skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Generalized conjugate gradient squared

Abstract

In order to solve non-symmetric linear systems of equations, the Conjugate Gradient Squared (CGS) is a well-known and widely used iterative method. In practice the method converges fast, often twice as fast as the Bi-Conjugate Gradient method. This is what you may expect, since CGS uses the square of the BiCG polynomial. However, CGS may suffer from its erratic convergence behavior. The method may diverge or the approximate solution may be inaccurate. BiCGSTAB uses the BiCG polynomial and a product of linear factors in an attempt to smoothen the convergence. In many cases, this has proven to be very effective. Unfortunately, the convergence of BiCGSTAB may stall when a linear factor (nearly) degenerates. BiCGstab({ell}) is designed to overcome this degeneration of linear factors. It generalizes BiCGSTAB and uses both the BiCG polynomial and a product of higher order factors. Still, CGS may converge faster than BiCGSTAB or BiCGstab({ell}). So instead of using a product of linear or higher order factors, it may be worthwhile to look for other polynomials. Since the BiCG polynomial is based on a three term recursion, a natural choice would be a polynomial based on another three term recursion. Possibly, a suitable choice of recursion coefficientsmore » would result in method that converges faster or as fast as CGS, but less erratic. It turns out that an algorithm for such a method can easily be formulated. One particular choice for the recursion coefficients leads to CGS. Therefore one could call this algorithm generalized CGS. Another choice for the recursion coefficients leads to BiCGSTAB. It is therefore possible to mix linear factors and some polynomial based on a three term recursion. This way one may get the best of both worlds. The authors will report on their findings.« less

Authors:
;  [1]
  1. Utrecht Univ. (Netherlands)
Publication Date:
Research Org.:
Front Range Scientific Computations, Inc., Boulder, CO (United States); USDOE, Washington, DC (United States); National Science Foundation, Washington, DC (United States)
OSTI Identifier:
219554
Report Number(s):
CONF-9404305-Vol.2
ON: DE96005736; TRN: 96:002321-0001
Resource Type:
Conference
Resource Relation:
Conference: Colorado conference on iterative methods, Breckenridge, CO (United States), 5-9 Apr 1994; Other Information: PBD: [1994]; Related Information: Is Part Of Colorado Conference on iterative methods. Volume 2; PB: 261 p.
Country of Publication:
United States
Language:
English
Subject:
99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; MATRICES; ITERATIVE METHODS; ASYMMETRY; CONVERGENCE; COMPUTER CALCULATIONS

Citation Formats

Fokkema, D.R., and Sleijpen, G.L.G. Generalized conjugate gradient squared. United States: N. p., 1994. Web.
Fokkema, D.R., & Sleijpen, G.L.G. Generalized conjugate gradient squared. United States.
Fokkema, D.R., and Sleijpen, G.L.G. 1994. "Generalized conjugate gradient squared". United States. doi:. https://www.osti.gov/servlets/purl/219554.
@article{osti_219554,
title = {Generalized conjugate gradient squared},
author = {Fokkema, D.R. and Sleijpen, G.L.G.},
abstractNote = {In order to solve non-symmetric linear systems of equations, the Conjugate Gradient Squared (CGS) is a well-known and widely used iterative method. In practice the method converges fast, often twice as fast as the Bi-Conjugate Gradient method. This is what you may expect, since CGS uses the square of the BiCG polynomial. However, CGS may suffer from its erratic convergence behavior. The method may diverge or the approximate solution may be inaccurate. BiCGSTAB uses the BiCG polynomial and a product of linear factors in an attempt to smoothen the convergence. In many cases, this has proven to be very effective. Unfortunately, the convergence of BiCGSTAB may stall when a linear factor (nearly) degenerates. BiCGstab({ell}) is designed to overcome this degeneration of linear factors. It generalizes BiCGSTAB and uses both the BiCG polynomial and a product of higher order factors. Still, CGS may converge faster than BiCGSTAB or BiCGstab({ell}). So instead of using a product of linear or higher order factors, it may be worthwhile to look for other polynomials. Since the BiCG polynomial is based on a three term recursion, a natural choice would be a polynomial based on another three term recursion. Possibly, a suitable choice of recursion coefficients would result in method that converges faster or as fast as CGS, but less erratic. It turns out that an algorithm for such a method can easily be formulated. One particular choice for the recursion coefficients leads to CGS. Therefore one could call this algorithm generalized CGS. Another choice for the recursion coefficients leads to BiCGSTAB. It is therefore possible to mix linear factors and some polynomial based on a three term recursion. This way one may get the best of both worlds. The authors will report on their findings.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = 1994,
month =
}

Conference:
Other availability
Please see Document Availability for additional information on obtaining the full-text document. Library patrons may search WorldCat to identify libraries that hold this conference proceeding.

Save / Share:
  • This paper discusses the application of preconditioned generalized conjugate gradient acceleration to fully implicit thermal simulation. The preconditioning step utilizes incomplete Gaussian elimination (IGE) to form an approximate factorization of the Jacobian matrix. IGE preconditioning and its implementation is discussed with respect to five, seven, nine or eleven-point finite difference approximations and optional well constraint equations. Numerical results were obtained using a thermal model allowing any number N/SUB c/ of components. The model's implicit formulation requires the solution of a linear system of equations in which each grid block has N /SUB c/ + 1 unknowns. Test problems involved combustion,more » steam drive and cyclic steam stimulation processes, some of which exhibited ill-conditioning, negative transmissabilities and high transmissability ratios. Numerical results indicate how different grid block orderings, different levels of incomplete factorization and different acceleration procedures affect convergence, storage and work requirements.« less
  • A numerical simulation of a reactor analysis usually involves the numerical solution of a very large and spare non-symmetry system of linear equations with the form Ax = b, (1) where matrix A and vector b are known, and vector x is the neutron flux to be found. For example, the system of linear equations can be obtained by discretizing the multidimensional multigroup neutron diffusion equations with the finite difference method. The purpose of this paper is to present the numerical results of some generalized conjugate gradient methods, derived from time dependent two-group two-dimensional diffusion equations. The numerical results aremore » an extension of our previous study. All of the computations are carried out on the Convex C3400 parallel and vector computer, which has six vector processors.« less
  • A generalized conjugate gradient method is considered for solving systems of linear equations having nonsymmetric coefficient matrices with positive-definite symmetric part. The method is based on splitting the matrix into its symmetric and skew-symmetric parts, and then accelerating the associated iteration by using conjugate gradients; the method simplifies in this case, as only one of the two usual parameters is required. The method is most effective for cases in which the symmetric part of the matrix corresponds to an easily solvable system of equations. Convergence properties are discussed, as well as an application to the numerical solution of elliptic partialmore » differential equations.« less
  • A generalized conjugate gradient method is considered for solving sparse, symmetric, positive-definite systems of linear equations, principally those arising from the discretization of boundary value problems for elliptic partial differential equations. The method is based on splitting off from the original coefficient matrix a symmetric, positive-definite one that corresponds to a more easily solvable system of equations, and then accelerating the associated iteration using conjugate gradients. Optimality and convergence properties are presented, and the relation to other methods is discussed. Several splittings for which the method seems particularly effective are also discussed; and for some, numerical examples are given. 1more » figure, 2 tables. (auth)« less
  • While the preconditioned conjugate gradient (PCG) method has been applied to the solution of symmetric linear systems of equations arising from the finite difference approximation of the multidimensional neutron diffusion equation and to systems occurring in other application areas, PCG and PCG-like methods do not seem to have been used for reactor thermal-hydraulic calculations (though they have been applied to other areas of fluid dynamics). Results obtained with the transient subchannel analysis code SWIRL suggest that there may be reason to do so. The SWIRL code uses a three-equation drift-flux model (mixture mass, mixture internal energy, and mixture momentum) withmore » the conservation form used for the axial and lateral momentum equations. The equations are integrated over a staggered spatial mesh, and when a semi-implicit temporal discretization is used, algebraic reduction yields a linear system of equations that must be solved at each time step for the pressure distribution. These results indicate that CG-like methods can be efficient and reliable for reactor thermal-hydraulic simulations. Nevertheless, further investigation of CG-like methods is needed.« less