DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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The conjugate-gradient recurrence

Definition

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

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

For every index $k$ with $r_{k} \neq 0$ , define

$α_{k} : = \frac{⟨ r_{k}, r_{k} ⟩}{⟨ A p_{k}, p_{k} ⟩}, x_{k + 1} : = x_{k} + α_{k} p_{k}, r_{k + 1} : = r_{k} - α_{k} A p_{k},$

and then

$β_{k} : = \frac{⟨ r_{k + 1}, r_{k + 1} ⟩}{⟨ r_{k}, r_{k} ⟩}, p_{k + 1} : = r_{k + 1} + β_{k} p_{k} .$

This is the conjugate-gradient recurrence.

Depends on

Used by

Dependency tree · two levels

5 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)