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Local convergence of Newton’s method on Lie groups and uniqueness balls. (English) Zbl 1307.65079

The article deals with the analysis of the Newton method for approximate solving nonlinear equations \(f(x) = 0\) on Lie groups. More precisely, let \(f:\;G \to {\mathfrak g}\) be a \(C^1\) a map from a Lie group \(G\) into \({\mathfrak g}\) the Lie algebra of \(G\). The Newton approximations \(x_n\) (\(n = 0,1,2,\dots\)) are defined as follows: \[ x_{n+1} = x_n \cdot \exp (-df_{x_n}^{-1}f(x_n)), \quad n = 0,1,2,\dots, \;x_0 \in G, \] (\(\exp:\;{\mathfrak g} \to G\) is the corresponding exponential mapping). It is assumed that \(T(x) = df_x^{-1} \circ f\) satisfies the following condition (the modified Kantorovich condition) \[ \|T(x \cdot \exp u) - T(x)\| \leq \int_r^{r + \|u\|} L(s) \, ds \quad (\rho(x_0,x) \leq r); \] here \(\|\cdot\|\) is the norm on \({\mathfrak g}\) associated with the inner product on \({\mathfrak g}\), the metric \(\rho(\cdot,\cdot)\) is defined by \[ \rho(x,y) =\inf \bigg\{\sum_{i=1}^k \|u_i\|:\;\text{there exists} \;k \geq 1 \;\text{and} \;u_1,\dots,u_k \in {\mathfrak g} \;\text{such that} \;y = x \cdot \exp u_1 \cdots \exp u_k\bigg\}. \] Also, it is assumed that \(f(x^*) = 0\). Under these conditions, the estimate for the radius of the unity ball is \(N(x^*,r) = x^* \exp (B(0,r_u))\) and for the radius of the convergence ball \(C(x^*,r_c)\). The special cases, when \(L(\cdot) = L\) (the usual Kantorovich condition) and \(L(s) = \dfrac{2\gamma}{(1 - \gamma s)^3}\) \(\bigg(0 < s < \dfrac1\gamma\bigg)\) are considered.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
22E30 Analysis on real and complex Lie groups
58C15 Implicit function theorems; global Newton methods on manifolds

Citations:

Zbl 1170.65040
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References:

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