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RISS 인기검색어
- 검색키워드
- 저자 : Gholam Ali Shaykhian (검색결과 4 건)
- 무료
- 기관 내 무료
- 유료
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Ill- versus well-posed singular linear systems: scope of randomized algorithms
S.K. Sen,Ravi P. Agarwal,Gholam Ali Shaykhian 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.3
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 suffi- ciently 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 reason- ably 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 domi- nating the deterministic approaches and, for some problems such as the traveling salesman problem, randomized algorithms are the only alterna- tives. 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 suffi- ciently 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 reason- ably 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 domi- nating the deterministic approaches and, for some problems such as the traveling salesman problem, randomized algorithms are the only alterna- tives.
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ILL-VERSUS WELL-POSED SINGULAR LINEAR SYSTEMS: SCOPE OF RANDOMIZED ALGORITHMS
Sen, S.K.,Agarwal, Ravi P.,Shaykhian, Gholam Ali The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.3
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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SHOULD PRUNING BE A PRE-PROCESSOR OF ANY LINEAR SYSTEM?
Sen, Syamal K.,Ramakrishnan, Suja,Agarwal, Ravi P.,Shaykhian, Gholam Ali The Korean Society for Computational and Applied M 2011 Journal of applied mathematics & informatics Vol.29 No.5
So far as a solution of the given consistent linear system is concerned many numerical methods - both mathematically non-iterative as well as iterative - have been reported in the literature over the last couple of centuries. Most of these methods consider all the equations including linearly dependent ones in the system and obtain a solution whenever it exists. Since linearly dependent equations do not add any new information to a system concerning a solution we have proposed an algorithm that identifies them and prunes them in the process of solving the system. The pruning process does not involve row/column interchanges as in the case of Gauss reduction with partial/complete pivoting. We demonstrate here that the use of pruning as an inbuilt part of our solution process reduces computational and storage complexities and also computational error.
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SHOULD PRUNING BE A PRE-PROCESSOR OF ANY LINEAR SYSTEM?
Syamal K. Sen,Suja Ramakrishnan,Ravi P. Agarwal,Gholam Ali Shaykhian 한국전산응용수학회 2011 Journal of applied mathematics & informatics Vol.29 No.5
So far as a solution of the given consistent linear system is concerned many numerical methods - both mathematically non-iterative as well as iterative - have been reported in the literature over the last couple of centuries. Most of these methods consider all the equations including linearly dependent ones in the system and obtain a solution whenever it exists. Since linearly dependent equations do not add any new information to a system concerning a solution we have proposed an algorithm that identifles them and prunes them in the process of solving the system. The pruning process does not involve row/column interchanges as in the case of Gauss reduction with partial/complete pivoting. We demonstrate here that the use of pruning as an inbuilt part of our solution process reduces computational and storage complexities and also computational error.
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