Variable metric conjugate gradient methods

Conference · Fri Jul 01 04:00:00 EDT 1994

OSTI ID:10165883

Barth, T; Manteuffel, T

1.1 Motivation. In this paper we present a framework that includes many well known iterative methods for the solution of nonsymmetric linear systems of equations, Ax = b. Section 2 begins with a brief review of the conjugate gradient method. Next, we describe a broader class of methods, known as projection methods, to which the conjugate gradient (CG) method and most conjugate gradient-like methods belong. The concept of a method having either a fixed or a variable metric is introduced. Methods that have a metric are referred to as either fixed or variable metric methods. Some relationships between projection methods and fixed (variable) metric methods are discussed. The main emphasis of the remainder of this paper is on variable metric methods. In Section 3 we show how the biconjugate gradient (BCG), and the quasi-minimal residual (QMR) methods fit into this framework as variable metric methods. By modifying the underlying Lanczos biorthogonalization process used in the implementation of BCG and QMR, we obtain other variable metric methods. These, we refer to as generalizations of BCG and QMR.

• Research Organization: Los Alamos National Lab., NM (United States)
• Sponsoring Organization: USDOE, Washington, DC (United States)
• DOE Contract Number: W-7405-ENG-36
• OSTI ID: 10165883
• Report Number(s): LA-UR--94-1602; CONF-9403142--1; ON: DE94014796
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200
CONVERGENCE
HERMITIAN MATRIX
ITERATIVE METHODS
MATHEMATICS AND COMPUTERS
NUMERICAL SOLUTION