TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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A bounded residual and derivative error give a local root certificate

Statement

For a linear map $A$ between Euclidean spaces, write $∥ A ∥_{o p} : = \sup_{∥ u ∥_{2} = 1} ∥ A u ∥_{2}$ . Let $m \geq 1$ , let $V \subseteq R^{m}$ be open, and let $G : V \to R^{m}$ be $C^{1}$ . Let $y \in V$ , $R > 0$ , and suppose the closed ball ${\overline{B}}_{2} (y, R)$ lies in $V$ . Assume that for some $L \geq 0$ , $∥ D G (u) - D G (v) ∥_{o p} \leq L ∥ u - v ∥_{2} (u, v \in {\overline{B}}_{2} (y, R)) .$ Let $\hat{g} \in R^{m}$ and $\hat{A} \in R^{m \times m}$ come with certified bounds $∥ \hat{g} - G (y) ∥_{2} \leq ϵ_{g}, ∥ \hat{A} - D G (y) ∥_{o p} \leq ϵ_{J},$ where $ϵ_{g}, ϵ_{J} \geq 0$ . Suppose $\hat{A}$ is invertible and $∥ {\hat{A}}^{- 1} ∥_{o p} \leq \hat{β}$ , with $\hat{β} ϵ_{J} < 1$ . Define $β : = \frac{\hat{β}}{1 - \hat{β} ϵ_{J}}, ρ : = ∥ \hat{g} ∥_{2} + ϵ_{g}, q : = β L R .$ If $q < 1$ and $β ρ \leq (1 - q) R$ , then $G$ has exactly one zero $y^{*}$ in ${\overline{B}}_{2} (y, R)$ and $∥ y^{*} - y ∥_{2} \leq \frac{β ρ}{1 - q} .$

Facts & Assumptions

Given: $G, V, y, R, L, \hat{g}, \hat{A}, ϵ_{g}, ϵ_{J}$ and $\hat{β}$ as in the Statement.

[F1]

A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point: a contraction of a nonempty complete metric space into itself has exactly one fixed point.

[F2]

$R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ : Euclidean $R^{m}$ is complete. A closed Euclidean ball is complete because every Cauchy sequence in it converges in $R^{m}$ and continuity of the norm keeps the limit in the ball.

[F3]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : the derivative of a composite is the composite of its derivatives.

[F4]

On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ : on a convex open set, a uniform derivative-norm bound implies the same Lipschitz bound on the map.

[F5]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: the Euclidean inner product satisfies Cauchy--Schwarz.

[F6]

For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ and Factorisation of $b^{n} - a^{n}$ , and the resulting Lipschitz estimate: if $0 \leq a < 1$ , then $a^{k} \to 0$ and $(1 - a) \sum_{j = 0}^{N} a^{j} = 1 - a^{N + 1}$ .

[F7]

Continuously differentiable maps, local inverses, and local diffeomorphisms: $C^{1}$ means total differentiability at every point and continuity of the first derivative matrix entries.

[F8]

The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder: the total derivative is the linear first-order approximation to a map at each point.

Proof

Given: The data and hypotheses in the Statement.

1.1givenalgebra

For matrices $X, Y$ , the definition of the operator norm gives $∥ X Y u ∥_{2} \leq ∥ X ∥_{o p} ∥ Y ∥_{o p} ∥ u ∥_{2}$ for every $u$ ; taking the supremum over unit $u$ proves submultiplicativity. Put $A : = D G (y)$ , $E : = A - \hat{A}$ , $P : = {\hat{A}}^{- 1} E$ , and $a : = \hat{β} ϵ_{J} < 1$ . Then $A = \hat{A} (I + P)$ and $∥ P^{k} ∥_{o p} \leq a^{k}$ by induction.

2.1F2F5F6step 1.1givenalgebra

The finite sums $S_{N} : = \sum_{j = 0}^{N} (- P)^{j}$ satisfy $(I + P) S_{N} = S_{N} (I + P) = I + (- 1)^{N} P^{N + 1}$ . By [F6], the tails are bounded by $a^{N + 1} / (1 - a) \to 0$ . Each matrix entry of $S_{N} - S_{M}$ is bounded in absolute value by $∥ S_{N} - S_{M} ∥_{o p}$ , by [F5] applied to standard unit vectors. Thus the entries are Cauchy; completeness [F2] gives a matrix limit $S$ . Taking entrywise limits in the finite matrix products gives $(I + P) S = S (I + P) = I$ . Also the finite-sum bound in [F6] gives $∥ S ∥_{o p} \leq (1 - a)^{- 1}$ by taking limits on each vector. Hence $A^{- 1} = S {\hat{A}}^{- 1}$ and $∥ A^{- 1} ∥_{o p} \leq β$ .

3.1F3F4F7F8givenstep 2.1algebra

Define $T (x) : = x - A^{- 1} G (x)$ on the ball. Its derivative is $D T (x) = A^{- 1} (A - D G (x))$ . For $x$ in the open ball, the Lipschitz hypothesis gives $∥ D T (x) ∥_{o p} \leq β L ∥ x - y ∥_{2} \leq q$ . The open ball is convex, so [F4] gives $∥ T (u) - T (v) ∥_{2} \leq q ∥ u - v ∥_{2}$ there; continuity extends this inequality to the closed ball.

4.1F1F2step 2.1step 3.1givenalgebra

Since $∥ T (y) - y ∥_{2} = ∥ A^{- 1} G (y) ∥_{2} \leq β ρ$ , every $x \in {\overline{B}}_{2} (y, R)$ satisfies $∥ T (x) - y ∥_{2} \leq ∥ T (x) - T (y) ∥_{2} + ∥ T (y) - y ∥_{2} \leq q R + β ρ \leq R .$ Thus $T$ maps the nonempty complete ball into itself. By [F1] it has a unique fixed point $y^{*}$ there, and invertibility of $A$ makes $T (y^{*}) = y^{*}$ equivalent to $G (y^{*}) = 0$ . Any zero of $G$ is a fixed point, so this zero is unique.

5.1F1step 3.1step 4.1givenalgebra∎

The contraction inequality and fixed-point identity give $∥ y^{*} - y ∥_{2} \leq ∥ T (y^{*}) - T (y) ∥_{2} + ∥ T (y) - y ∥_{2} \leq q ∥ y^{*} - y ∥_{2} + β ρ,$ which rearranges to the stated error bound because $1 - q > 0$ .

Remarks

For an augmented pseudo-arclength residual, use this certificate only when the ball lies in the local curve-chart neighborhood and the corrector hyperplane is transverse to the curve. Failure of either strict contraction condition gives no certificate; a finite Newton budget alone certifies neither existence nor nonexistence of a root.

Depends on

Used by

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66 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eugene L. Allgower and Kurt Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics 13 (1990) (standard reference, not scraped)
K. A. Cliffe, A. Spence, and S. J. Tavener, The numerical analysis of bifurcation problems with application to fluid mechanics, Acta Numerica 9 (2000), 39–131 (standard reference, not scraped)