An Augmented Conjugate Gradient Method for Solving Consecutive Symmetric Positive Definite Linear Systems

Jocelyne Erhel (1) , Frédéric Guyomarc'H (1)

1 ALADIN - Algorithms Adapted to Intensive Numerical Computing

Many scientific applications require one to solve successively linear systems Ax = b with different right-hand sides b and a symmetric positive definite matrix A. The conjugate gradient method applied to the first system generates a Krylov subspace which can be efficiently recycled thanks to orthogonal projections in subsequent systems. A modified conjugate gradient method is then applied with a specific initial guess and initial descent direction and a modified descent direction during the iterations. This paper gives new theoretical results for this method and proposes a new version. Numerical experiments show the efficacy of our method even for quite different right-hand sides.

Mots clés: conjugate gradient; Krylov subspace; orthogonal projection

Domaines: Calcul parallèle, distribué et partagé [cs.DC]

Jocelyne Erhel, Frédéric Guyomarc'H. An Augmented Conjugate Gradient Method for Solving Consecutive Symmetric Positive Definite Linear Systems. SIAM Journal on Matrix Analysis and Applications, 2000, 21 (4), pp.1279-1299. ⟨10.1137/S0895479897330194⟩. ⟨inria-00523682⟩