TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

M-node collocation has local defect order M

Statement

Let $M \geq 1$ , let $0 \leq c_{1} < \dots < c_{M} \leq 1$ be fixed distinct nodes, and fix an exact IVP solution $y$ on an interval containing $[t, t + h_{0}]$ for some $h_{0} > 0$ , with initial state $u = y (t)$ . Let $D \subseteq R \times R^{d}$ be open with $f : D \to R^{d}$ an IVP vector field on a neighborhood of a fixed compact convex tube. Assume each scalar coordinate of $f$ has continuous partial derivatives through total order $M$ , bounded on the tube, and is $L$ -Lipschitz in state there. Smoothness and derivative statements for vector-valued functions are componentwise; scalar Taylor is applied to each coordinate. Assume the exact path, stages, collocation polynomial, and comparison segments stay in the tube. In particular, the composite $g_{h} (s) = f (t + h s, y (t + h s))$ satisfies $∥ g_{h}^{(M)} ∥_{\infty} \leq C_{F} h^{M}$ uniformly; the stated componentwise smoothness and bounded exact-path derivatives imply this bound. Assume directly that for every sufficiently small $h > 0$ the collocation stage equation has a unique solution in the tube. Then its endpoint map has uniform exact-start local defect $O (h^{M + 1})$ as $h ↓ 0$ . The constant depends on the fixed nodes, including their Lebesgue constant $Λ_{M}$ ; no uniform-in- $M$ estimate is asserted.

Facts & Assumptions

Given: An exact IVP solution on $[t, t + h]$ , its collocation nodes, the stated smoothness and tube assumptions, and the small-step collocation branch.

[F1]

First-order systems, initial value problems, and solutions on intervals: the exact path satisfies $y^{'} (s) = f (s, y (s))$ .

[F2]

Collocation tableau from distinct quadrature nodes: $Q_{i j} = \int_{0}^{c_{i}} ℓ_{j} (s) d s$ , $b_{j} = \int_{0}^{1} ℓ_{j} (s) d s$ , and the endpoint is $u + h \sum_{j} b_{j} F_{j} (U)$ .

[F3]

Collocation quadrature data are well defined: the node integrals, $Q$ , $b$ , and $Λ_{M}$ are finite.

[F4]

Multivariable Taylor formula with a Lagrange remainder along a line segment: scalar Taylor's formula on the compact convex tube gives a uniform remainder for each coordinate, controlled by the next bounded derivative.

[F5]

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , and A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: vector derivatives and integrals are componentwise; applying the scalar FTC to each coordinate gives exact path increments, and continuous coordinate derivatives are Riemann integrable.

[F6]

Chain sum product and composition rules for Banach derivatives and Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : the one-variable coordinate paths use the $C^{M}$ convention of the scalar smoothness definition, and repeated chain and product rules apply componentwise to $g_{h}$ ; smoothness of $f$ is the partial-derivative assumption in the Statement.

[F7]

Cardinal-polynomial interpolation and Cardinal interpolation functions are polynomials of the stated degrees give the cardinal sum $I_{M}$ , its degree bound $M - 1$ , and the nodal values $I_{M} g_{h} (c_{j}) = g_{h} (c_{j})$ ; Existence and uniqueness of the cardinal-polynomial interpolant, with $n = M - 1$ , supplies uniqueness for these $M$ nodal data.

[A1]

The exact trajectory, exact stage values, collocation stages and the segments used in the Lipschitz estimates lie in the common tube; the constants are uniform as $h ↓ 0$ .

[A2]

The composite $g_{h}$ has the uniform derivative bound in the Statement. This follows by repeated chain and product rules: every normalized-time derivative contributes a factor $h$ , while derivatives of $f$ and the exact path are bounded on the fixed tube.

Proof

1.1A2F2F3F4F6F7algebra

Put $g_{h} (s) = f (t + h s, y (t + h s))$ . The exact path derivatives used here are bounded on the compact tube: $y^{'} = f (t, y)$ is bounded there, and differentiating this identity successively expresses each higher derivative of each scalar coordinate of $y$ as a finite sum of products of bounded partial derivatives of $f$ and already bounded lower derivatives of $y$ . This induction gives bounded derivatives through order $M$ under the stated componentwise $C^{M}$ hypothesis. Repeated chain rules [F6] then show coordinatewise $∥ g_{h}^{(M)} ∥_{\infty} \leq C h^{M}$ : every differentiation in normalized time contributes one factor $h$ . Let $p_{M - 1}$ be the degree- $(M - 1)$ Taylor polynomial of each coordinate of $g_{h}$ at $0$ . By [A2] and scalar Taylor's remainder [F4], $∥ g_{h} - p_{M - 1} ∥_{\infty} \leq C h^{M}$ . For each real component polynomial $p$ of degree at most $M - 1$ , $p$ and $I_{M} p$ have that degree bound and the same $M$ nodal data, so uniqueness in [F7] gives $I_{M} p = p$ . Thus interpolation reproduces $p_{M - 1}$ componentwise by [F7]. The cardinal formula and [F3] give $∥ I_{M} v ∥_{\infty} \leq Λ_{M} ∥ v ∥_{\infty}$ coordinatewise, hence $∥ g_{h} - I_{M} g_{h} ∥_{\infty} \leq (1 + Λ_{M}) C h^{M} .$

2.1F1F2F5F7step 1.1algebra

Let $Y_{i} = y (t + c_{i} h)$ be the exact stage values and $u = y (t)$ . Coordinatewise FTC and the interpolation values in [F7] give their collocation residual $r_{i} = Y_{i} - u - h \sum_{j} Q_{i j} f (t + c_{j} h, Y_{j}) = h \int_{0}^{c_{i}} (g_{h} (s) - I_{M} g_{h} (s)) d s .$ Step 1.1 yields $∥ r ∥_{\infty, 2} \leq C_{M} h^{M + 1}$ , where $C_{M}$ includes $1 + Λ_{M}$ .

3.1A1F2F3step 2.1algebra

The fixed matrix $Q$ is finite by [F3], so for fixed $L$ and sufficiently small $h$ , $h L ∥ Q ∥_{\infty} \leq 1 / 2$ . Subtract the exact-stage residual equation from the collocation equation for the unique branch $U^{*}$ assumed in the Statement; the row-sum norm and state Lipschitz bound give $∥ Y - U^{*} ∥_{\infty, 2} \leq ∥ r ∥_{\infty, 2} + h L ∥ Q ∥_{\infty} ∥ Y - U^{*} ∥_{\infty, 2} .$ Thus $∥ Y - U^{*} ∥_{\infty, 2} \leq C_{M} h^{M + 1} / (1 - h L ∥ Q ∥_{\infty})$ .

4.1A1A2F2F5F7step 1.1step 3.1algebra∎

Applying the scalar FTC to each coordinate gives $y (t + h) = u + h \int_{0}^{1} g_{h} (s) d s$ by [F5]. Also $\int_{0}^{1} I_{M} g_{h} (s) d s = \sum_{j} b_{j} g_{h} (c_{j})$ by [F2] and [F7]. Add and subtract this interpolant integral and the endpoint evaluated at exact stages to obtain $y (t + h) - Ψ_{h} (u) = h \int_{0}^{1} (g_{h} - I_{M} g_{h}) (s) d s + h \sum_{j} b_{j} (f (t + c_{j} h, Y_{j}) - f (t + c_{j} h, U_{j}^{*})) .$ The first term is $O (h^{M + 1})$ by step 1.1. The second is bounded by $h L (\sum_{j} ∣ b_{j} ∣) ∥ Y - U^{*} ∥_{\infty, 2} = O (h^{M + 2})$ by step 3.1, so the local defect is $O (h^{M + 1})$ .

Depends on

Used by

Dependency tree · two levels

86 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

Mathew F. Causley and David C. Seal, On the convergence of spectral deferred correction methods, Communications in Applied Mathematics and Computational Science 14 (2019), §2.1, Lemma 2.1 (standard reference, not scraped)
Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner, Solving Ordinary Differential Equations I, 2nd ed., Chapter II §II.7, Theorem 7.10 (standard reference, not scraped)