AbstractThe conjugate gradient (CG) method for optimization and equation solving is described, along with three principal families of algorithms derived from it. In each case, a foundational CG algorithm is formulated mathematically and followed by a brief discussion of refinements and variants within its family. The hierarchical nature of the subject matter is highlighted, because each foundational CG algorithm presented is a descendant of one earlier in the list from which it inherits fundamental algorithmic properties. Copyright © 2009 John Wiley & Sons, Inc.This article is categorized under: Algorithms and Computational Methods > Numerical Methods