TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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 differences have first- and second-order error bounds

Statement

Let $I \subseteq R$ be open, $x \in I$ , and $h > 0$ . For the forward bound assume $[x, x + h] \subset I$ , $f \in C^{2}$ on a neighborhood of that interval, and $∣ f^{''} (y) ∣ \leq M_{2}$ there. For the central bound assume $[x - h, x + h] \subset I$ , $f \in C^{3}$ on a neighborhood of that interval, and $∣ f^{'''} (y) ∣ \leq M_{3}$ there. Then

$∣ D_{h}^{+} f (x) - f^{'} (x) ∣ \leq \frac{M_{2} h}{2}, ∣ D_{h}^{0} f (x) - f^{'} (x) ∣ \leq \frac{M_{3} h^{2}}{6} .$

If each function evaluation has absolute error at most $δ \geq 0$ as in Forward and central difference stencils with bounded evaluation errors, then

$∣ {\hat{D}}_{h}^{+} f (x) - f^{'} (x) ∣ \leq \frac{M_{2} h}{2} + \frac{2 δ}{h}, ∣ {\hat{D}}_{h}^{0} f (x) - f^{'} (x) ∣ \leq \frac{M_{3} h^{2}}{6} + \frac{2 δ}{h} .$

For the central stencil the evaluation contribution is in fact at most $δ / h$ ; the displayed $2 δ / h$ is a uniform conservative bound for both stencils.

Facts & Assumptions

Given: The interval, step, smoothness, derivative bounds and evaluation-error hypotheses in the statement.

[F1]

Forward and central difference stencils with bounded evaluation errors: $D_{h}^{+} f (x) = (f (x + h) - f (x)) / h$ and $D_{h}^{0} f (x) = (f (x + h) - f (x - h)) / (2 h)$ ; each inexact value is $f (y) + e_{y}$ with $∣ e_{y} ∣ \leq δ$ .

[F2]

The Lagrange and Cauchy forms of Taylor's remainder and Taylor polynomials and their remainders: for a $C^{n + 1}$ function on the interval, $f (y)$ is its degree- $n$ Taylor polynomial at $a$ plus $f^{(n + 1)} (ξ) (y - a)^{n + 1} / (n + 1)!$ for some intermediate $ξ$ .

[F3]

Absolute value in an ordered field and Basic properties of the absolute value: absolute value is nonnegative, and $∣ - r ∣ = ∣ r ∣$ for real $r$ .

[F4]

The triangle inequality: $∣ r + s ∣ \leq ∣ r ∣ + ∣ s ∣$ for real $r, s$ .

Proof

Given: The hypotheses listed above.

1.1givenF1F2algebra

Apply [F2] with $n = 1$ , $a = x$ , and $y = x + h$ . Then $f (x + h) = f (x) + h f^{'} (x) + h^{2} f^{''} (ξ) / 2$ for some $ξ \in (x, x + h)$ . Substituting into [F1] gives $D_{h}^{+} f (x) - f^{'} (x) = h f^{''} (ξ) / 2$ , so the forward truncation bound follows.

1.2givenF1F2F3F4algebra

Apply [F2] with $n = 2$ to $y = x + h$ and $y = x - h$ . Subtracting the two expansions cancels the equal even-order terms and gives $D_{h}^{0} f (x) - f^{'} (x) = h^{2} (f^{'''} (ξ_{+}) + f^{'''} (ξ_{-})) / 12$ for intermediate points $ξ_{+} \in (x, x + h)$ and $ξ_{-} \in (x - h, x)$ . The derivative bound and [F3]–[F4] give magnitude at most $M_{3} h^{2} / 6$ .

2.1givenF1F3F4step 1.1algebra

Write the forward evaluation errors as $e_{+} = e_{x + h}$ and $e_{0} = e_{x}$ . By [F1], ${\hat{D}}_{h}^{+} - D_{h}^{+} = (e_{+} - e_{0}) / h$ ; [F3]–[F4] bound its magnitude by $2 δ / h$ . The triangle inequality with step 1.1 proves the forward inexact bound.

3.1givenF1F3F4step 1.2algebra∎

Write the central errors as $e_{+} = e_{x + h}$ and $e_{-} = e_{x - h}$ . Then ${\hat{D}}_{h}^{0} - D_{h}^{0} = (e_{+} - e_{-}) / (2 h)$ , whose magnitude is at most $δ / h$ by [F1], [F3] and [F4], hence at most $2 δ / h$ . Combining with step 1.2 proves the stated central inexact bound.

Depends on

Used by

Dependency tree · two levels

18 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)