dimensional, structurally stable dynamical systems. Further, we will assume that the region of interest of phase space can be enclosed by a closed transversal curve (i.e., a closed curve along which the velocity vector is neither tangent nor zero, so it always traverses the curve from ... |
2.1 Construction of the Diagrams The diagrams encode information about the limit sets of the system, which in two dimensions can only be stable or unstable nodes (or foci, which are topologically equivalent), stable or unstable limit cycles, or saddle points. We will represent those with the symbols of Table 1. |
In Table 1 we have introduced two quantities associated to limit sets: the index and the repulsion. These can be computed easily from the diagrams: the index is the number of arrows above its symbol minus the arrows below, and the repulsion is the number of outgoing arrows minus the incoming ones. Their interpretation ... |
The shape of a symbol encodes the type of limit set it represents, and the filling its repulsion. For nodes and cycles, the repulsion is directly related to their (in)stability. |
Notice that there are various ways in which the three arrows connecting the square saddle point symbol can be oriented. It is forbidden to have all the arrows departing from the square, or all converging, but the other possibilities are allowed. These represent different ways in which the saddle’s invariant manifolds c... |
If a system has several saddle points, we shall represent them with a compound symbol as shown in the last two entries of Table 1 (unless they are separated by a limit cycle so that one is inside the limit cycle and the other outside, in which case we treat them individually). In those symbols, the undirected lines sho... |
Every well-formed diagram consists of any number of the symbols in Table 1, connected by arrows so that there is only one arrow unmatched, at the top of the diagram. Each connecting arrow should have a definite direction (upward or downward), so two symbols can only be connected if the directions of their arrows match.... |
Suppose we know from experiments or simulations what the stable limit sets of a two-dimensional dynamic system are. Then it is possible to obtain all the phase portraits compatible with these stable solutions by adding unstable sets (i.e. unstable nodes, unstable cycles or saddle points) in a way such that the resultin... |
2.2 Interpretation of the Diagrams as Phase Portraits The interpretation of the diagrams is as follows: each arrow in the diagram represents a family of closed transversal curves in the phase plane, that enclose all the limit sets in the branch of the diagram below that arrow (see insets in Fig. ). An arrow pointing do... |
Notice that every symbol in Table 1 has one arrow on top and a number of arrows at the bottom. Accordingly, the transversal curves in phase space will delimit regions with an outer boundary and a number of holes inside. In each of these regions we draw the limit sets associated with the corresponding symbol in the diag... |
To complete the phase portrait, trajectories that cross all the transversals in the direction they define should be sketched. Of these, the ones along the saddles’ invariant manifolds are the most interesting. Two begin in each saddle and two end, constituting its unstable and stable manifold respectively. Each pair ap... |
In the case of multiple saddle points, we do not attempt to assign repulsions to them individually, because it cannot always be done without ambiguity (see § 2.4 ). A repulsion can be assigned, however, to the whole set of saddles. By doing this, our treatment abandons a complete description of the connectivities of th... |
The only other topological ambiguity left when reconstructing the phase portrait is the sense in which the limit cycles circulate. If this is relevant for a certain application, it can be easily incorporated to the method by using different symbols for clockwise- or counterclockwise-circulating cycles. For simplicity, ... |
Thus, a specific diagram represents a class of phase portraits, all of which equivalent up to topological equivalence and, if several saddles are present, heteroclinic bifurcations. In the following section we will show the converse statement: that every phase portrait satisfying the hypotheses made at the beginning of... |
2.3 Obtaining the Diagram from the Phase Portrait Given a phase portrait, the diagram that represents it can be constructed following algorithmic steps. We need to identify the closed transversals on the phase portrait and associate an arrow in the diagram to each of them. The arrows then define the limit set symbols a... |
For every stable or unstable node, a sufficiently small transversal can always be found that encircles it, we draw one of these with the appropriate orientation for each node present (inward flow for stable, outward for unstable). Similarly, for every stable or unstable limit cycle a pair of closed transversals can alw... |
After this process, the phase space will be divided by the transversals in regions enclosing a single node, a single cycle or a nonzero number of saddle points. That is, matching one of the entries of Table 1. The full diagram can now be constructed following the hierarchy induced by the distribution of these regions. ... |
Thus, we have an algorithmic way of identifying any phase portrait with a specific diagram, and any diagram with a specific class of phase portraits. The diagrams may be used to consistently generate and classify phase portraits in a highly qualitative approach, and restrictions about the system’s limit sets can be nat... |
2.4 Index and Repulsion We can now give an interpretation to the index and repulsion introduced in Table 1. In planar dynamical systems, the index of a closed curve is defined as the amount of counterclockwise revolutions that the vector field does as one travels counterclockwise once around the curve. In particular, t... |
We define the repulsion only for regions bounded by closed transversals, and it is directly the number of transversals through which the flow leaves the region minus the number through which it enters it. The repulsion of a limit set is the repulsion of a region bounded by closed transversals that contains it and no ot... |
Both quantities are additive: the index of the union of two disjoint regions is the sum of their indexes, and similarly for the repulsion. The interesting case is when two symbols adjacent in the diagram are considered together: then the arrow that connects them is not counted when computing these quantities. However, ... |
An important observation is that for the class of dynamical systems we are considering, i.e. whose relevant phase diagram can be bounded by a single closed transversal, the total index is [MATH] and the total repulsion is [MATH] if the flow enters the transversal, or [MATH] if it exits it. These quantities can also be ... |
Bifurcations 3.1 Continuous Transitions between Diagrams Varying the system’s parameters continuously may change its behavior qualitatively, a process known as bifurcation. Then, in crossing a bifurcation the diagram describing the system should change, and do so somehow “continuously”. Interestingly, the diagram forma... |
By changing a diagram continuously we mean either shrinking the length of a connecting arrow until it disappears, or alternatively creating a new zero-length arrow and enlarging it. Since the arrows imply the limit set symbols according to Table 1, these must be updated as the arrow configuration changes. |
Recall that each arrow represents a family of closed transversals that separate the limit sets associated to the symbols in its ends. An arrow length approaching zero is interpreted as the two involved limit sets approaching each other in phase space, so that a transversal curve should be finely tuned to separate them.... |
Depending on the type of limit sets that are connected to the arrow involved, different kinds of bifurcation can occur. Table 2 shows all the possible connections between symbols that can be made using the entries of Table 1, and the transition to a different diagram that takes place when the involved arrow shrinks to ... |
Notice that the amount and orientation of all the arrows external to each partial diagram remains unchanged in the bifurcations. This yields conservation laws for the index and repulsion of the system, which must be constant for all parameter values as long as the hypotheses made at the beginning of Section still hold.... |
3.2 A Representation of Bifurcations in Parameter Space We will now introduce a representation of codimension 1 bifurcations in parameter space, that explicits which limit sets are created or destroyed in crossing the bifurcation. This helps proposing plausible bifurcation diagrams from partial knowledge of the behavio... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.