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Newton maps are uniform contractions near a point with invertible derivative

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be open, let $f : U \to R^{n}$ be $C^{1}$ , and let $a \in U$ . Suppose $A : = D f (a)$ is invertible and put $B : = A^{- 1}$ . Then there are $R > 0$ , $0 \leq q < 1$ , and $C > 0$ such that $\overline{B} (a, R) \subseteq U$ , $∥ B v ∥_{2} \leq C ∥ v ∥_{2}$ , and, for every $y \in R^{n}$ , the Newton map

$T_{y} (x) : = x + B (y - f (x))$

satisfies

$∥ T_{y} (x) - T_{y} (z) ∥_{2} \leq q ∥ x - z ∥_{2} (x, z \in \overline{B} (a, R)) .$

Moreover $D f (x)$ is invertible for every $x \in \overline{B} (a, R)$ and

$∥ D f (x)^{- 1} v ∥_{2} \leq \frac{C}{1 - q} ∥ v ∥_{2} .$

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 $∥ L h ∥_{2} \leq K ∥ h ∥_{2}$ for some $K \geq 0$ ).

[L2]

The entries of $D f$ 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: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ ).

[L4]

Every $R^{n}$ , $n \geq 1$ , is complete, so its contractions have unique fixed points ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ , 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 $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , 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 $∥ B v ∥_{2} \leq C ∥ v ∥_{2}$ . Matrix-entry continuity [L2], [L6], and [L5] give $R > 0$ such that $\overline{B} (a, R) \subset B (a, 2 R) \subseteq U$ and $∥ B (D f (w) - A) v ∥_{2} \leq \frac{1}{2} ∥ v ∥_{2}$ for $w \in B (a, 2 R)$ and $v \in R^{n}$ . Fix $q : = 1 / 2$ .

L1L2L5L6algebra

2.1

The chain rule gives $D T_{y} (w) = I - B D f (w) = B (A - D f (w))$ , independently of $y$ . The convex open ball $B (a, 2 R)$ contains the closed ball, so [L3] and step 1.1 yield $∥ T_{y} (x) - T_{y} (z) ∥_{2} \leq q ∥ x - z ∥_{2}$ for $x, z \in \overline{B} (a, R)$ .

step 1.1L3

3.1

Fix $w$ in the ball, put $L : = D f (w)$ , and fix $v \in R^{n}$ . The map $S_{v} (u) : = u + B (v - L u)$ is a contraction of the complete space $R^{n}$ 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 $L u = v$ . Thus $L$ is surjective. If $L u = 0$ , 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 $L u = v$ and step 1.1, $∥ u ∥_{2} \leq ∥ u - B L u ∥_{2} + ∥ B v ∥_{2} \leq q ∥ u ∥_{2} + C ∥ v ∥_{2}$ . Therefore $∥ L^{- 1} v ∥_{2} \leq C (1 - q)^{- 1} ∥ v ∥_{2}$ .

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

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Sources

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