DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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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.
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The $m$ th Krylov Galerkin iterate in the affine space $x_{0} + K_{m} (A, r_{0})$ for a Hermitian positive-definite system

Definition

Let $A$ be Hermitian positive definite, let $A x = b$ , let $x_{0}$ be an initial guess, and put

$r_{0} : = b - A x_{0} .$

For $m \geq 0$ , an element $x_{m} \in x_{0} + K_{m} (A, r_{0})$ is the $m$ th Krylov Galerkin iterate when its residual

$r_{m} : = b - A x_{m}$

is orthogonal to $K_{m} (A, r_{0})$ in the standard inner product. The next proposition identifies this residual condition with $A$ -orthogonality of the error $x_{*} - x_{m}$ .

Depends on

Used by

Dependency tree · two levels

7 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)

The mmth Krylov Galerkin iterate in the affine space x0+Km(A,r0)x0​+Km​(A,r0​) for a Hermitian positive-definite system