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Title: A matrix analysis of conjugate gradient algorithms

This paper explores the relationships between the conjugate gradient algorithms Orthodir, Orthomin, and Orthores. To facilitate this exploration, a matrix formulation for each algorithm is given. It is shown that Orthodir directly computes a Hessenberg matrix H{sub k} at step k. Orthores also computes a Hessenberg matrix, G{sub k}, which is similar to a Hessenberg matrix obtained from H{sub k} by perturbing its last column. (This perturbation vanishes at convergence.) Orthomin, on the other hand, computes a UL and LU factorization of the perturbed H{sub k} and G{sub k}, respectively. The breakdown of Orthomin and Orthores are interpreted in terms of these underlying matrix factorizations. A connection with Lanczos algorithms is also examined, as is the special case of B-normal(1) matrices (for which efficient three-term CG algorithms exist).
Authors:
 [1] ;  [2]
  1. Lawrence Livermore National Lab., CA (United States)
  2. Eidgenoessische Technische Hochschule, Zurich (Switzerland)
Publication Date:
OSTI Identifier:
10193408
Report Number(s):
UCRL-JC--113560; CONF-9303237--1
ON: DE94002546
DOE Contract Number:
W-7405-ENG-48
Resource Type:
Conference
Resource Relation:
Conference: 9. symposium on precondition conjugate gradients,Yokohama (Japan),2 Mar 1993; Other Information: PBD: Apr 1993
Research Org:
Lawrence Livermore National Lab., CA (United States)
Sponsoring Org:
USDOE, Washington, DC (United States)
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ITERATIVE METHODS; ALGORITHMS; MATHEMATICS; MATRICES; ORTHOGONAL TRANSFORMATIONS; EIGENVALUES 990200; MATHEMATICS AND COMPUTERS