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Newton maps are uniform contractions near a point with invertible derivative

Statement

Let $n\ge1$ , let $U\subseteq\mathbb R^n$ be open, let $f:U\to\mathbb R^n$ be $C^1$ , and let $a\in U$ . Suppose $A:=Df(a)$ is invertible and put $B:=A^{-1}$ . Then there are $R>0$ , $0\le q<1$ , and $C>0$ such that $\overline B(a,R)\subseteq U$ , $\|Bv\|_2\le C\|v\|_2$ , and, for every $y\in\mathbb R^n$ , the Newton map

$T_y(x):=x+B(y-f(x))$

satisfies

$\|T_y(x)-T_y(z)\|_2\le q\|x-z\|_2\qquad(x,z\in\overline B(a,R)).$

Moreover $Df(x)$ is invertible for every $x\in\overline B(a,R)$ and

$\|Df(x)^{-1}v\|_2\le\frac C{1-q}\|v\|_2.$

Facts & Assumptions

Given: The dimensions, open set, $C^1$ map, point, and invertible derivative in the statement.

[L1]

Euclidean linear maps have finite matrices and a global norm bound (Every Euclidean linear map has a unique matrix and satisfies $\|Lh\|_2\le K\|h\|_2$ for some $K\ge0$ ).

[L2]

The entries of $Df$ are continuous by the definition of $C^1$ (Continuously differentiable maps, local inverses, and local diffeomorphisms).

[L3]

Total derivatives obey the linear algebra and chain rules, and a uniform derivative bound on a convex open set gives the corresponding Lipschitz bound (Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives, The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ , On a convex open set, a uniform bound $\|Df(z)v\|_2\le M\|v\|_2$ implies $\|f(y)-f(x)\|_2\le M\|y-x\|_2$ ).

[L4]

Every $\mathbb R^n$ , $n\ge1$ , is complete, so its contractions have unique fixed points ( $\mathbb{R}$ and $\mathbb{R}^n$ for $n \ge 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $\mathbb{R}$ , A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point).

[L5]

Metric balls have their open and closed forms, and openness supplies a closed ball about $a$ contained in $U$ (Open ball, closed ball and sphere in a metric space, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

[L6]

The Euclidean norm satisfies the finite-dimensional Cauchy--Schwarz estimate (Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$ , the parallelogram law and polarisation).

[L7]

Invertibility means having a two-sided linear inverse (Invertible Euclidean linear maps).

Proof

technique · contraction

1.1

By [L1], choose $C>0$ with $\|Bv\|_2\le C\|v\|_2$ . Matrix-entry continuity [L2], [L6], and [L5] give $R>0$ such that $\overline B(a,R)\subset B(a,2R)\subseteq U$ and $\|B(Df(w)-A)v\|_2\le\tfrac12\|v\|_2$ for $w\in B(a,2R)$ and $v\in\mathbb R^n$ . Fix $q:=1/2$ .

L1L2L5L6algebra

2.1

The chain rule gives $DT_y(w)=I-BDf(w)=B(A-Df(w))$ , independently of $y$ . The convex open ball $B(a,2R)$ contains the closed ball, so [L3] and step 1.1 yield $\|T_y(x)-T_y(z)\|_2\le q\|x-z\|_2$ for $x,z\in\overline B(a,R)$ .

step 1.1L3

3.1

Fix $w$ in the ball, put $L:=Df(w)$ , and fix $v\in\mathbb R^n$ . The map $S_v(u):=u+B(v-Lu)$ is a contraction of the complete space $\mathbb R^n$ with constant $q$ , by the same estimate as step 2.1. By [L4] it has a unique fixed point $u$ , and the fixed-point equation is equivalent to $Lu=v$ . Thus $L$ is surjective. If $Lu=0$ , both $u$ and $0$ are fixed by $S_0$ , so uniqueness gives $u=0$ ; hence $L$ is injective. The solution map is linear by uniqueness, so it is $L^{-1}$ .

step 1.1L4L7

4.1

From $Lu=v$ and step 1.1, $\|u\|_2\le\|u-BLu\|_2+\|Bv\|_2 \le q\|u\|_2+C\|v\|_2$ . Therefore $\|L^{-1}v\|_2\le C(1-q)^{-1}\|v\|_2$ .

step 1.1step 3.1algebra

5.1

Steps 1.1--4.1 give every asserted constant, contraction estimate, invertibility claim, and inverse bound.

step 1.1step 2.1step 3.1step 4.1∎

Depends on

Used by

The Euclidean inverse function theorem Theorem

Dependency tree · next 3 levels

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Sources

J. Lebl, Basic Analysis II, §8.5 Inverse and implicit function theorems (standard reference, not scraped)