LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The one-stage update has an explicit Jacobian-approximation defect

Statement

Use the one-stage map of The Rosenbrock–Euler step map with an exact or approximate Jacobian with the same $t, y, h, γ, f$ but matrices $J$ and $\tilde{J}$ . If $I_{d} - γ h J$ and $I_{d} - γ h \tilde{J}$ are invertible, with inverse norms bounded by $K$ and $\tilde{K}$ , then $∥ Φ_{γ, h}^{\tilde{J}} (t, y) - Φ_{γ, h}^{J} (t, y) ∥_{2} \leq γ h^{2} K \tilde{K} ∥ \tilde{J} - J ∥_{2} ∥ f (t, y) ∥_{2} .$ This is a one-step perturbation bound; it alone gives no uniform stability or convergence claim for a sequence of approximate Jacobians.

Facts & Assumptions

Given: The one-stage maps and inverse bounds of the Statement.

[F1]

The Rosenbrock–Euler step map with an exact or approximate Jacobian: $Φ_{γ, h}^{J} (t, y) = y + h (I_{d} - γ h J)^{- 1} f (t, y)$ when the shifted matrix is invertible.

[F2]

Perturbing a shifted Jacobian perturbs its inverse explicitly: the two shifted inverses differ in norm by at most $γ h K \tilde{K} ∥ \tilde{J} - J ∥_{2}$ .

Proof

Given: The data in the Statement.

1.1F1algebra

Subtract the two update formulas in [F1]. The common $y$ cancels, leaving $h [(I_{d} - γ h \tilde{J})^{- 1} - (I_{d} - γ h J)^{- 1}] f (t, y)$ .

2.1step 1.1F2algebra∎

Apply [F2] to the matrix difference in step 1.1 and the induced-norm inequality to its product with $f (t, y)$ . Multiplication by the outer $h$ gives the stated $γ h^{2} K \tilde{K} ∥ \tilde{J} - J ∥_{2} ∥ f (t, y) ∥_{2}$ bound.

Depends on

Used by

Rosenbrock–Euler has an explicit one-step defect and solve-residual allowance Lemma

Dependency tree · two levels

8 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

WIAS Berlin, Numerical Methods for Stiff Ordinary Differential Equations, §2.4 Rosenbrock Methods (standard reference, not scraped)
Joachim Rang and Lutz Angermann, New Rosenbrock methods of order 3 for PDAEs of index 2 (standard reference, not scraped)