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Everything about Conjugate gradient method

Mathematical optimization algorithm

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 other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Conjugate gradient methodA 5-minute orientation
  2. Skim a recent paper: An efficient hybrid conjugate gradient method for unconstrained optimizationInfluential work since 2022
  3. Pick up a classic book: Conjugate gradient type methods for ill-posed problemsThe most-reprinted book on the subject
  4. Read the landmark paper: Restart procedures for the conjugate gradient methodCited 1,379 times
  5. Go sideways: Biconjugate gradient methodA closely related topic
  6. Go sideways: Anderson accelerationA closely related topic

đź“– Definitions

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🗂️ The subject at a glance

  1. Description of the problem addressed by conjugate gradients
  2. Derivation as a direct method
  3. As an iterative method
  4. Finite Termination Property
  5. Convergence properties
  6. The preconditioned conjugate gradient method
  7. The flexible preconditioned conjugate gradient method
  8. Vs. the locally optimal steepest descent method
  9. Conjugate gradient method as optimal feedback controller for double integrator
  10. Conjugate gradient on the normal equations
  11. Conjugate gradient method for complex Hermitian matrices
  12. Advantages and disadvantages

🏆 Landmark papers (most cited)

🆕 Influential since 2022

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