AlgorithmAlgorithm: AI-adaptedProof: 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.

Fourier diagonal solver for the periodic centered-difference Poisson equation

Definition

Input an odd $N \geq 3$ periodic grid as in Periodic uniform PIC grid and inner product and a real grid source $s$ . If $\overset{ˉ}{s} \neq 0$ , return the status incompatible-source. Otherwise compute the finite discrete Fourier coefficients ${\hat{s}}_{k} = N^{- d} \sum_{j} s_{j} e^{- 2 π i k \cdot j / N}$ . Set ${\hat{ϕ}}_{0} = 0$ and, for each $k \neq 0$ , set ${\hat{ϕ}}_{k} = {\hat{s}}_{k} / λ_{h} (k)$ with $λ_{h} (k)$ from Periodic centered-difference Poisson solvability and zero-mean gauge. Return the inverse finite Fourier transform $ϕ_{j} = \sum_{k \in K_{N}} {\hat{ϕ}}_{k} e^{2 π i k \cdot j / N}$ and status success. All arithmetic here is exact real or complex arithmetic; an implementation using floating-point FFTs requires a separate perturbation analysis.

Depends on

Used by

Dependency tree · one level

2 results within one dependency step 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

Benjamin Seibold, MIT 18.336 Lecture 5, Finite Difference Approximation (standard reference, not scraped)
A. Myers, P. Colella, and B. Van Straalen, A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov–Poisson Equation, Section 2.2 (standard reference, not scraped)