GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Hybrid Krylov methods for nonlinear systems of equations
SIAM Journal on Scientific and Statistical Computing
On nonlinear generalized conjugate gradient methods
Numerische Mathematik
SIAM Journal on Numerical Analysis
SIAM Journal on Matrix Analysis and Applications
Evaluating derivatives: principles and techniques of algorithmic differentiation
Evaluating derivatives: principles and techniques of algorithmic differentiation
Testing Unconstrained Optimization Software
ACM Transactions on Mathematical Software (TOMS)
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
CUTEr and SifDec: A constrained and unconstrained testing environment, revisited
ACM Transactions on Mathematical Software (TOMS)
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The solution of large scale nonlinear problems can be tackled by solving successive small scale nonlinear problems on subspaces of the original space. We propose here a natural method to construct small dimensional subspaces of the original space, that is an extension to the nonlinear case of Krylov spaces. In the linear case, we show that the Krylov spaces are the subspaces generated by the successive derivatives at the origin of a particular mapping. An extension of this description leads to the definition of algorithms that generalize to the nonlinear case the relaxation methods and GMRES. These methods are also generalizations to a higher degree of Newton's method and Newton's method with line search, and a local fast convergence rate is obtained. Numerical simulations are conducted in two test cases from the CUTEr test set.