• 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


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.



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