  We describe a Lagrange-Newton framework for the derivation of learning rules
with desirable convergence properties and apply it to the case of principal
component analysis (PCA). In this framework, a Newton descent is applied to an
extended variable vector which also includes Lagrange multipliers introduced
with constraints. The Newton descent guarantees equal convergence speed from
all directions, but is also required to produce stable fixed points in the
system with the extended state vector. The framework produces "coupled" PCA
learning rules which simultaneously estimate an eigenvector and the
corresponding eigenvalue in cross-coupled differential equations. We
demonstrate the feasibility of this approach for two PCA learning rules, one
for the estimation of the principal, the other for the estimate of an arbitrary
eigenvector-eigenvalue pair (eigenpair).
