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Smale’s point estimate theory for Newton’s method on Lie groups. (English) Zbl 1170.65040

Many problems from applied mathematics can be formulated as computing zeroes of mappings or vector fields on a Riemannian manifold. The best known of them is the Newton’s method, another one is Smale’s $\alpha$- and $\gamma$-theory. On the other hand, some numerical problems such as symmetric eigenvalue problems, ordinary differential equations on manifolds can be considered as problems on Lie groups.

In the present paper the authors study the convergence for a variant of the Newton’s method defined as follows:

${x}_{n+1}={x}_{n}·exp\left(-d{f}_{{x}_{n}}^{-1}f\left({x}_{n}\right)\right)$

where $df$ is the derivative of $f$ and is defined in terms of the exponential map and independent of the Riemannian metric. Unlike the notion of the $\gamma$-condition on the Riemann manifolds, this condition is defined via the one parameter subgroup and so is independent of the Riemannian connection. They establish the generalized $\alpha$- and $\gamma$-theory. As an application two initial value problems are presented on the special orthogonal group SO$\left(N,ℝ\right)$.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 58C15 Implicit function theorems and global Newton methods on manifolds 34G20 Nonlinear ODE in abstract spaces
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