A Novel Parallel Algorithm Based on the Gram-Schmidt Method for Tridiagonal Linear Systems of Equations

This paper introduces a new parallel algorithm based on the Gram-Schmidt orthogonalization method. This parallel algorithm can find almost exact solutions of tridiagonal linear systems of equations in an efficient way. The system of equations is partitioned proportional to number of processors, and each partition is solved by a processor with a minimum request from the other partitions' data. The considerable reduction in data communication between processors causes interesting speedup. The relationships between partitions approximately disappear if some columns are switched. Hence, the speed of computation increases, and the computational cost decreases. Consequently, obtained results show that the suggested algorithm is considerably scalable. In addition, this method of partitioning can significantly decrease the computational cost on a single processor and make it possible to solve greater systems of equations. To evaluate the performance of the parallel algorithm, speedup and efficiency are presented. The results reveal that the proposed algorithm is practical and efficient

Parallel Sparse Modified Gram-Schmidt QR Decomposition

This is a post-peer-review, pre-copyedit version of an article published in Lecture Notes in Computer Science. The final authenticated version is available online at: https://doi.org/10.1007/3-540-61142-8_609[Abstract] We present a parallel computational method for the QR decomposition with column pivoting of a sparse matrix by means of Modified Gram-Schmidt orthogonalization. Nonzero elements of the matrix M to be decomposed are stored in a one-dimensional doubly linked list data structure. We discuse a strategy to reduce fill-in in order to get memory savings and decrease the computation times. As an application of QR decomposition, we describe the least squares problem. This algorithm was designed for a message passing multiprocessor and has been evaluated on a Cray T3D, using the Harwell-Boeing sparse matrix collection.Comisión Interministerial de Ciencia y Tecnología; TIC92-0942-C03European Commision; ERB-CHGE-CT92-000

Parallel Sparse Modified Gram-Schmidt QR Decomposition

. We present a parallel algorithm for the QR decomposition with column pivoting of a sparse matrix by means of Modified GramSchmidt orthogonalization. Nonzero elements of the matrix M to decompose are stored in a one-dimensional doubly linked list data structure. A strategy to reduce fill-in is discussed to get memory savings and decrease the computation times. As an application of QR decomposition, we describe the least squares problem. This algorithm has been designed for a message passing multiprocessor and we evaluate it on the Cray T3D supercomputer using the Harwell-Boeing sparse matrix collection. 1 Introduction QR factorization is a direct method in matrix algebra which involves the decomposition of a matrix M of dimensions A \Theta B (A B) into the product of an orthogonal matrix Q (Q T = Q \Gamma1 ) and an upper triangular matrix R. QR decomposition has many applications in linear algebra to solve linear systems of equations, least squares problems (LSP), linear program..