Analysis of the Truncated Conjugate Gradient Method for Linear Matrix Equations
Valeria Simoncini, Yue Hao
Abstract
The matrix-oriented version of the conjugate gradient (CG) method can be used to approximate the solution to certain linear matrix equations. To limit memory consumption, low-rank reduction of the factored iterates is often employed, possibly leading to disruption of the regular convergence behavior. We analyze the properties of the method in the matrix regime and identify the quantities that are responsible for early termination, usually stagnation, when truncation is in effect. Moreover, we illustrate relations between CG and a projection technique directly applied to the same matrix equation.
Topics & Concepts
MathematicsConjugate gradient methodMatrix (chemical analysis)Conjugate residual methodDerivation of the conjugate gradient methodNonlinear conjugate gradient methodApplied mathematicsTruncation (statistics)Iterated functionRank (graph theory)Convergence (economics)Mathematical analysisMatrix functionProjection (relational algebra)Symmetric matrixMathematical optimizationAlgorithmGradient descentCombinatoricsStatisticsComputer scienceEigenvalues and eigenvectorsQuantum mechanicsMaterials scienceMachine learningPhysicsEconomic growthArtificial neural networkEconomicsComposite materialMatrix Theory and AlgorithmsAdvanced Optimization Algorithms ResearchModel Reduction and Neural Networks