Solving Linear Systems
Frédéric Magoulès, François‐Xavier Roux, Guillaume Houzeaux · 2015
Within the scientific codes, the solution of large linear systems is often the most expensive, both in terms of memory and computation time. This chapter presents an introduction to the two categories of resolution methods used for solving large linear systems: direct methods and iterative methods. Direct methods are based on the initial factorization of the matrix and easy-to-solve products of matrices by a direct algorithm, in other words, an algorithm that requires no transformation of the matrix and which gives the result in a number of exactly predictable operations. The disadvantage of direct methods lies in the high costs in terms of the number of arithmetic operations, and also, in terms of the memory footprint of the matrices that were initially sparse, from the initial phase of factorization. The advantage of iterative methods is that they involve matrix-vector operations, and require nothing more than non-zero coefficients of the matrix.