Parallel implementation of the block conjugate gradient algorithm

Abstract The conjugate gradient algorithm is well-suited for vector computation but, because of its many synchronization points and relatively short message packets, is more difficult to implement for parallel computation. In this work we introduce a parallel implementation of the block conjugate gradient alhorithm. In this algorithm, we carry a block of vectors along at each iteration, reducing the number of iterations and increasing the length of each message. On machines with relatively costly message passing, this algorithm is a significant improvement over the standard conjugate gradient algorithm.

[1]  J. Cullum,et al.  A block Lanczos algorithm for computing the q algebraically largest eigenvalues and a corresponding eigenspace of large, sparse, real symmetric matrices , 1974, CDC 1974.

[2]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[3]  C. Paige.  Computational variants of the Lanczos method for the eigenproblem , 1972 .

[4]  Mark K. Seager,et al.  Parallelizing conjugate gradient for the CRAY X-MP , 1986, Parallel Comput..

[5]  N. Abdelmalek.  Round off error analysis for Gram-Schmidt method and solution of linear least squares problems , 1971 .

[6]  Richard R. Underwood.  An iterative block Lanczos method for the solution of large sparse symmetric eigenproblems , 1975 .

[7]  E. Wachspress.  Extended Application of Alternating Direction Implicit Iteration Model Problem Theory , 1963 .

[8]  G. Stewart,et al.  Assignment and scheduling in parallel matrix factorization , 1986 .

[9]  D. O’Leary.  The block conjugate gradient algorithm and related methods , 1980 .

[10]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[11]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[12]  Å. Björck.  Solving linear least squares problems by Gram-Schmidt orthogonalization , 1967 .

[13]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

[14]  O. Axelsson.  Solution of linear systems of equations: Iterative methods , 1977 .

[15]  J. Reid.  Large Sparse Sets of Linear Equations , 1973 .