Abstract
This paper is a survey on universal algorithms for solving the matrix Bellman equations over semirings and especially tropical and idempotent semirings. However, original algorithms are also presented. Some applications and software implementations are discussed.
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References
Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic Press, New York
Baccelli FL, Cohen G, Olsder GJ, Quadrat JP (1992) Synchronization and linearity: an algebra for discrete event systems. Wiley, Hoboken
Backhouse RC, Carré BA (1975) Regular algebra applied to path-finding problems. J Inst Math Appl 15:161–186
Barth W, Nuding E (1974) Optimale Lösung von Intervalgleichungsystemen. Comput Lett 12:117–125
Butkovič P (2010) Max-linear systems: theory and algorithms. Springer, Berlin
Butkovič P, Schneider H, Sergeev S (2011) Z-matrix equations in max algebra, nonnegative linear algebra and other semirings. http://www.arxiv.org/abs/1110.4564
Carré BA (1971) An algebra for network routing problems. J Inst Math Appl 7:273–294
Carré BA (1979) Graphs and networks. Oxford University Press, Oxford
Cechlárová K, Cuninghame-Green RA (2002) Interval systems of max-separable linear equations. Linear Alg Appl 340(1–3):215–224
Cuninghame-Green RA (1979) Minimax algebra, volume 166 of lecture notes in economics and mathematical systems. Springer, Berlin
Faddeev DK, Faddeeva VN (2002) Computational methods of linear algebra. Lan’, St. Petersburg. 3rd ed., in Russian
Fiedler M, Nedoma J, Ramík J, Rohn J, Zimmermann K (2006) Linear optimization problems with inexact data. Springer, New York
Golan J (2000) Semirings and their applications. Kluwer, Dordrecht
Golub GH, van Loan C (1989) Matrix computations. John Hopkins University Press, Baltimore and London
Gondran M (1975) Path algebra and algorithms. In: Roy B (ed), Combinatorial programming: methods and applications, Reidel, Dordrecht, pp 137–148
Gondran M, Minoux M (1979) Graphes et algorithmes. Éditions Eylrolles, Paris
Gondran M, Minoux M (2010) Graphs, dioids and semirings. Springer, New York a.o.
Gunawardena J (eds) (1998) Idempotency. Cambridge University Press, Cambridge
Hardouin L, Cottenceau B, Lhommeau M, Le Corronc E (2009) Interval systems over idempotent semiring. Linear Alg Appl 431:855–862
Kolokoltsov VN, Maslov VP (1997) Idempotent analysis and its applications. Kluwer, Dordrecht
Kreinovich V, Lakeev A, Rohn J, Kahl P (1998) Computational complexity and feasibility of data processing and interval computations. Kluwer, Dordrecht
Krivulin NK (2006) Solution of generalized linear vector equations in idempotent linear algebra. Vestnik St.Petersburg Univ Math 39(1):23–36
Lehmann DJ (1977) Algebraic structures for transitive closure. Theor Comp Sci 4:59–76
Litvinov GL (2007) The Maslov dequantization, idempotent and tropical mathematics: a brief introduction. J Math Sci 141(4):1417–1428. http://www.arxiv.org/abs/math.GM/0507014
Litvinov GL, Maslov VP (1996) Idempotent mathematics: correspondence principle and applications. Russ Math Surv 51:1210–1211
Litvinov GL, Maslov VP (1998) The correspondence principle for idempotent calculus and some computer applications. In: Gunawardena J (ed) Idempotency, Cambridge University Press, Cambridge, pp 420–443. http://www.arxiv.org/abs/math.GM/0101021
Litvinov GL, Maslova EV (2000) Universal numerical algorithms and their software implementation. Progr Comput Softw 26(5):275–280. http://www.arxiv.org/abs/math.SC/0102114Z
Litvinov GL, Sobolevskiĭ AN (2000) Exact interval solutions of the discrete bellman equation and polynomial complexity in interval idempotent linear algebra. Dokl Math 62(2):199–201. http://www.arxiv.org/abs/math.LA/0101041
Litvinov GL, Sobolevskiĭ AN (2001) Idempotent interval analysis and optimization problems. Reliab Comput 7(5):353–377. http://www.arxiv.org/abs/math.SC/0101080
Litvinov GL, Maslov VP, Rodionov AYa (2000) A unifying approach to software and hardware design for scientific calculations and idempotent mathematics. International Sophus Lie Centre, Moscow. http://www.arxiv.org/abs/math.SC/0101069
Litvinov GL, Maslov VP, Shpiz GB (2001) Idempotent functional analysis. An algebraic approach. Math Notes 69(5):696–729. http://www.arxiv.org/abs/math.FA/0009128
Litvinov GL, Rodionov AYa, Tchourkin AV (2008) Approximate rational arithmetics and arbitrary precision computations for universal algorithms. Int J Pure Appl Math 45(2):193–204. http://www.arxiv.org/abs/math.NA/0101152
Litvinov GL, Maslov VP, Rodionov AYa, Sobolevskiĭ AN (2011) Universal algorithms, mathematics of semirings and parallel computations. Lect Notes Comput Sci Eng 75:63–89. http://www.arxiv.org/abs/1005.1252
Lorenz M (1993) Object oriented software: a practical guide. Prentice Hall Books, Englewood Cliffs, N.J.
Maslov VP (1987a) A new approach to generalized solutions of nonlinear systems. Soviet Math Dokl 42(1):29–33
Maslov VP (1987b) On a new superposition principle for optimization process. Uspekhi Math Nauk [Russian Math Surveys] 42(3):39–48
Mikhalkin G (2006) Tropical geometry and its applications. In: Proc ICM 2:827–852. http://www.arxiv.org/abs/math.AG/0601041
Moore RE (1979) Methods and applications of interval analysis. SIAM Studies in Applied Mathematics. SIAM, Philadelphia
Myškova H (2005) Interval systems of max-separable linear equations. Linear Alg Appl 403:263–272
Myškova H (2006) Control solvability of interval systems of max-separable linear equations. Linear Alg Appl 416:215–223
Neumaier A (1990) Interval methods for systems of equations. Cambridge University Press, Cambridge
Pohl I. (1997) Object-Oriented Programming Using C++ , 2nd ed. Addison-Wesley, Reading
Rote G (1985) A systolic array algorithm for the algebraic path problem. Comput Lett 34:191–219
Sedgewick R (2002) Algorithms in C++ . Part 5: graph algorithms, 3rd ed. Addison-Wesley, Reading
Sergeev S (2011) Universal algorithms for generalized discrete matrix Bellman equations with symmetric Toeplitz matrix. Tambov University Reports, ser. Natural and Technical Sciences 16(6):1751–1758. http://www.arxiv.org/abs/math/0612309
Simon I (1988) Recognizable sets with multiplicities in the tropical semiring. Lect Notes Comput Sci 324:107–120
Sobolevskiĭ AN (1999) Interval arithmetic and linear algebra over idempotent semirings. Dokl Math 60:431–433
Stepanov A, Lee M (1994) The standard template library. Hewlett-Packard, Palo Alto
Tchourkin AV, Sergeev SN (2007) Program demonstrating how universal algorithms solve discrete Bellman equation over various semirings. In: Litvinov G, Maslov V, Sergeev S (eds) Idempotent and tropical mathematics and problems of mathematical physics (Volume II), Moscow. French-Russian Laboratory J.V. Poncelet. http://www.arxiv.org/abs/0709.4119
Viro O (2001) Dequantization of real algebraic geometry on logarithmic paper. In: 3rd European Congress of Mathematics: Barcelona, July 10–14, 2000. Birkhäuser, Basel, pp 135. http://www.arxiv.org/abs/math/0005163
Viro O (2008) From the sixteenth hilbert problem to tropical geometry. Jpn J Math 3:1–30
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The authors are grateful to the anonymous referees for a number of important corrections in the paper.
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Communicated by A. D. Nola.
This work is supported by the RFBR-CRNF grant 11-01-93106 and RFBR grant 12-01-00886-a.
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Litvinov, G.L., Rodionov, A.Y., Sergeev, S.N. et al. Universal algorithms for solving the matrix Bellman equations over semirings. Soft Comput 17, 1767–1785 (2013). https://doi.org/10.1007/s00500-013-1027-5
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DOI: https://doi.org/10.1007/s00500-013-1027-5