Universal algorithms, mathematics of semirings and parallel computations

Litvinov, G. L.; Maslov, V. P.; Rodionov, A. Ya.; Sobolevski, A.

doi:10.48550/arxiv.1005.1252

Cited by 2 publications

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“…This definition allows for some pairs of nodes to be disconnected if the corresponding element of the matrix A is 0 and for some channels to be "loops" with coincident ends if the matrix A has nonzero diagonal elements. This concept is convenient for analysis of parallel and distributed computations and design of computing media and networks (see, e.g., [5,45,69,97]).…”

Section: Dequantization Of Geometrymentioning

confidence: 99%

“…The closure operation for matrix semirings Mat n (S) can be defined and computed in terms of the closure operation for S (see Subsection 11.3 above); some methods are described in [1,2,7,14,15,[26][27][28][29]33,37,51,59,83,84,87]. One such method is described below (LDM-factorization), see [45].…”

Section: Universal Algorithms Of Linear Algebra Over Semiringsmentioning

confidence: 99%

“…A systolic array is a 'homogeneous' computing medium consisting of elementary processors, where the general scheme and processor connections are simple and regular. Every elementary processor pumps data in and out performing elementary operations in a such way that the corresponding data flow is kept up in the computing medium; there is an analogy with the blood circulation and this is a reason for the term "systolic", see e.g., [40,42,44,45,66,83,84,87]. Some systolic processors for the general algebraic path problem are presented in [83,84,87].…”

Section: The Correspondence Principle For Hardware Designmentioning

confidence: 99%

“…The history of the subject is discussed, e.g., in [39], with extensive bibliography. See also [15,17,18,20,22,[40][41][42]45].…”

Section: Introductionmentioning

confidence: 99%

“…See V.P. Maslov's original formulation in [63][64][65], as well as [6,14,15,17,18,20,22,33,[39][40][41][42][43]45], and below.…”

Section: Introductionmentioning

confidence: 99%

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Idempotent/Tropical Analysis, the Hamilton–Jacobi and Bellman Equations

Litvinov

2013

Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

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Tropical and idempotent analysis with their relations to the Hamilton-Jacobi and matrix Bellman equations are discussed. Some dequantization procedures are important in tropical and idempotent mathematics. In particular, the Hamilton-Jacobi-Bellman equation is treated as a result of the Maslov dequantization applied to the Schrödinger equation. This leads to a linearity of the Hamilton-Jacobi-Bellman equation over tropical algebras. The correspondence principle and the superposition principle of idempotent mathematics are formulated and examined. The matrix Bellman equation and its applications to optimization problems on graphs are discussed. Universal algorithms for numerical algorithms in idempotent mathematics are investigated. In particular, an idempotent version of interval analysis is briefly discussed.In dear memory of my beloved wife Irina.

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Section: Dequantization Of Geometrymentioning

confidence: 99%

Section: Universal Algorithms Of Linear Algebra Over Semiringsmentioning

confidence: 99%

Section: The Correspondence Principle For Hardware Designmentioning

confidence: 99%

“…The history of the subject is discussed, e.g., in [39], with extensive bibliography. See also [15,17,18,20,22,[40][41][42]45].…”

Section: Introductionmentioning

confidence: 99%

“…See V.P. Maslov's original formulation in [63][64][65], as well as [6,14,15,17,18,20,22,33,[39][40][41][42][43]45], and below.…”

Section: Introductionmentioning

confidence: 99%

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Idempotent/Tropical Analysis, the Hamilton–Jacobi and Bellman Equations

Litvinov

2013

Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

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Idempotent and tropical mathematics; complexity of algorithms and interval analysis

Litvinov

2013

Computers & Mathematics with Applications

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A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics usually constructions and algorithms are more simple with respect to their traditional analogs. We especially examine algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operations and other "good" operations. Every algorithm of this type has an interval version. The complexity of this interval version coincides with the complexity of the initial algorithm. The interval version of an algorithm of this type gives exact interval estimates for the corresponding output data. Algorithms of linear algebra over idempotent and semirings are examined. In this case, basic algorithms are polynomial as well as their interval versions. This situation is very different from the traditional linear algebra, where basic algorithms are polynomial but the corresponding interval versions are NP-hard and interval estimates are not exact.

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Universal algorithms, mathematics of semirings and parallel computations

Cited by 2 publications

References 0 publications

Idempotent/Tropical Analysis, the Hamilton–Jacobi and Bellman Equations

Idempotent/Tropical Analysis, the Hamilton–Jacobi and Bellman Equations

Idempotent and tropical mathematics; complexity of algorithms and interval analysis

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