Machine learningKrylov Subspace Iterative

Conjugate Gradient Method

The Conjugate Gradient (CG) Method is an iterative algorithm for solving large sparse symmetric positive-definite linear systems Ax = b, developed by Hestenes and Stiefel in 1952. It is one of the most widely used iterative solvers in scientific computing because it converges in at most n iterations for an n × n matrix and typically requires far fewer.

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Sources

  1. Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6), 409–436. DOI: 10.6028/jres.049.044
  2. Saad, Y. (2003). Iterative Methods for Sparse Linear Systems (2nd ed.). SIAM. DOI: 10.1137/1.9780898718003
  3. Nocedal, J., & Wright, S. J. (2006). Numerical Optimization (2nd ed.). Springer. DOI: 10.1007/978-0-387-40065-5

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Referenced by

ScholarGateConjugate Gradient Method (Conjugate Gradient Method for Linear Systems). Retrieved 2026-06-04 from https://scholargate.app/en/numerical-methods/conjugate-gradient-method