Inexact Block Jacobi--Broyden Methods for Solving Nonlinear Systems of Equations

We describe a parallelizable Jacobi-type block Broyden algorithm for solving nonlinear systems of equations. We study the conditions under which the algorithm is locally convergent. Basically, the Jacobian matrix in a Newton algorithm is replaced by a block Broyden-like matrix which can be easily calculated by a recurrence relation. A family of nonlinear solvers could be generated combining it with some iterative or direct linear solvers. Different initializations for the block Broyden matrix are proposed, as well as several partitioning schemes, producing an effective block partitioning of the Jacobian matrix. Coupled with the restarted version of the linear GMRES(m) algorithm, the method parallelizes very well and gives good CPU time savings, which are shown by several numerical tests.