LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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Perturbing a shifted Jacobian perturbs its inverse explicitly

Statement

Let $d \geq 1$ , $h, γ > 0$ , and $J, \tilde{J} \in R^{d \times d}$ . Set $A = I_{d} - γ h J$ and $\tilde{A} = I_{d} - γ h \tilde{J}$ . If both matrices are invertible, then ${\tilde{A}}^{- 1} - A^{- 1} = γ h {\tilde{A}}^{- 1} (\tilde{J} - J) A^{- 1} .$ Consequently, if $∥ A^{- 1} ∥_{2} \leq K$ and $∥ {\tilde{A}}^{- 1} ∥_{2} \leq \tilde{K}$ , then $∥ {\tilde{A}}^{- 1} - A^{- 1} ∥_{2} \leq γ h K \tilde{K} ∥ \tilde{J} - J ∥_{2} .$ Here the induced matrix norm is the one defined in A small shifted Jacobian has an explicit inverse bound.

Facts & Assumptions

Given: The stated matrices and inverse bounds.

[F1]

Induced matrix norms are compatible with matrix-vector multiplication, submultiplicative, and satisfy ||I|| = 1: the induced Euclidean matrix norm is submultiplicative: $∥ A B ∥_{2} \leq ∥ A ∥_{2} ∥ B ∥_{2}$ .

[F2]

A small shifted Jacobian has an explicit inverse bound: if $∥ γ h B ∥_{2} \leq q < 1$ , then $I - γ h B$ is invertible and its inverse norm is at most $(1 - q)^{- 1}$ .

Proof

Given: The data in the Statement.

1.1givenalgebra

Multiplying out the right-hand side gives ${\tilde{A}}^{- 1} (A - \tilde{A}) A^{- 1} = {\tilde{A}}^{- 1} A A^{- 1} - {\tilde{A}}^{- 1} \tilde{A} A^{- 1} = {\tilde{A}}^{- 1} - A^{- 1}$ . Also $A - \tilde{A} = γ h (\tilde{J} - J)$ , proving the identity.

2.1step 1.1F1algebra

Apply the induced-norm inequality in [F1] twice to the identity in step 1.1 and then use the two inverse bounds. This gives the claimed estimate.

3.1F2step 2.1algebra∎

If $∥ γ h J ∥_{2} \leq q < 1$ and $∥ γ h \tilde{J} ∥_{2} \leq \tilde{q} < 1$ , [F2] supplies $K = (1 - q)^{- 1}$ and $\tilde{K} = (1 - \tilde{q})^{- 1}$ , so the estimate is explicit under these sufficient smallness conditions.

Depends on

Used by

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Sources

WIAS Berlin, Numerical Methods for Stiff Ordinary Differential Equations, §2.4 Rosenbrock Methods (standard reference, not scraped)
Daniel Kressner, Advanced Numerical Analysis, §3.5 Rosenbrock methods (standard reference, not scraped)