Abstract

CHAPTER 3. BIFURCATION ANALYSIS 56
3.6 Householder Continuation
This method based on [37] was implemented in LOCA. The idea is to create an embed-
ding E : R M → R M+1 such that for each y ∈ R M , Ey2 = y2 and Ey is orthogonal
to the subspace created by the span of ˜t. Householder projections achieve this in a
computationally efficient fashion. Also, Ey satisfies the one dimensional constraint
(3.7) automatically for all y ∈ R M , and thus the constraint (3.7) is automatically
built into each Newton step. Let Q : RM+1 → RM be the matrix Q = W ′ (f) ∂W
. ∂V
This is the Jacobian matrix of W with respect to the both the state variable f and
the parameter V . Now the composition QE is a linear operator that maps vectors
in R M to vectors in R M . The idea is to solve the linear system QE˜s = −W ( f,V)
for ˜s ∈ R M and call its embedding the Newton step s = E˜s. An algorithm of this
method is as follows:
• Compute embedding E given tangent vector ˜t
• Use GMRES to solve QE˜s = −W ( f,V) for ˜s
• Set the Newton step s to E˜s.
This method requires only the construction of E (which can be done quickly), and one
linear solve with the coefficient matrix of QE. Since the bordering algorithms require
two linear solves, we expect a factor of two speed up from using the Householder
continuation instead of the bordering algorithm. Further, the linear solves within the
Householder continuation will be better conditioned than the linear solves within the