Solving a positive definite system of linear equations via the matrix exponential

We present a new direct algorithm for solving a system of linear equations with a positive definite matrix by discretizing a continuous-time dynamical system for a large sampling time. The obtained algorithm is highly fine-grain parallelizable and its computational complexity grows logarithmically with respect to the condition number of the system of linear equations. When the parallelism is fully exploited, the algorithm is shown to be more efficient in terms of computational speed in comparison to other popular methods for solving a positive definite system of linear equations, especially for large and ill-conditioned problems.

[1]  George A. Constantinides,et al.  A High Throughput FPGA-based Floating Point Conjugate Gradient Implementation , 2008, ARC.

[2]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[3]  George A. Constantinides,et al.  An FPGA-based implementation of the MINRES algorithm , 2008, 2008 International Conference on Field Programmable Logic and Applications.

[4]  J. Ortega.  Stability of Difference Equations and Convergence of Iterative Processes , 1973 .

[5]  Timothy A. Davis,et al.  The university of Florida sparse matrix collection , 2011, TOMS.

[6]  Paul Tseng.  Control perspectives on numerical algorithms and matrix problems , 2008, Math. Comput..

[7]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[8]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[9]  Thomas A. Manteuffel,et al.  A Comparison of Adaptive Chebyshev and Least Squares Polynomial Preconditioning for Hermitian Positive Definite Linear Systems , 1992, SIAM J. Sci. Comput..

[10]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[11]  Graham C. Goodwin,et al.  Digital control and estimation : a unified approach , 1990 .

[12]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[13]  C. Loan.  Computing integrals involving the matrix exponential , 1978 .

[14]  Moody T. Chu,et al.  Linear algebra algorithms as dynamical systems , 2008, Acta Numerica.

[15]  Jacques Laminie,et al.  Differential equations and solution of linear systems , 2005, Numerical Algorithms.

[16]  George A. Constantinides,et al.  A High Throughput FPGA-Based Floating Point Conjugate Gradient Implementation for Dense Matrices , 2010, TRETS.

[17]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[18]  George A. Constantinides,et al.  Tutorial paper: Parallel architectures for model predictive control , 2009, 2009 European Control Conference (ECC).

[19]  Husheng Li,et al.  High Performance Reconfigurable Computing for Cholesky Decomposition , 2009 .