Speaker : Sebastian Schlenkrich Abstract: Efficient similarity factorization of rank-1 modifications For the solution of stiff ODEs by implicit methods one has to solve sequences of algebraic systems, whose Jacobian is a multiple of the identity plus the 'system' Jacobian of the right hand side. The latter can be successivley approximated by a new 'two-sided rank one' formula. It has been observed to acheive an only moderate increase in the number of iterations compared to Newton's method. However, so far we have only been able to achieve the expected reduction in the linear algebra effort per step from O(n^3) to O(n^2) as long as the size of the integration time-step is kept constant. Otherwise one has to add varying multiples of the identity to the approximate system Jacobian and then solve linear systems in that matrix. To achieve this with an effort of O(n^2) per iteration seems to require updating a similarity reduction of the system Jacobian to Hessenberg or even diagonal form. As we have found no O(n^2) algorithms for updating such factorizations exactly in the literature, we discuss ideas to approximate the update with enough accuracy to still yield rapid convergence of the underlying implicit equation solver.