DEMiCs: A Software Package for Computing the Mixed Volume Via Dynamic Enumeration of all Mixed Cells

Mizutani, Tomohiko

doi:10.1007/978-0-387-78133-4_5

Cited by 21 publications

(34 citation statements)

References 19 publications

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“…Dynamic enumeration [15] applies a greedy algorithm to compute all mixed cells and its implementation in DEMiCs [14] produces the mixed cells at a faster pace than MixedVol [8]. Table 4 shows times for the mixed volume computation with DEMiCs [14] in a pipelined version of the polyhedral homotopies. Table 5: Wall clock times in seconds for all stages of the solver on cyclic 12roots.…”

Section: Pipelined Polyhedral Homotopiesmentioning

confidence: 99%

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A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

Verschelde

2018

Developments in Language Theory

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A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods are applied to compute a numerical irreducible decomposition. Load balancing and pipelining are techniques in a parallel implementation on a computer with multicore processors. The application of the parallel algorithms is illustrated on solving the cyclic n-roots problems, in particular for n = 8, 9, and 12.

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Section: Pipelined Polyhedral Homotopiesmentioning

confidence: 99%

“…The software used in this paper is PHCpack [20], which provides a numerical irreducible decomposition [18]. For the mixed volume computation, MixedVol [8] and DEMiCs [14] are used. An introduction to the homotopy continuation methods for computing positive dimensional solution sets is described in [19].…”

Section: Introductionmentioning

confidence: 99%

A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

Verschelde

2018

Developments in Language Theory

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“…Algorithm 3 and Algorithm 2 have recursion trees with different shapes. In the setting of mixed volume computation it was observed in [26] that the tree of Algorithm 3 has far fewer vertices. Figure 3 demonstrates the difference between the tree traversal of static enumeration and dynamic enumeration.…”

Section: Dynamic Enumerationmentioning

confidence: 99%

“…In [12], a total polynomial-time algorithm is presented to compute the edge skeleton of a polytope. Free dedicated software packages to compute mixed volumes are MixedVol [17] and DEMiCS [26]. In [26], the computation of the mixed volume for the cyclic 16-roots problem was reported for the first time.…”

Section: Introductionmentioning

confidence: 99%

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Computing Tropical Prevarieties in Parallel

Jensen

Sommars

Verschelde

2017

Proceedings of the International Workshop on Parallel Symbolic Computation

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The computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the power series developments of the solutions. The computation of the power series may start as soon as one pretropism is available, so our parallel computation of the tropical prevariety has an application in a pipelined solver.We present a parallel implementation of dynamic enumeration. Our first distributed memory implementation with forked processes achieved good speedups, but quite often resulted in large variations in the execution times of the processes. The shared memory multithreaded version applies work stealing to reduce the variability of the run time. Our implementation applies the thread safe Parma Polyhedral Library (PPL), in exact arithmetic with the GNU Multiprecision Arithmetic Library (GMP), aided by the fast memory allocations of TCMalloc.Our parallel implementation is capable of computing the tropical prevariety of the cyclic 16-roots problem. We also report on computational experiments on the n-body and n-vortex problems; our computational results compare favorably with Gfan.

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Dynamic Decomposition of Tropical Prevarieties for Celestial Mechanics

Jensen

2024

Mathematical Software – ICMS 2024

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DEMiCs: A Software Package for Computing the Mixed Volume Via Dynamic Enumeration of all Mixed Cells

Cited by 21 publications

References 19 publications

A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

Computing Tropical Prevarieties in Parallel

Dynamic Decomposition of Tropical Prevarieties for Celestial Mechanics

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