Alphabeta MathHow 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.
In floating-point arithmetic, CG can lose exact residual orthogonality, -conjugacy, and the finite-termination guarantee
Remark
The orthogonality and finite-termination theorems on this page are exact arithmetic statements. In floating-point arithmetic, rounding errors perturb the three-term recurrence, so later residuals need not remain exactly orthogonal and later search directions need not remain exactly -conjugate. Once those identities drift, the argument that forces termination by the relative grade no longer applies verbatim. Practical CG often still converges well, but the exact algebraic structure is only approximate.
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Dependency tree · two levels
8 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
- Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems (standard reference, not scraped)
- Jonathan Richard Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain (standard reference, not scraped)