Martin H. Gutknecht, Lanczos-type solvers for non-symmetric linear systems of equations, Technical report, Swiss center for scienti c computing, ETH Zurich, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

..... until convergence x n = x n 1 q n , the vector x n does not occur in other statements) Aq n Most Krylov subspace iterative methods, including the conjugate gradient method (CG) 13] the biconjugate gradient method (BiCG) 4, 14] CGS [21] and BiCGSTAB [24] fit in this framework (see [2, 12, 17] for other methods) In exact arithmetic, the recursively defined r n in Algorithm 1 is exactly the residual for the approximate solution x n , because b Ax n 1 r n 1 = Ax 0 r 0 = 0. In a floating point arithmetic, however, the round o# patterns for x n and r n will be di#erent. It ....

....of Manitoba Research Development Fund and the Natural Sciences and Engineering Research Council of Canada. 835 ence on the iteration process. This leads to the well known situation that b Ax n and r n may di#er significantly. This phenomenon has been extensively discussed in the literature; see [20, 11, 12, 19] and the references cited therein. Indeed, if we denote the computed results of x n , r n by x n , r n , respectively (but we still use q n to denote the computed update vector of the algorithm) then we have ) 1) Aq n ) r n 1 ) 2) where f l(z) denotes the ....

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M. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, Acta Numer., 6 (1997), pp. 271--397.

.... Research Development Fund and from Natural Sciences and Engineering Research Council of Canada Most Krylov subspace iterative methods, including the conjugate gradient method (CG) 12] the bi conjugate gradient method (Bi CG) 4, 13] CGS [19] and BiCGSTAB [22] fit in this framework (see [2, 11, 16] for other methods) In exact arithmetic, the recursively defined r n in Algorithm 1 is exactly the residual for the approximate solution x n , because b Gamma Ax n Gamma r n = b Gamma Ax n Gamma1 Gamma r n Gamma1 = b Gamma Ax 0 Gamma r 0 = 0. In a floating point arithmetic, however, the ....

....or in other words, computational errors to x n do not force the method to correct, since x n has no influence on the iteration process. This leads to the well known situation that b Gamma Ax n and r n may differ significantly. This phenomenon has been extensively discussed in the literature, see [10, 11, 18] and the references cited there. Indeed, if we denote the computed results of x n ; r n by x n ; r n , respectively (but we still use q n to denote the computed update vector of the algorithm) then we have x n = f l(x n Gamma1 q n ) x n Gamma1 q n n ; j n j ujx n j O(u ) 1) ....

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M. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, Acta Numerica 6, pp. 271-397 (1997).

....process may break down before the Krylov space becomes stationary, and in finite arithmetic this leads to numerical instabilities in the case on near breakdowns. This problem is addressed by so called look ahead techniques for the non Hermitian Lanczos process (cf. Freund et al. 13] Gutknecht [23] and the references therein) which result in a pair of block biorthogonal bases and a Hessenberg matrix which is as close to tridiagonal form as possible while maintaining stability. Remark 5.6. We have mentioned that Krylov subspace QMR QOR methods have the advantage of being able to work with ....

M. H. Gutknecht. Lanczos-type solvers for nonsymmetric linears systems of equations. Acta Numerica, pages 271--397, 1997.

....norm to a general p norm, though possible, would lead to long recurrences rather than to short recurrences. Due to the Faber Manteuffel theorem [2] such a method would be senseless because long recurrences can be used to minimize not only a factor of the residual but the residual itself; see [5, 8, 9, 12] for an overview of Krylov subspace methods. We will refer to the above mentioned generalization of the standard QMR approach as the p norm QMR approach. In this note, it is shown that methods based on the p norm QMR approach that still involve short recurrences can be derived if the upper ....

M. H. Gutknecht, Lanczos-Type Solvers for Nonsymmetric Linear Systems of Equations, in Acta Numerica 1997, Cambridge University Press, Cambridge, 1997, pp. 271--397.

....process may break down before the Krylov space becomes stationary, and in finite arithmetic this leads to numerical instabilities in the case on near breakdowns. This problem is addressed by so called look ahead techniques for the non Hermitian Lanczos process (cf. Freund et al. 13] Gutknecht [23] and the references therein) which result in a pair of block biorthogonal bases and a Hessenberg matrix which is as close to tridiagonal form as possible while maintaining stability. Remark 5.6. We have mentioned that Krylov subspace QMR QOR methods have the advantage of being able to work with ....

M. H. Gutknecht. Lanczos-type solvers for nonsymmetric linears systems of equations. Acta Numerica, pages 271--397, 1997.

.... methods based on the Arnoldi process, like GMRES ( 20] or truncated versions of it (e.g. GMRES(m) or squared methods based on the two sided Lanczos process, like CGS ( 23] BiCGStab ( 24] BiCGStab( 22] or TFQMR ( 10] For an overview of Lanczos type solvers, one can also refer to [13]. Assuming we use one of these solvers, we only need to know how to approximate matrix vector products of C and S. In the case of C, this can be done using the differencing scheme C(x; y)w Psi 2 1 w (0; x; y) where (61) Psi w (r; x; y) Phi(x h 2 r; y) Gamma Phi(x; y) h ....

Gutknecht, M. H. : Lanczos-Type Solvers for Nonsymmetric Linear Systems of Equations, Technical Report TR-97-04, CSCS/SCSC, ETH Zurich (1997).

....the old basis and a vector resulting from applying the coefficient matrix A to the basis vector generated in the previous iteration. Thus, the dimension of the Krylov subspace is sought to be increased by one and the process can be expressed by implicitly generating an upper Hessenberg matrix; see [5, 8, 11] for excellent surveys on Krylov subspace methods. Examples of procedures to generate a basis of the Krylov subspace include the Arnoldi [1] and the Lanczos [9] algorithms. In the Arnoldi case the basis is orthonormal but expensive in terms of storage and computation; each computation of a new ....

....the standard QMR approach that is contained for p = 2, is not applicable to more general minimization problems involving upper Hessenberg matrices. The necessity to generalize the p norm QMR approach to upper Hessenberg matrices stems from the Lanczos algorithm that needs lookahead techniques [8] for reasons of numerical stability in which an upper Hessenberg matrix replaces the tridiagonal or bidiagonal matrix found in the formulation of the classical Lanczos algorithm. In this note, the extension to upper Hessenberg matrices is established for the particular case p = 1 and thus ....

M. H. Gutknecht, Lanczos-Type Solvers for Nonsymmetric Linear Systems of Equations, in Acta Numerica 1997, Cambridge University Press, Cambridge, 1997, pp. 271--397.

....is also possible in the algorithms presented in [5, 6, 7] for curing breakdowns and near breakdowns in Lanczos process since they directly derive from the theory of FOP. Changing the auxiliary vectors is, maybe, feasible in the other existing look ahead algorithms for treating this problem; see [10] for a review. Our choices of the auxiliary vectors can also be made in the conjugate gradient algorithm (CG) of Hestenes and Stiefel [12] or any other algorithm, for implementing Lanczos method when the matrix is symmetric positive definite. The CG corresponds to the choice U k j Q k and V k j ....

M.H. Gutknecht, Lanczos--type solvers for nonsymmetric linear systems of equations, Acta Numerica, 6 (1997) 271--397.

....(r 0 = b) where P n is a polynomial of degree n and called the BiCG polynomial. In the past few years, several efficient product Krylov subspace methods such as CGS [13] and BiCGSTAB [15] have been developed to accelerate convergence in BiCG and to avoid multiplications by the transpose of A (see [1, 8, 10] for a general review and comparison) These product type methods are based on constructing the product of the BiCG polynomial and some other accelerating polynomials. In comparison with BiCG, which requires 2n matrix vector multiplications at iteration n, CGS uses the same amount of ....

M. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, Acta Numerica 6, pp. 271-397 (1997).

....update for a number of successive iteration steps [197] This does not affect the numerical stability of the scheme. It can even be used to improve the accuracy of the iteration method, especially if it is done for irregularly converging methods such as CGS, in a way known as reliable updating [115, 184]. The main problem in this case is that if large errors are introduced into the iterates x i (that are not usually computed explicitly) they can swamp any meaningful information previously present in the iterates, from which it is impossible to recover later (see, for instance, 107, Chapter ....

M. H. Gutknecht. Lanczos-type solvers for nonsymmetric linear systems of equations. In Acta Numerica 1997, pages 271--397. Cambridge University Press, Cambridge, 1997.

.... Research Development Fund and from Natural Sciences and Engineering Research Council of Canada Most Krylov subspace iterative methods, including the conjugate gradient method (CG) 12] the bi conjugate gradient method (Bi CG) 4, 13] CGS [19] and BiCGSTAB [22] fit in this framework (see [2, 11, 16] for other methods) In exact arithmetic, the recursively defined r n in Algorithm 1 is exactly the residual for the approximate solution x n , because b Gamma Ax n Gamma r n = b Gamma Ax n Gamma1 Gamma r n Gamma1 = b Gamma Ax 0 Gamma r 0 = 0. In a floating point arithmetic, however, the ....

....in other words, computational errors to x n do not force the method to correct, since x n has no influence on the iteration process. This leads to the well known situation that b Gamma Ax n and r n may differ significantly. This phenomenon has been extensively discussed in the literature, see [10, 11, 18] and the references cited there. Indeed, if we denote the computed results of x n ; r n by x n ; r n , respectively (but we still use q n to denote the computed update vector of the algorithm) then we have x n = f l(x n Gamma1 q n ) x n Gamma1 q n n ; j n j ujx n j O(u 2 ) 1) ....

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M. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, Acta Numerica 6, pp. 271-397 (1997).

....inaccurate recurrence coefficients (and, hence, inaccurate eigenvalue approximations) and, possibly, low ultimate accuracy of the approximate solution of the linear system. For all these shortcomings, there are at least partial remedies. Breakdowns can be overcome by look ahead; see [7] and references given there. The transpose and the second matrix vector product can be avoided by squaring BICG, as suggested by Sonneveld with his (hi)conjugate gradient squared (CGS) algorithm [12] This algorithm still needs two matrix vector products per iteration step but we gain two ....

....results from an incorporated one dimensional local residual minimization. This paper is devoted to a set of algorithms that realize a particular LTPM first proposed 1993 by Zhang[15] where a 2 dimensional minimization is incorporated in each step, and which was therefore called BICGxMR2 in [7]. In one version, which independently was also proposed by Cao [1] and Gutknecht[6] the implementation only requires a one line modification of BICGSTAB2 [5] which does such a minimization in every other step. But Zhang also proposed a version called GPBICG that is fully based on coupled ....

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M. H. Gutknecht. Lanczos-type solvers for nonsymmetric linear systems of equations. Acta Numerica, 6:271-397, 1997.

....(GMRes) iterates. The relevant theoretical relationship between the convergence behavior of CG and CR, as well as FOM and GMRes is by now well understood through the work of Brown, Cullum, Freund, Greenbaum, Gutknecht, Weiss, and others; see, in particular, Cullum and Greenbaum [1] and Gutknecht [2]. This relationship approximately carries over to BiCG and QMR, and makes us understand why peaks in the residual norm plot of BiCG are matched by plateaux in the one of QMR. The new results in this talk on the limitations of smoothing processes (which also apply to MinRes and, in a certain ....

M. H. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, Acta Numerica, 6 (1997), pp. 271-397.

....m = o due to the nonsingularity of A. There are yet other cases where the nite termination property holds. For nding iterates and residuals satisfying (2) 6) we could apply the GramSchmidt process as follows. A proof is easily found by adapting the one of the similar result in Section 4. 3 of [4]. Theorem 2 If de ned for all n up to m, the iterates x n and the residuals r n of a Krylov space solver characterized by (1) satisfy, for n = 0; 1; m 1, x n 1 : y n x n n;n x n 1 n 1;n x 0 0;n = n 1;n ; 9) r n 1 : Ay n r n n;n r n 1 n 1;n ....

....the approximate solution satisfying (29) need not exist at all steps n. This potential hazard makes these methods less attractive to many users, although there is a simple way to get around this problem, namely by keeping track of the normalization of the residual polynomial separately; see, e.g. [4] for the corresponding adjustment of the biconjugate gradient method, which encounters the same problem. There is a strong connection between orthogonal residual methods and minimal residual methods, which was discovered by Weiss in his thesis [1] and will be discussed later in this paper. For ....

M. H. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, Acta Numerica 6 (1997) 271-397.

.... the Chebyshev iteration [20, 18, 16] the second order Richardson iteration [18] which is the stationary form of the Chebyshev iteration) the three term version of the conjugate gradient (CG) method [20, 13] and the three term version (BiORes) of the unsymmetric or two sided Lanczos method [15, 12] (which is a variation of the biconjugate gradient or BiCG method) see also [13] CG and BiCG have even better known versions that are instead based on coupled twoterm recursions which involve in addition to the iterates and their residuals also direction vectors p n : r n 1 = r n Gamma Ap n ....

....[21] still use the first two of these recursions, but have a more complex update formula for the direction vectors. In principle, the version (2) can be obtained from the three term version (1) by an LU decomposition of the tridiagonal matrix with coefficients fi n Gamma1 , ff n , and fl n [3, 12] in the (n 1)st column. The folklore confirmed by many experiments is that implementations based on coupled two term recursions are less affected by roundoff than those based on three term recursions. The fact that in exact arithmetic the vectors r n generated by (1) are the true ....

[Article contains additional citation context not shown here]

M. H. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, Acta Numerica, 6 (1997), pp. 271--397.

.... minimization of the residual in some norm is incompatible with short recurrences (Faber and Manteu el, 1984; Faber and Manteu el, 1987; Voevodin, 1983) The Lanczos type methods including BiCG feature short recurrences, but no minimization property. For an overview of Lanczos type methods see (Gutknecht, 1997). c Martin H. Gutknecht, September 2, 1999 22 References 23 ....

M. H. Gutknecht (1997), `Lanczos-type solvers for nonsymmetric linear systems of equations', Acta Numerica 6, 271-397.

.... in many papers, see, for example, 1 3,7,14 16] 2 The biconjugate gradient method and some related methods The rst Lanczos type method, introduced in 1952 by Lanczos [33] as the complete algorithm for minimized iterations , is essentially what we now call the (standard) BiOMin form [27] of the the biconjugate gradient (BiCG) method [8] It is fully analogous to the classical Hestenes Stiefel version of the conjugate gradient (CG) method for Hermitian positive de nite systems [30] which is also referred to as OMin algorithm for CG. Unlike CG, which is restricted to Hermitian ....

....satis ed automatically: the corresponding orthogonality is inherited at least in exact arithmetic. By eliminating the direction vectors from the recurrences of Algorithm 1 we obtain the BiORes form of the BiCG method, where the Lanczos vectors are generated by three term recurrences; see, e.g. [27] for this connection, which is based on an LU decomposition of a tridiagonal matrix: Algorithm 2 (BiORes form of the BiCG method) To solve Ax = b, choose an initial approximation x 0 , set y 0 : b Ax 0 , and choose e y 0 such that 0 : he y 0 ; y 0 i 6= 0. Set 1 : 0. Then, for n = 0; 1; ....

[Article contains additional citation context not shown here]

M. H. Gutknecht. Lanczos-type solvers for nonsymmetric linear systems of equations. Acta Numerica, 6:271-397, 1997.

....minimization of the residual in some norm is incompatible with short recurrences (Faber and Manteuffel, 1984; Faber and Manteuffel, 1987; Voevodin, 1983) The Lanczos type methods including BiCG feature short recurrences, but no minimization property. For an overview of Lanczos type methods see (Gutknecht, 1997). c flMartin H. Gutknecht, November 16, 1998 References 23 ....

M. H. Gutknecht (1997), `Lanczos-type solvers for nonsymmetric linear systems of equations', Acta Numerica 6, 271--397.

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Martin H. Gutknecht, Lanczos-type solvers for non-symmetric linear systems of equations, Technical report, Swiss center for scienti c computing, ETH Zurich, 1997.

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M. H. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, in Acta Numerica,

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M. H. Gutknecht, Lanczos-type solvers for nonsymmetric linear systems of equations, in Acta Numerica,

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