Generalized conjugate gradient squared
Abstract
In order to solve nonsymmetric linear systems of equations, the Conjugate Gradient Squared (CGS) is a wellknown and widely used iterative method. In practice the method converges fast, often twice as fast as the BiConjugate 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 »
 Authors:
 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):
 CONF9404305Vol.2
ON: DE96005736; TRN: 96:0023210001
 Resource Type:
 Conference
 Resource Relation:
 Conference: Colorado conference on iterative methods, Breckenridge, CO (United States), 59 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 nonsymmetric linear systems of equations, the Conjugate Gradient Squared (CGS) is a wellknown and widely used iterative method. In practice the method converges fast, often twice as fast as the BiConjugate 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 =
}

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