DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Forward and central difference stencils with bounded evaluation errors

Definition

Let $I \subseteq R$ be open, $x \in I$ , and $h > 0$ . When $x + h \in I$ , the forward difference of $f : I \to R$ is

$D_{h}^{+} f (x) : = \frac{f (x + h) - f (x)}{h} .$

When $x - h, x + h \in I$ , the central difference is

$D_{h}^{0} f (x) : = \frac{f (x + h) - f (x - h)}{2 h} .$

An evaluation with absolute error bound $δ \geq 0$ is a value $\hat{f} (y) = f (y) + e_{y}$ with $∣ e_{y} ∣ \leq δ$ at each requested point $y$ . The corresponding inexact quotients are obtained by replacing each $f (y)$ in the displayed stencil by $\hat{f} (y)$ . The bound is absolute and applies separately to each call; no correlation or relative-error assumption is included.

Depends on

Absolute value in an ordered field

Used by

Dependency tree · two levels

2 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

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, finite differences (standard reference, not scraped)
Andreas Buttenschoen, Introduction to Scientific Computing, Numerical Differentiation (standard reference, not scraped)