## 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

Zbl 1170.65040
Full Text:

### References:

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