# zbMATH — the first resource for mathematics

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:
$x_{n+1}=x_{n}\cdot \exp(-df^{-1}_{x_{n}}f(x_{n}))$
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$$(N,\mathbb R)$$.

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators 58C15 Implicit function theorems; global Newton methods on manifolds 34G20 Nonlinear differential equations in abstract spaces
Full Text:
##### References:
 [1] Adler, R.; Dedieu, J.P.; Margulies, J.; Martens, M.; Shub, M., Newton method on Riemannian manifolds and a geometric model for human spine, IMA J. numer. anal., 22, 1-32, (2002) [2] Edelman, A.; Arias, T.A.; Smith, T., The geometry of algorithms with orthogonality constraints, SIAM J. matrix anal. appl., 20, 303-353, (1998) · Zbl 0928.65050 [3] Gabay, D., Minimizing a differentiable function over a differential manifold, J. optim. theory appl., 37, 177-219, (1982) · Zbl 0458.90060 [4] Smith, S.T., Optimization techniques on Riemannian manifolds, (), 113-146 · Zbl 0816.49032 [5] S.T. Smith, Geometric optimization method for adaptive filtering, Ph. D. Thesis, Harvard University, Cambridge, MA, (1993) [6] Udriste, C., () [7] Kantorovich, L.V., On Newton method for functional equations, Dokl. acad. nauk., 59, 1237-1240, (1948) [8] Kantorovich, L.V.; Akilov, G.P., Functional analysis, (1982), Pergamon Oxford · Zbl 0484.46003 [9] Ferreira, O.P.; Svaiter, B.F., Kantorovich’s theorem on newton’s method in Riemannian manifolds, J. complexity., 18, 304-329, (2002) · Zbl 1003.65057 [10] Blum, L.; Cucker, F.; Shub, M.; Smale, S., Complexity and real computation, (1997), Springer-Verlag New York [11] Smale, S., Newton’s method estimates from data at one point, (), 185-196 [12] Dedieu, J.P.; Priouret, P.; Malajovich, G., Newton’s method on Riemannian manifolds: covariant alpha theory, IMA J. numer. anal., 23, 395-419, (2003) · Zbl 1047.65037 [13] Li, C.; Wang, J.H., Newton’s method on Riemannian manifolds: smale’s point estimate theory under the $$\gamma$$-condition, IMA J. numer. anal., 26, 228-251, (2006) · Zbl 1094.65052 [14] Wang, X.H.; Han, D.F., Criterion $$\alpha$$ and newton’s method in weak condition, Chinese J. numer. appl. math., 19, 96-105, (1997) [15] Alvarez, F.; Bolte, J.; Munier, J., A unifying local convergence result for newton’s method in Riemannian manifolds, Found. comput. math., 8, 197-226, (2008) · Zbl 1147.58008 [16] Mahony, R.E., The constrained Newton method on a Lie group and the symmetric eigenvalue problem, Linear algebra appl., 248, 67-89, (1996) · Zbl 0864.65032 [17] Mahony, R.E.; Manton, J., The geometry of the Newton method on non-compact Lie groups, J. global optim., 23, 309-327, (2002) · Zbl 1019.22005 [18] Munthe-Kaas, H., High order runge – kutta methods on manifold, Appl. numer. math., 29, 115-127, (1999) · Zbl 0934.65077 [19] Owren, B.; Welfert, B., The Newton iteration on Lie groups, BIT, numer., 40, 121-145, (2000) · Zbl 0957.65054 [20] Varadarajan, V.S., () [21] Warner, F.W., () [22] DoCarmo, M.P., Riemannian geometry, (1992), Birkhauser Boston [23] Wang, X.H., Convergence on the iteration of Halley family in weak conditions, Chinese sci. bull., 42, 552-555, (1997) · Zbl 0884.30004 [24] Wang, X.H.; Han, D.F., On the dominating sequence method in the point estimates and smale’s theorem, Scientia sinica ser. A., 33, 135-144, (1990) · Zbl 0699.65046 [25] Helgason, S., Differential geometry Lie groups symmetric spaces, (1978), Academic Press Inc. New York · Zbl 0451.53038 [26] Hall, B.C., ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.