Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

ILL-VERSUS WELL-POSED SINGULAR LINEAR SYSTEMS: SCOPE OF RANDOMIZED ALGORITHMS

  • Sen, S.K. (Department of Mathematical Sciences, Florida Institute of Technology) ;
  • Agarwal, Ravi P. (Department of Mathematical Sciences, Florida Institute of Technology) ;
  • Shaykhian, Gholam Ali (National Aeronautics and Space Administration (NASA), Engineering Directorate, NE-C1, Kennedy Space Center)
  • Published : 2009.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

The linear system Ax = b will have (i) no solution, (ii) only one non-trivial (trivial) solution, or (iii) infinity of solutions. Our focus will be on cases (ii) and (iii). The mathematical models of many real-world problems give rise to (a) ill-conditioned linear systems, (b) singular linear systems (A is singular with all its linearly independent rows are sufficiently linearly independent), or (c) ill-conditioned singular linear systems (A is singular with some or all of its strictly linearly independent rows are near-linearly dependent). This article highlights the scope and need of a randomized algorithm for ill-conditioned/singular systems when a reasonably narrow domain of a solution vector is specified. Further, it stresses that with the increasing computing power, the importance of randomized algorithms is also increasing. It also points out that, for many optimization linear/nonlinear problems, randomized algorithms are increasingly dominating the deterministic approaches and, for some problems such as the traveling salesman problem, randomized algorithms are the only alternatives.

Keywords

References

  1. S.K. Sen and S. Sen, Linear systems: relook, concise algorithms, and Matlab programs, Academic Studies - National Journal of Jyoti Research Academy1(1) (2007), 1-8.
  2. S.K. Sen and S.S. Prabhu, Optimal iterative schemes for computing Moore-Penrose matrix inverse, Internat. J. Systems Sci. 8 (1976), 748-753.
  3. E.A. Lord, V. Ch. Venkaiah, and S.K. Sen, A concise algorithm to solve under-/over-determined linear systems, Simulation 54 (1990), 239-240. https://doi.org/10.1177/003754979005400503
  4. M. Haahr, Introduction to randomness and random numbers, http://www.random.org/essay.html 1999.
  5. S. Galanti and A. Jung, Low-discrepancy sequences: Monte Carlo simulation of option prices, The Journal of Derivatives 5 (1997), 63-83. https://doi.org/10.3905/jod.1997.407985
  6. T. Samanta, Random Number Generators: MC Integration and TSP-solving by Simu- lated Annealing, Genetic and Ant System Approaches, Ph.D. thesis, Florida Institute of Technology (Department of Mathematical Sciences), Melbourne, Florida, USA 2006.
  7. A. Reese, Quasi- Versus Pseudo-random Numbers with Applications to Nonlinear Optimization, Ph.D. thesis, Florida Institute of Technology (Department of Mathematical Sciences), Melbourne, Florida, USA 2006.
  8. S.K. Sen, T. Samanta, and A. Reese, Quasi- versus pseudo-random generators: Discrepancy, complexity and integration-error based comparison, International Journal of Innovative Computing, Information and Control 2 3 (2006), 621-651.
  9. J.H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi dimensional integrals, Numerische Mathematik 2 (1960), 84-90. https://doi.org/10.1007/BF01386213
  10. I.M. Sobol, On the distribution of points in a cube and the approximate evaluation of integrals, U.S.S.R. Computational Mathematics and Mathematical Physics 7 (1967) 86- 112. https://doi.org/10.1016/0041-5553(67)90144-9
  11. H. Faure, Discrepance de suites associees a un systeme de numeration (en dimension s), Acta Arithmetica, XLZ (1982), 337-351.
  12. H. Niederreiter, Low discrepancy and low dispersion sequences, Journal of Number Theory 30 (1988),51-70. https://doi.org/10.1016/0022-314X(88)90025-X
  13. B.L. Fox, Algorithm 647: Implementation and relative efficiency of quasirandom sequence generators, ACM Transactions on Mathematical Software 12 4 (1986), 362-376. https://doi.org/10.1145/22721.356187
  14. P. Bratley, B.L. Fox, and H. Niederreiter, Algorithm 738: Programs to generate Nieder- reiter's low-discrepancy sequences, ACM Transactions on Mathematical Software 20 4 (1994), 494-495. https://doi.org/10.1145/198429.198436
  15. H. Niederreiter, Random number generation and quasi-Monte Carlo methods, SIAM, Pennsylvania, 1992.
  16. P.K. Sarkar and M.A. Prasad, A comparative study of pseudo and quasi random sequences for the solution of integral equations, Journal of Computational Physics 68 (1987), 66-88. https://doi.org/10.1016/0021-9991(87)90045-3
  17. P. Shirley, Discrepancy as a quality measure for simple distributions, Proceedings of Eurographics 91 (1991), 183-193.
  18. J. Struckmeier, Fast generation of low discrepancy sequences, Journal of Computational and Applied Mathematics, 61 (1995), 29-41. https://doi.org/10.1016/0377-0427(94)00054-5
  19. L. Kocis and W. Whiten, Computational investigations of low-discrepancy sequences, ACM Transactions on Mathematical Software 23 2 (1997), 266-294. https://doi.org/10.1145/264029.264064
  20. S.G. Henderson, B.A. Chiera, and R.M. Cooke, Generating "dependent" quasi-random numbers, Proceedings of the 32nd Conference on Winter Simulation, (2000), 527-536.
  21. S. Haupt and R. Haupt, it Practical Genetic Algorithms, Wiley, New York, 1998.
  22. M. Mascagni and A. Karaivanova, Matrix computations using quasirandom numbers, Springer Verlag Lecture Notes in Computer Science, 552-559, 2000.
  23. J. Banks, J. Carson, and B. Nelson, Discrete-event System Simulation (2nd edition), Prentice-Hall, New Jersey, 1996.
  24. T. Samanta and S.K. Sen, Pseudo- versus quasi-random generators in heuristics for traveling salesman problem (2008), to appear.
  25. M. Junger, G. Reinelt, and G. Renaldi, The traveling salesman problem, Operations Research and Management Sciences 7 (1995),225-330. https://doi.org/10.1016/S0927-0507(05)80121-5
  26. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, anf D.B. Shmoys, The Traveling Salesman Problem A Guided Tour of Combinatorial Optimization, Wiley, New York, 1985.
  27. G. Reinelt, it The Traveling Salesman: Computational Solutions for TSP Applications, Springer-Verlag, New York, 1994.
  28. Georgia Tech TSP page, http://www.tsp.gatech.edu/.
  29. W.L. Winston, Operations Research: Applications and Algorithms (4th Edition), Thomson, Belmont, California, 2004.
  30. M. Dorigo, Optimization, Learning, and Natural Algorithms, Ph.D. thesis (in Italian), Politecnico di Milano, Italy, 1992.
  31. M. Dorigo, V. Maniezzo, and A. Colorni, The ant system: Optimization by a colony of cooperating agents, IEEE Trans. On Systems, Man, and Cybernetics-Part B 26 1 (1996), 29-41. https://doi.org/10.1109/3477.484436
  32. L.M. Gambardella and M. Dorigo, Solving symmetric and asymmetric TSPs by ant colonies. Proceedings of the IEEE Conference on Evolutionary Computation, ICEC96, Nagoya, Japan, 622-627, 1996.
  33. H. Chi, M. Mascagni, and T. Warnock, On the optimal Halton sequence, Mathematics and Computers in Simulation 70(2005), 9-21. https://doi.org/10.1016/j.matcom.2005.03.004
  34. H. Niederreiter, Low-discrepancy and low dispersion sequences, Journal of Number Theory 30 (1988),51-70. https://doi.org/10.1016/0022-314X(88)90025-X
  35. P. Bratley, B.L. Fox, and H. Niederreiter, Implementation and test of low-discrepancy sequences, ACM Transections on Modeling and Computer Simulation 2 3 (1992), 195-213. https://doi.org/10.1145/146382.146385
  36. V. Lakshmikantham, S.K. Sen, and T. Samanta, Comparing random number generators using Monte Carlo integration, International Journal of Innovative Computing, Information, and Control 1 2 (2005), 143-165.
  37. D.E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison-Wesley, Reading, MA, 2nd Edition, 1981.
  38. S.K. Park and K.W. Miller, Random number generators: Good ones are hard to find, Communications of the ACM 31 (1988), 1192-1201. https://doi.org/10.1145/63039.63042
  39. MP-TESTDATA - The TSPLIB Symmetric Traveling Salesman Problem Instances, http://elib.zib.de/pub/Packages/mp-testdata/tsp/tsplib/tsp/ ,1995.
  40. A. Ralston and E.D. Reilly, Jr. eds., Encyclopedia of Computer Science and Engineering (2nd ed.), Van Nostrand Reinhold, New York, 1983.
  41. V. Lakshmikantham and S.K. Sen, Computational Error and Complexity in Science and Engineering, Elsevier, Amsterdam, 2005.
  42. E.V. Krishnamurthy and S.K. Sen it Introductory Theory of Computer Science, Affiliated East-West Press, New Delhi, 2004.
  43. Chidanand Rajghatta, The Times of India, TNN, India hosts world's fourth fastest supercomputer, The Times of India Daily News Paper, reported on November 13, 2007 at 2143 hours Indian Standard Time from Washington.
  44. B.Nobel and J.W. Daniel, Applied Linear Algebra (3rd ed), Prentice-Hall, New Jersey, 1998.
  45. W.L. Winston, Operations Research: Applications and Algorithms (4th ed), Thomson, Belmont, California, 2004.
  46. E.V. Krishnamurthy and S.K. Sen, Numerical Algorithms: Computations in Science and Engineering, Affiliated East-West Press, New Delhi, 2001.