Voevodin, V.V., The problem of non-selfadjoint generalization of the conjugate gradient method has been closed, USSR Comput. Math. Math. Phys., 23 (1983), 143-144.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= RtRH = H. The question whether for unsymmetric matrices A the upper Hessenberg matrix can take on a banded form with a small bandwidth is of practical importance, since such a form limits the length of the recurrence for the v vectors in (4) This question was answered largely negatively by [25] and more generally by [6] the conjugate gradient algorithm gives a tridiagonal matrix if the spectrum of A lies on a line in the complex plane; for other matrices the bandwidth is a large action of the matrix size. 4.2 Bi orthogonalization If the sequence s, is based on a Krylov recurrence ....

V.V. Voevodin. The problem of non-self-adjoint generalization of the con- jugate gradient method is closed. USSR Compui. Maltis. Malll. Plly,s., 23:143 144, 1983. 30

....[71] for more general architectures, see Demmel, Heath and Van der Vorst [67] 2.3. 5 BiConjugate Gradient (BiCG) The Conjugate Gradient method is not suitable for nonsymmetric systems because the residual vectors cannot be made orthogonal with short recurrences (for proof of this see Voevodin [213] or Faber and Manteuffel [96] The GMRES method retains orthogonality of the residuals by using long recurrences, at the cost of a larger storage demand. The BiConjugate Gradient method takes another approach, replacing the orthogonal sequence of residuals by two mutually orthogonal sequences, at ....

V. Voevodin, The problem of non-self-adjoint generalization of the conjugate gradient method is closed, U.S.S.R. Comput. Maths. and Math. Phys., 23 (1983), pp. 143--144.

....t RH = H. ffl The question whether for unsymmetric matrices A the upper Hessenberg matrix can take on a banded form with a small bandwidth is of practical importance, since such a form limits the length of the recurrence for the r i vectors in (4) This question was answered largely negatively by [25] and more generally by [6] the conjugate gradient algorithm gives a tridiagonal matrix if the spectrum of A lies on a line in the complex plane; for other matrices the bandwidth is a large fraction of the matrix size. 4.2 Bi orthogonalization If the sequence fs ng is based on a Krylov ....

V.V. Voevodin. The problem of non-self-adjoint generalization of the conjugate gradient method is closed. USSR Comput. Maths. Math. Phys., 23:143--144, 1983.

....based on the A T A orthonormal basis of the Krylov subspaces K n (A; r 0 ) Section 3 describes implementation details. Section 4 presents results of numerical experiments. Section 5 contains concluding remarks. Methods based on (1. 5) require, in general, full term recurrences [10] see also [26]. Recently, a lot of effort has been devoted to the methods for nonsymmetric (non Hermitian) systems using short recurrences (see [13] and the papers referred to there) which generate in most cases good, but not optimal, approximate solutions. These methods are not within the subject of this ....

V.V. Voevodin, The Problem of a Non-Selfadjoint Generalization of the Conjugate Gradient Method Has Been Closed, USSR Comput. Math. and Math. Phys. 23, pp. 143-144, 1983

....applicable to nonsymmetric or indefinite linear systems. CG has two fundamental properties: it minimizes the residual vector along each iteration step, and the residual vectors satisfy a three term recurrence. It is not possible to maintain both properties for a nonsymmetric linear system [16] [49]. There are several ways to deal with this situation. One can maintain both properties by considering the normal equations instead of the original system. The new matrices A T A (CGNR) or AA T (CGNE) are symmetric positive definite, and the CG algorithm can therefore be applied to them. This ....

V. Voevodin, The problem of a non-selfadjoint generalization of the conjugate gradient method has been closed, USSR Comput. Math. and Math. Phys. 23 (1983) 143-144.

....[68] for more general architectures, see Demmel, Heath and Van der Vorst [64] 2.3. 5 BiConjugate Gradient (BiCG) The Conjugate Gradient method is not suitable for nonsymmetric systems because the residual vectors cannot be made orthogonal with short recurrences (for proof of this see Voevodin [208] or Faber and Manteuffel [92] The GMRES method retains orthogonality of the residuals by using long recurrences, at the cost of a larger storage demand. The BiConjugate Gradient method takes another approach, replacing the orthogonal sequence of residuals by two mutually orthogonal sequences, at ....

V. Voevodin, The problem of non-self-adjoint generalization of the conjugate gradient method is closed, U.S.S.R. Comput. Maths. and Math. Phys., 23 (1983), pp. 143--144.

....fulfill (i) and (ii) simultaneously. This result is due to Faber and Manteuffel [10,11] who have shown that, except for a few anomalies, CG type algorithms with (i) and (ii) exist only for matrices of the special form A = e i (T oeI) where T = T H ; 2 R; oe 2 C; 1:4) see also Voevodin [55] and Joubert and Young [35] Note that the class (1:4) consists of just the shifted and rotated Hermitian matrices. We remark that the important subclass of real nonsymmetric matrices A = I Gamma S; where S = GammaS T is real; 1:5) is contained in (1:4) with e i = i, oe = Gammai, and T ....

V.V. Voevodin, The problem of a non-selfadjoint generalization of the conjugate gradient method has been closed, USSR Comput. Math. and Math. Phys. 23 (1983), pp. 143--144.

No context found.

Voevodin, V.V., The problem of non-selfadjoint generalization of the conjugate gradient method has been closed, USSR Comput. Math. Math. Phys., 23 (1983), 143-144.

No context found.

V.V. Voevodin (1983), "The problem of a non-selfadjoint generalization of the conjugate gradient method has been closed", USSR Comput. Math. and Math.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC