🗊Презентация A block version of Gmres, Bicg, Bicgstab for linear systems with multiple right-hand sides

A BLOCK VERSION OF GMRES, BICG, BICGSTAB FOR LINEAR SYSTEMS WITH MULTIPLE RIGHT-HAND SIDES

Research aim: Research a block version of GMRES, BICG, BICGSTAB for linear systems with multiple right-hand side.

The need for computer modeling of increasingly complex structures has led to the need to solve large linear systems. The need for computer modeling of increasingly complex structures has led to the need to solve large linear systems. All methods for solving linear systems can be divided into two classes: direct and iterative. Methods that lead to the solution for a finite number of arithmetic operations. Iterative methods are called, which should be obtained as a result of infinite repetition. Many iterative methods are based on an iteration loop that accesses the coefficient matrix A once per loop and performs a matrix-vector multiply. To reduce data movement in the algorithm, our approach is to modify the algorithm such that more than one matrix-vector product occurs for a single memory access of A. To this end, we investigated alternatives for solving a single right-hand side system based on solving a corresponding block linear system AX = B, where X and B are both groups of vectors.

Algorithm B-LGMRES

Algorithm Bl-BIC

Algorithm Bl-BICGSTAB

We compared the performance (in term of flops) of the Bl-BiCGSTAB algorithm, the Bl-GMRES algorithm and the BiCGSTAB algorithm applied to each single right-hand side. For all the tests the matrix was an random matrix. In Table 2 we list the effectiveness of Bl-BiCGSTAB measured by the ratios and where We compared the performance (in term of flops) of the Bl-BiCGSTAB algorithm, the Bl-GMRES algorithm and the BiCGSTAB algorithm applied to each single right-hand side. For all the tests the matrix was an random matrix. In Table 2 we list the effectiveness of Bl-BiCGSTAB measured by the ratios and where =flops(Bl-BiCGSTAB)/(flops(BiCGSTAB)) =flops(Bl-BiCGSTAB)/(flops(Bl-GMRES))

Comparison of various block methods for solving linear systems

Сonclusion The block methods are of great practical value in applications involving linear systems with multiple right-hand sides. However, they are not as well studied from the theoretical point of view.