GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Scientific computing: an introduction with parallel computing
Scientific computing: an introduction with parallel computing
Iterative solution methods
Matrix computations (3rd ed.)
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
A parallel solution of large-scale heat equation based on distributed memory hierarchy system
ICA3PP'10 Proceedings of the 10th international conference on Algorithms and Architectures for Parallel Processing - Volume Part II
| Hi-index | 0.00 |
The conjugate gradient method has been used extensively to solve a given set of (symmetric) positive definite linear equations instead of Gaussian's elimination based direct methods, especially for very large systems when parallel solution environment is preferred. The method itself can take many iterations to converge. In order to reduce the number of iterations, i.e., in order to accelerate the rate of convergence of the method, the set of linear equations at hand is preconditioned. We present the implementation results of a previously published approximate inverse preconditioner for the conjugate gradient method. The preconditioner is based on an approximate inverse of the coefficient matrix obtained from a linear combination of matrix-valued Chebyshev polynomials. We implement and test the proposed method on a Sun SMP machine. Because the preconditoner itself contains mainly matrix-matrix products and conjugate gradient method contains mostly matrix-vector products, convincing results are obtained in terms of both speedup and scalability.