Enlarging the convergence ball
of Newton’s method on Lie groups
February 16, 2013.
Dedicated to prof. I. Păvăloiu on the occasion of his 75th anniversary
We present a local convergence analysis of Newton’s method for approximating a zero of a mapping from a Lie group into its Lie algebra. Using more precise estimates than before [ 55 , 56 ] and under the same computational cost, we obtain a larger convergence ball and more precise error bounds on the distances involved. Some examples are presented to further validate the theoretical results.
MSC. 65G99, 65J15, 47H17, 49M15, 90C90.
Keywords. Newton’s method, Lie group, Lie algebra, Riemannian manifold, convergence ball, Kantorovich hypothesis.
1 Introduction
In this paper, we are concerned with the problem of approximating a zero
The study of numerical algorithms on manifolds for solving eigenvalue or optimization problems on Lie groups [ 1 ] – [ 12 ] , [ 33 ] – [ 36 ] , [ 54 ] – [ 57 ] is very important in Computational Mathematics. Newton-type methods are the most popular iterative procedures used to solve equations, when these equations contain differentiable operators. The study about convergence matter of Newton’s method is usually centered on two types: semilocal and local convergence analysis. The semilocal convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of Newton’s method; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls. There is a plethora of studies on the weakness and/or extension of the hypothesis made on the underlying operators; see for example (cf. [ 7 , 11 , 16 ] and references theiren). A local as well as a semilocal convergence of Newton-type methods has been given by several authors under various conditions [ 2 ] – [ 58 ] . Recently in [ 15 ] , we presented a finer semilocal convergence analysis for Newton’s method than in earlier studies [ 7 , 32 , 34 , 36 , 54 , 55 , 56 ] .
Newton’s method with initial point
Newton’s method is undoubtedly the most popular method for generating a sequence
Larger convergence ball;
and
Tighter error bounds on the distances involved.
The necessary background on Lie groups can be found in [ 7 , 11 , 15 , 16 ] and the references therein. The paper is organized as follows. In Section 2, we present the local convergence analysis of Newton’s method. Finally, numerical examples are given in the concluding Section 3.
2 Local convergence analysis for Newton’s method
We shall study the semilocal/local convergence of Newton’s method. In the rest of the paper we assume
By convention
the open ball centered at
We need the definition of Lipschitz continuity for a mapping.
Let
holds for any
holds for any
We then say
holds in general and
Let us show that (2.3) indeed implies (2.4). If (2.3) holds, then for
There exist
Using (2.3) and the triangle inequality, we obtain in turn that
which implies (2.4).
We need a Banach type lemma on invertible mappings.
Suppose that
It follows from (2.7) and the Banach lemma on invertible operators
[
7
]
,
[
32
]
that
Next, we study the convergence domain of Newton’s method around a zero
Let
for all
We also have the following identity
In view of (2.10) and (2.12), we get that
Moreover, by (2.11) and (2.13), we obtain that
That completes the proof of Lemma 2.3.
The proof of Lemma 2.3 reduces to
[
56
,
Lemma 3.2
]
if
Then, it is can easily be seen that
We have the local convergence result for Newton’s method.
Let
Set
We shall show mapping
Using Lemma 2.2, we have that
Set
Then, by (2.16), we obtain that
and
by the choice of
That completes the proof of Theorem 2.5.
The proofs of remaining results in this Section are omitted, since they can be obtained from the corresponding ones in [ 56 ] .
Let
Let
Let
Let
Let
for each
with and some .Linear mapping
exists and
This assertion follows from the hypothesis that
satisfies the -Lipschitz condition at on .Let
, , . Then, we have . Using the -Lipschitz condition, we get in turn thatwhich shows (ii).
We have by (ii) that
Hence,
exists andWe have that
Hence, we get that
Therefore, we obtain that
We have that
. That is, we can getThe result now follows from (2.20), (2.21) and
The proof of Lemma 2.10 is complete.
Let us define parameter
We have that
Let
and
The proof of Theorem 2.11 is complete.
Let
Let
and
Then, sequence
Let
Let
(a) The local results reduce to the corresponding ones in
[
56
]
if
(b) The local results if
(c) It is obvious that finer results can be immediately obtained if similar conditions as the semilocal case (see
[
15
,
Section 1,
L⋆
instead
L0
]
) are used instead of Kantorovich condition for
3 Numerical examples
In this Section we present two numerical examples in the more general setting of a nonlinear equation on a Hilbert space where
Let
Let
Then, the Fréchet derivative of
Notice that we have
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