Abstract

APPENDIX B. TRILINOS 136
Minimum Residual (GMRES). Other packages within Trilinos provides incomplete
factorization preconditioning and multilevel preconditioning.
B.3.5 Anasazi
Anasazi is the eigensolver package within Trilinos. It currently uses a block implicitly
restarted Arnoldi method for the solve. This method finds the eigenvalues with largest
magnitude first. The linear solves needed by the eigensolver are also performed by
AztecOO.
B.4 Contribution
My main contribution to LOCA was the enhancement of LOCA’s stability calcula-
tions. The stability calculations involve solving the eigenproblem ˆ Jv = λv, where
ˆJ is the Jacobian matrix. Originally, LOCA had Anasazi solve the eigenproblem
ˆJ −1 v = λv. Now ˆ J −1 v = λv implies that 1
λ v = ˆ Jv. So the eigenvectors of ˆ J −1 are
the same as the eigenvectors of ˆ J, but the eigenvalues of ˆ J −1 are the inverses of the
eigenvalues of ˆ J. Since Anasazi first finds the eigenvectors of ˆ J −1 corresponding to
the largest magnitude eigenvalues, Anasazi will return eigenvectors of the Jacobian
ˆJ that have eigenvalues closest to the origin. This is a useful feature if one expects
turning point bifurcations in their continuation. The eigensolves will track the eigen-
values that are nearest to the origin, and will therefore find the eigenvalue that crosses
the real axis through the origin and creates the turning point.