Abstract

APPENDIX D. GMRES AND ARNOLDI ITERATIVE METHODS 151
matrix-free which means the matrix D and its inverse D −1 are not computed or stored.
Therefore, iterative methods can get a good approximate solution to these subprob-
lems for less computer memory and time than direct methods that must store and
calculate the matrix D and its inverse D −1 . Both of the iterative methods considered
find approximate solutions in a Krylov subspace. This description of these methods
are based on [19] (linear solver) and [25] (eigensolver). Note that this appendix does
not give an exhaustive look at these methods, and one should read the mentioned
references if one wants a more thorough explanation of the methods.
We will first consider the linear solver GMRES for the Newton-step linear sub-
problems of the form
Da = b,
where D ∈ R M×M is the preconditioned Jacobian matrix. For solving this linear
system, one starts with an initial iterate a 0 ∈ R M and computes the initial residual
r 0 = b − Da 0 . Then the j-dimensional Krylov subspace, denoted by Kj, used in
solving the linear system is defined as
Kj =span{r 0 ,Dr 0 ,D 2 r 0 ,...,D j−1 r 0 }.
At the j-th GMRES iteration, GMRES finds the linear least-squares solution of the
given linear subproblem over the j-th dimensional Krylov subspace Kj. Leta j ∈ Kj