Iterative Algorithms for Solution of Large Sparse Systems of Linear Equations on Hypercubes
IEEE Transactions on Computers
Leapfrog variants of iterative methods for linear algebraic equations
Journal of Computational and Applied Mathematics - Special issue on iterative methods for the solution of linear systems
The algebraic eigenvalue problem
The algebraic eigenvalue problem
s-step iterative methods for symmetric linear systems
Journal of Computational and Applied Mathematics
The symmetric eigenvalue problem
The symmetric eigenvalue problem
A Parallel Solution Method for Large Sparse Systems of Equations
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Parallel solution of a traffic flow simulation problem
Parallel Computing
Efficient biorthogonal Lanczos algorithm on message passing parallel computer
MTPP'10 Proceedings of the Second Russia-Taiwan conference on Methods and tools of parallel programming multicomputers
A study on the efficient parallel block lanczos method
CIS'04 Proceedings of the First international conference on Computational and Information Science
Communication-Efficient algorithms for numerical quantum dynamics
PARA'10 Proceedings of the 10th international conference on Applied Parallel and Scientific Computing - Volume 2
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The Lanczos algorithm is most commonly used in approximating a small number of extreme eigenvalues and eigenvectors for symmetric large sparse matrices. Main memory accesses for shared memory systems or global communications (synchronizations) in message passing systems decrease the computation speed. In this paper, the standard Lanczos algorithm is restructured so that only one synchronization point is required; that is, one global communication in a message passing distributed-memory machine or one global communication in a message passing distributed-memory machine or one global memory sweep in a shared-memory machine per each iteration is required. We also introduce the s-step Lanczos method for finding a few eigenvalues of symmetric large sparse matrices in a similar way to the s-step Conjugate Gradient method [2], and we prove that the s-step method generates reduction matrices which are similar to reduction matrices generated by the standard method. One iteration of the s-step Lanczos algorithm corresponds to s iterations of the standard Lanczos algorithm. The s-step method has improved data locality, minimized global communication and has superior parallel properties to the standard method. These algorithms are implemented on a 64-node NCUBE/seven hypercube and a CRAY-2, and performance results are presented.