Abstract

APPENDIX B. TRILINOS 138
A very useful property [9] of this transformation is that if we choose ¯σ = −¯µ, ¯µ>0,
then the Cayley transformation maps the eigenvalues of the Jacobian with negative
real part to the interior of the unit circle in the complex-plane and maps the eigenval-
ues of the Jacobian with positive real part to the exterior of the unit circle. Therefore,
if our Jacobian matrix has eigenvalues with positive real part, then Anasazi would
quickly detect them when solving the Cayley transformed eigenproblem.
Another extension made to LOCA’s stability calculation was the ability to handle
a mass matrix ˆ M. This frequently occurs in a finite element discretization, where
numerically one will solve the ODE
ˆM dx
dt
= F (x)
where x ∈ R N and F : R N → R N is a nonlinear equation with ˆ J as its Jacobian
matrix. The eigenvalues that determine the stability of the steady-state solution of
this equation are calculated from the generalized eigenproblem ˆ Jv = λ ˆ Mv.
There are analogous shift-invert and Cayley transformations for the generalized
eigenproblem. The generalized shift-invert transformation is given by ˜ B = ( ˆ J −
¯µ ˆ M) −1 , and the generalized Cayley transformation is given by ˜ C =( ˆ J − ¯µ ˆ M) −1 ( ˆ J −
¯σ ˆ M). Here, Anasazi solves the transformed generalized eigenproblem ˜ Bv = ɛs ˆ Mv
or ˜ Cv = ɛc ˆ Mv. These generalized transformations have the same properties as the
regular shift-invert and Cayley transformations previously mentioned.
My contributions to LOCA were: