Abstract

CHAPTER 2. TEMPORAL INTEGRATION 35
Then the Krylov subspace for this problem at iteration j is
Kj =span(r 0 ,Dr 0 ,D 2 r 0 ,...,D j−1 r 0 ).
So the jth Krylov iterate is the solution of the least squares problem. GMRES
terminates after reducing the relative residual by a specific tolerance η ∈ (0, 1], that
is
b − Da j 2
r 0 2
For the case of using GMRES to solve for the Newton step sm,whereD is the Jacobian
matrix F ′ (zm), b = F (zm), and the initial iterate is a 0 = 0, then this reduces to the
inexact-Newton condition. The standard theorem for convergence of GMRES [19] is
given by
Theorem 2.3. Let D be a nonsingular matrix in R M×M . Then GMRES will converge
to the solution of the linear equation Da = b for y, b ∈ R M in at most M iterations.
At each iteration j, aj dimensional least squares problem is solved to find a j ,and
we must compute and store a basis for the Krylov space Kj. Appendix D explains
how this j dimensional least squares problem is formed. The storage cost is j vectors,
and the computation of the new basis vector is through a matrix-vector product
of D with a vector. The advantage of using GMRES is that it does not require
the coefficient matrix, but only a way of evaluating the matrix-vector product of
the coefficient. Therefore this method is called matrix-free. In particular, we are
≤ η