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Universal Algorithms for Solving the Matrix Bellman Equations Over Semirings

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Universal Algorithms for Solving the Matrix Bellman Equations Over Semirings Universal algorithms for solving the matrix Bellman equations over semirings G. L. Litvinov, A. Ya. Rodionov, S. N. Sergeev & A. N. Sobolevski Soft Computing A Fusion of Foundations, Methodologies and Applications ISSN 1432-7643 Soft Comput DOI 10.1007/s00500-013-1027-5 1 23 Your article is protected by copyright and all rights are held exclusively by Springer- Verlag Berlin Heidelberg. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your work, please use the accepted author’s version for posting to your own website or your institution’s repository. You may further deposit the accepted author’s version on a funder’s repository at a funder’s request, provided it is not made publicly available until 12 months after publication. 1 23 Author's personal copy Soft Comput DOI 10.1007/s00500-013-1027-5 FOUNDATIONS Universal algorithms for solving the matrix Bellman equations over semirings G. L. Litvinov • A. Ya. Rodionov • S. N. Sergeev • A. N. Sobolevski Ó Springer-Verlag Berlin Heidelberg 2013 Abstract This paper is a survey on universal algorithms computer representation of real numbers are provided by for solving the matrix Bellman equations over semirings various modifications of floating point arithmetics, and especially tropical and idempotent semirings. How- approximate arithmetics of rational numbers (Litvinov ever, original algorithms are also presented. Some appli- et al. 2008), interval arithmetics etc. The difference cations and software implementations are discussed. between mathematical objects (‘‘ideal’’ numbers) and their finite models (computer representations) results in com- putational (for instance, rounding) errors. 1 Introduction An algorithm is called universal if it is independent of a particular numerical domain and/or its computer represen- Computational algorithms are constructed on the basis of tation (Litvinov and Maslov 1996, 1998, 2000; Litvinov certain primitive operations. These operations manipulate et al. 2000). A typical example of a universal algorithm is data that describe ‘‘numbers.’’ These ‘‘numbers’’ are ele- the computation of the scalar product (x, y) of two vectors ments of a ‘‘numerical domain,’’ that is, a mathematical x ¼ðx1; ...; xnÞ and y ¼ðy1; ...; ynÞ by the formula ðx; yÞ¼ object such as the field of real numbers, the ring of integers, x1y1 þÁÁÁþxnyn: This algorithm (formula) is independent different semirings etc. of a particular domain and its computer implementation, In practice, elements of the numerical domains are since the formula is well-defined for any semiring. It is clear replaced by their computer representations, that is, by that one algorithm can be more universal than another. For elements of certain finite models of these domains. example, the simplest Newton–Cotes formula, the rectan- Examples of models that can be conveniently used for gular rule, provides the most universal algorithm for numerical integration. In particular, this formula is valid Communicated by A. D. Nola. also for idempotent integration [that is, over any idempotent semiring, see (Kolokoltsov and Maslov 1997; Litvinov This work is supported by the RFBR-CRNF grant 11-01-93106 and 2007)]. Other quadrature formulas (for instance, combined RFBR grant 12-01-00886-a. trapezoid rule or the Simpson formula) are independent of computer arithmetics and can be used (for instance, in the G. L. Litvinov (&) Á A. N. Sobolevski A.A. Kharkevich Institute for Information Transmission iterative form) for computations with arbitrary accuracy. In Problems, and National Research University Higher School contrast, algorithms based on Gauss–Jacobi formulas are of Economics, Moscow, Russia designed for fixed accuracy computations: they include e-mail: [email protected] constants (coefficients and nodes of these formulas) defined A. Ya. Rodionov Á S. N. Sergeev with fixed accuracy. (Certainly, algorithms of this type can Moscow Center for Continuous Mathematical Education, be made more universal by including procedures for com- Moscow, Russia puting the constants; however, this results in an unjustified complication of the algorithms.) S. N. Sergeev University of Birmingham, School of Mathematics, Modern achievements in software development and Edgbaston B15 2TT, UK mathematics make us consider numerical algorithms and 123 Author's personal copy G. L. Litvinov et al. their classification from a new point of view. Conventional mechanics; in fact, the two principles are closely connected numerical algorithms are oriented to software (or hard- (see Litvinov 2007; Litvinov and Maslov 1996, 1998). In a ware) implementation based on floating point arithmetic sense, the traditional mathematics over numerical fields can and fixed accuracy. However, it is often desirable to per- be treated as a ‘quantum’ theory, whereas the idempotent form computations with variable (and arbitrary) accuracy. mathematics can be treated as a ‘classical’ shadow (or For this purpose, algorithms are required that are inde- counterpart) of the traditional one. It is important that the pendent of the accuracy of computation and of the specific idempotent correspondence principle is valid for algorithms, computer representation of numbers. In fact, many algo- computer programs and hardware units. rithms are independent not only of the computer repre- In quantum mechanics the superposition principle sentation of numbers, but also of concrete mathematical means that the Schro¨dinger equation (which is basic for the (algebraic) operations on data. In this case, operations theory) is linear. Similarly in idempotent mathematics the themselves may be considered as variables. Such algo- (idempotent) superposition principle (formulated by V. rithms are implemented in the form of generic programs P. Maslov) means that some important and basic problems based on abstract data types that are defined by the user in and equations that are nonlinear in the usual sense (for addition to the predefined types provided by the language. instance, the Hamilton-Jacobi equation, which is basic for The corresponding program tools appeared as early as in classical mechanics and appears in many optimization Simula-67, but modern object-oriented languages (like problems, or the Bellman equation and its versions and C??, see, for instance (Lorenz 1993; Pohl 1997)) are more generalizations) can be treated as linear over appropriate convenient for generic programming. Computer algebra idempotent semirings, see Maslov (1987a, b). algorithms used in such systems as Mathematica, Maple, Note that numerical algorithms for infinite dimensional REDUCE, and others are also highly universal. linear problems over idempotent semirings (for instance, A different form of universality is featured by iterative idempotent integration, integral operators and transforma- algorithms (beginning with the successive approximation tions, the Hamilton–Jacobi and generalized Bellman method) for solving differential equations (for instance, equations) deal with the corresponding finite-dimensional methods of Euler, Euler–Cauchy, Runge–Kutta, Adams, a approximations. Thus idempotent linear algebra is the basis number of important versions of the difference approxi- of the idempotent numerical analysis and, in particular, the mation method, and the like), methods for calculating discrete optimization theory. elementary and some special functions based on the Carre´ (1971, 1979) (see also Gondran 1975; Gondran and expansion in Taylor’s series and continuous fractions (Pade´ Minoux 1979, 2010) used the idempotent linear algebra to approximations). These algorithms are independent of the show that different optimization problems for finite graphs computer representation of numbers. can be formulated in a unified manner and reduced to solving The concept of a generic program was introduced by Bellman equations, that is, systems of linear algebraic many authors; for example, in Lehmann (1977) such pro- equations over idempotent semirings. He also generalized grams were called ‘program schemes.’ In this paper, we principal algorithms of computational linear algebra to the discuss universal algorithms implemented in the form of idempotent case and showed that some of these coincide with generic programs and their specific features. This paper is algorithms independently developed for solution of optimi- closely related to Litvinov (2007), Litvinov and Maslov zation problems. For example, Bellman’s method of solving (1996, 1998), Litvinov and Maslova (2000), Litvinov et al. the shortest path problem corresponds to a version of Jaco- (2000, 2011), Sergeev (2011), in which the concept of a bi’s method for solving a system of linear equations, whereas universal algorithm was defined and software and hardware Ford’s algorithm corresponds to a version of Gauss–Seidel’s implementation of such algorithms was discussed in con- method. We briefly discuss Bellman equations and the cor- nection with problems of idempotent mathematics, see, for responding optimization problems on graphs, and use the instance, Kolokoltsov and Maslov (1997), Litvinov and ideas of Carre´ to obtain new universal algorithms. We stress Sobolevskiıˇ(2001), Mikhalkin (2006), Viro (2001, 2008). that these well-known results can be interpreted as a mani- The so-called idempotent correspondence principle, see festation of the idempotent superposition principle. Litvinov and Maslov (1996, 1998), linking this mathematics Note that many algorithms for solving the matrix Bellman with the usual mathematics
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