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Preconditioned conjugate‐ and secant‐Newton methods for non‐linear problems

1989; Wiley; Volume: 28; Issue: 6 Linguagem: Inglês

10.1002/nme.1620280606

1097-0207

Manolis Papadrakakis, Charis J. Gantes,

Iterative Methods for Nonlinear Equations

Abstract The preconditioned conjugate gradient (CG) method is becoming accepted as a powerful tool for solving the linear systems of equations resulting from the application of the finite element method. Applications of the non‐linear algorithm are mainly confined to the diagonally scaled CG. In this study the coupling of preconditioning techniques with non‐linear versions of the conjugate gradient and quasi‐Newton methods creates a set of conjugate‐ and secant‐Newton methods for the solution of non‐linear problems. The preconditioning matrices used to improve the ellipticity of the problem and to reduce the computer storage requirements are obtained by the application of the partial preconditioning and the partial elimination techniques. Both techniques use a drop‐off parameter ψ to control the computer storage demands of the method, making it more versatile for any computer hardware environment. Consideration is given to the development of a highly effective stability test for the line search minimization routine, which computes accurate values without much effort. This results in a beneficiary effect not only on the convergence properties of the methods but on their efficiency as well.