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Then there exist a matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that spec(A) = {[[mu].sub.1],..., [[mu].sub.n]} and the iterates of the restarted Arnoldi method with restart length m do not converge to g(A)b.
It is easily possible to construct the matrix A in such a way that the (implicitly generated) FOM(m) iterates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for t in some interval [[t.sub.1], [t.sub.2]] diverge (by prescribing increasing residual norms), and one would surely expect the corresponding Arnoldi approximation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for g(A)b to inherit this behavior and diverge in this case as well.
The error norm is therefore decreasing initially, until the FOM iterates for the underlying (implicitly solved) linear systems with large t all have converged.
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the errors of the FOM(m) iterates for the systems (4.2).
The jth FOM iterate for the linear system (1.1) is then given by
The resulting approximation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is then used as an additive correction to the iterate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e.,
This approach can be continued until the resulting iterate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] fulfills a prescribed stopping criterion (e.g., a residual norm below some given tolerance).
Precisely, provided that the jth FOM iterate for Ax = b is defined, it holds
In step 6, we compute a next iterate [x.sub.+] by performing the standard backtracking line search global strategy described in Algorithm 2.2.
If the user sets the variable INFORM to 1, then the package uses reverse communication to obtain the multiplication of the Hessian matrix at the current iterate by a given vector.
For example, what's the ultimate behavior of chaotic trajectories when there are infinitely many iterates? For a given function, do different trajectories always form roughly the same pattern of dots?
If the same step sizes are chosen, and the corrector step is not performed, then the same iterate [x.sup.1] = [[bar.y].sup.1] is obtained.