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Conjugate gradient method

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or…

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Overview

Description of the problem addressed by conjugate gradients

Derivation as a direct method

As an iterative method

Convergence properties

The preconditioned conjugate gradient method

Vs. the locally optimal steepest descent method

Conjugate gradient method as optimal feedback controller for double integrator

Conjugate gradient on the normal equations

Conjugate gradient method for complex Hermitian matrices

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Conjugate gradient method

Nodes67
Edges66
Triples71
Avg. degree1.97
Density0.029851
Components1

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Conjugate gradient method

Top relations

related to The resulting algorithm · 10
Conjugate gradient method → Ab, And, CG, However, Krylov, Seemingly, That, The, Therefore, This
related to Conjugate gradient on the normal equations · 9
Conjugate gradient method → As, ATA, ATb, CGN, CGNR, Finding, However, The, Therefore
related to As an iterative method · 7
Conjugate gradient method → Ax, Az, If, So, Starting, This, We
related to Finite Termination Property · 6
Conjugate gradient method → In, It, Therefore, These, This, Under
related to Derivation as a direct method · 5
Conjugate gradient method → Arnoldi/Lanczos, Despite, The, These, We
related to Vs. the locally optimal steepest descent method · 5
Conjugate gradient method → However, In, SPD, Thus, With
see also · 5
Conjugate gradient method → BiCG, Biconjugate, CGS, Conjugate, Linear
related to Convergence properties · 4
Conjugate gradient method → As, In, Krylov, The
related to Practical convergence · 4
Conjugate gradient method → If, In, Krylov, The
related to The preconditioned conjugate gradient method · 4
Conjugate gradient method → If, In, It, The

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Important terminology

displaystyle mathbf method conjugate gradient solution matrix algorithm residual convergence symmetric used using system vector may exact preconditioner iterations positive-definite

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Conjugate gradient methodis aalgorithm for the numerical solution of particular systems of linear equations0.90text
the Cholesky decompositioninstance ofapplicable to sparse systems that are too large to be handled by a direct implementation or other direct methods0.80text
energy minimizationinstance ofLarge sparse systems often arise when numerically solving partial differential equations or optimization problems.The conjugate gradient method can also be used to solve unconst…0.80text
Conjugate gradient methodrelated to Advantages and disadvantagesThe0.60section
Conjugate gradient methodrelated to Advantages and disadvantagesNemirovsky0.60section
Conjugate gradient methodrelated to Advantages and disadvantagesBenTal0.60section
Conjugate gradient methodrelated to As an iterative methodIf0.60section
Conjugate gradient methodrelated to As an iterative methodSo0.60section
Conjugate gradient methodrelated to As an iterative methodThis0.60section
Conjugate gradient methodrelated to As an iterative methodWe0.60section
Conjugate gradient methodrelated to As an iterative methodAz0.60section
Conjugate gradient methodrelated to As an iterative methodAx0.60section

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