R. Fletcher, Conjugate gradient methods for indefinite systems, Proc. Dundee Conference Numer. Anal., ed. G. A. Watson, Springer-Verlag, New York, 1976, 73-89.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....while other formulations, such as the method of moments, such a singularity has to be handled with caution [7 9] When the inhomogeneous body is discretized into a regular grid, the resultant equation has a block Toeplitz structure. An iterative solver such as BiCG (bi conjugate gradient) method [10,11] can be used to solve for the solution of the matrix equation iteratively. When an iterative solver is used, the main cost of seeking the solution is the cost of performing a matrix vector multiplication. Exploiting the block Toeplitz structure, we can perform the matrix vector multiplication in ....

R. Fletcher, "Conjugate gradient methods for indefinite systems," in Numerical Analysis Dundee 1975.

.... initial approximation x 0 ; r 0 = b Ax 0 ; For n = 1, 2, 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 ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Proceedings of the Dundee Conference on Numerical Analysis,

....SYMMLQ [74] However, for very large unsymmetric systems, the SOR methods, and the method of Chebyshev [102, 54, 53] nicely tuned by Manteuffel [68] were still the methods of choice. Due to several poorly understood numerical problems, the two sided Lanczos method, and a special variant Bi CG [41], were not so popular at that time, but slowly more robust variants of Krylov subspace methods, with longer recursion formulas, entered the field. We mention GENCG [32] FOM [84] ORTHOMIN [103] ORTHODIR and ORTHORES [62] In the mid eighties we see the start of the popularity of Krylov subspace ....

....TFQMR, FGMRES, and GMRESR. Most of these methods have been proposed in the last ten years. GMRES, proposed in 1986 by Saad and Schultz [84] is the most robust of them, but, in terms of work per iteration step it is also the most expensive. Bi CG, which was suggested by Fletcher in 1977 [41], is a relatively inexpensive alternative, but it has problems with respect to convergence: the so called breakdown situations. This aspect has received much attention in the past period. Parlett et al. [75] introduced the notion of look ahead, in order to overcome breakdowns, and this was further ....

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R. Fletcher. Conjugate gradient methods for indefinite systems, volume 506 of Lecture Notes Math., pages 73--89. Springer-Verlag, Berlin--Heidelberg-- New York, 1976.

....popular one of this type. In 1952, Lanczos [21] used bi orthogonality relations to reduce iteratively a matrix to tri diagonal form and he suggested how to solve non symmetric linear systems from the reduced tri diagonal form. Later, his ideas were adapted for a variety of methods, such as Bi CG [11], QMR [13] and others. In 1989, Sonneveld [38] introduced a squared Bi CG algorithm , the CG S algorithm: he used the observation that the computational effort in the Bi CG algorithm to get bi orthogonality can be used for an additional reduction of the residuals. The squared QMR algorithm [12] ....

....; r k Gamma1 Gamma r k ) Gamma (r k ; r k ) r k Gamma1 ; r k Gamma1 ) For the scalar ff k the following relations are useful. They follow from applying (64) r k ; u k ) r k ; r k ) and (r k ; c k ) c k ; r k ) and this leads to Bi CG (Bi Conjugate Gradients [21, 11]) in Algorithm 6. Each Bi CG step is cheap. Estimate (58) may give some theoretical information on the speed of convergence. However, we don t have a priori information on C(C) and since we wish to keep k relatively small, we will never have this. It was not our purpose to obtain the sequence ....

Fletcher R.: Conjugate gradient methods for indefinite systems. In G. Watson, ed., Proc. of the Dundee Biennial Conference on Numerical Analysis (Springer-Verlag, New York, 1975).

....BiCG methods. The methods in this class iteratively compute, for a given initial guess x 0 , approximate solutions x k for which the residual r k = b Gamma Ax k can be written formally as r k = q k (A)r k ; 1. 2) in which q k is a polynomial of degree k with q(0) 1, and r k is the kth BiCG [2, 5] residual. In BiCGstab( the polynomial q k is chosen as a product of locally minimizing polynomials of degree . More precisely, for k = m , with P the space of polynomials p of degree with p(0) 1, q k = p m; q k Gamma , with p such that kp(A)q k Gamma (A)r k k 2 is minima ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis Dundee

.... Research supported by grants from University of Manitoba 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 ....

R. Fletcher, Conjugate Gradient Methods for Indefinite Systems, in Proc. Dundee Conference on Numerical Analysis, 1975.

....matrix A is symmetric and positive definite. The coefficients of the recurrence relationships for the residuals r k are given by ratios of scalars products. Since A is symmetric positive definite, their numerators are always nonzero. This algorithm was extended to an arbitrary matrix by Fletcher [29]. It is the biconjugate gradient algorithm. In this algorithm, divisions by zero can occur due to a zero scalar product. Such a situation is called a breakdown and the algorithm has to be stopped. Other algorithms for the implementation of the Lanczos method can be derived by using other ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, G.A. Watson ed., LNM vol. 506, Springer--Verlag, Berlin, 1976, pp. 73--89.

....cedex, France. E mail: Claude.Brezinski univ lille1.fr Dipartimento di Matematica, Universit a degli Studi della Calabria, Arcavacata di Rende, 87036 Rende (CS) Italy. E mail: m. redivo zaglia unical.it influences the numerical stability of the biconjugate gradient algorithm of Fletcher [5] for implementing Lanczos method for linear systems [4] 2 Formal orthogonal polynomials Let c be a linear functional on the space of polynomials defined by its moments c i = c(x ) i 0 and let fP k g (with degree P k k) be the family of formal orthogonal polynomials (FOP) with respect to ....

.... [15] for implementing another method, also due to Lanczos, for solving a system of linear equations [9] Lanczos Orthores is known (see [6] to be less numerically stable than the implementation called Lanczos Orthomin [14] equivalent to the biconjugate gradient (BiCG) algorithm of Fletcher [5]) which is based on two coupled recurrences quite similar to (3) So, we will now give an algorithm, based on (3) for transforming a matrix into a similar tridiagonal one. This algorithm was already given by Lanczos [9] q k = Q k (A)y ; q )z; we immediately obtain from (3) p k 1 = Aq k ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, Dundee 1975.

.... k ) ff k 1 = Gamma(y; AV k (A)r k 1 ) y; AV k (A)p k ) This algorithm is due to Vinsome [27] For the choice U k j V k j P k , it is called Lanczos Orthomin [28] and is equivalent to the bi conjugate gradient (BCG) of Lanczos [21, 22] which was written under an algorithmic form by Fletcher [14]. This algorithm is also known under the name of BIOMIN [18] 3.3 Lanczos Orthodir We immediately obtain from the relations (9) with z 0 = r 0 = b Gamma Ax 0 and z Gamma1 = 0. Using the definition of the linear functional c, we have k 1 = y; U k (A)r k ) y; AU k (A)z k ) fl k 1 = y; ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, Dundee 1975.

....fi fi fi fi fi fi fi fi is different from zero. The polynomials P k can be recursively computed in different ways leading to the various Lanczos type algorithms known as Lanczos Orthores, Biores [28] Lanczos Orthodir, Biodir [28, 42] Lanczos Orthomin [40, 42] Biomin [26] biconjugate gradient [23], and so on. A unified presentation and derivation of all these methods can be based on the theory of formal orthogonal polynomials [14] An interesting property of Lanczos method is its finite termination, namely that 9k n such that r k = 0 and x k = x = A b. A variant of Lanczos ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, G.A. Watson ed., LNM 506, Springer Verlag, Berlin, 1976, pp. 73--89.

....Gradient method. We note also that the smoothing procedure is a particular case of a more general method, the hybrid procedure, introduced and studied by C. Brezinski and M. Redivo Zaglia [3] In particular, we will obtain a relationship between the residual vectors of the Arnoldi s [11, 4] BCG [5, 8], Hessenberg [7, 19] algorithm for solving linear systems and the residual vectors of the GMRES [12, 4] QMR [9] and CMRH [13] algorithm respectively. We also remark that all the previous methods can be derived from the Generalized Hessenberg method. Throughout the paper, all vectors and ....

....j;k b j ; v = v Gamma h j;k b j h k 1;1 = 1; b k 1 = u; b k 1 = v; j k 1 = y k 1 ; u) end. By imposing the Galerkin condition r k K k (y; A ) we see that the Conjugate Gradient type method reduces to BiConjugate Gradient method (BCG) proposed by Lanczos [8] and later by Fletcher [5]. Finally, if we impose the least squares condition r k Z k AK k (r 0 ; A) we see that the Hybrid method reduces to the QMR method (without look ahead ) 6, 9] We now recall that L. Zhou and H.F. Walker have shown in [20] that the QMR method is obtained from the BCG method in the following ....

R. Fletcher, Conjugate Gradient methods for indefinite systems, in Numerical Analysis, G. A. Watson ed. LNM 506, Springer Verlag, Berlin, 1976, pp. 73--89.

.... the expanded system leads to the following algorithm z 0 = r 0 = b Gamma Ax 0 z 0 = r x n 1 = x n Gamma n z n n = Gamma(r z n 1 = r n 1 fi n 1 z n fi n 1 = r n 1 ; r n 1 fi n 1 z This algorithm is the biconjugate gradient algorithm (BCG) due to Fletcher [17]. It is also called Lanczos Orthomin [47] see also [30] It is essentially identical to the complete algorithm for minimized iterations given by Lanczos [35] The vectors computed by Lanczos algorithm are related to ours by r n = a n p n ; r n = a n p n ; z n = a n q n ; z n = a n q ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, Dundee 1975.

....of the underlying methods. However this case is very much convenient for a parallel computation of r n . If both methods are not independent, then the cost of one iteration of the hybrid procedure can be lowered. This is, in particular, the case for the biconjugate gradient algorithm (BCG) [13] and the CGS [22] since, in both methods, the constants appearing in the recurrence relations are the same and thus they have to be computed only once. Then, a coupled implementation of the BCG and the CGS only requires 3 matrix by vector multiplications (instead of 4) and, moreover, A is no ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, Dundee 1975.

....7. 2 The HBMRZ and the HBMRZ stab In this variant, the polynomial P k 1 satisfy the following relation k 1 ( A k 1 P k 1 ( B k 1 P k ( 21) where and B k 1 = With this choice, we recover exactly the BCG, written under an algorithmic form by Fletcher [19] (see also [7, p.91] Using this relation, and computing the polynomial w k ( as in Theorem 1, we obtain the following algorithm Algorithm HBMRZ (A; b; x 0 ; y) 0 (y; r 0 ) 2. While r k 6= 0 do 0 = 0 then Impossible to use the HBMRZ. Stop. 0 (y; z k ) 14 6. A k 1 ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, G.A. Watson ed., LNM 506 (Springer, Berlin, 1976) pp. 73--89.

....degli Studi di Padova, via Gradenigo 6 a, I 35131 Padova, Italy. Email: elen elett1.dei.unipd.it 1 E. Stiefel [33] around the same period. Extensions to the nonsymmetric case were given in [32] but the method only became widely known in 1975 with the biconjugate gradient algorithm of R. Fletcher [24]. An enormous literature on the L anczos method exists and it is not our purpose here neither to give a list of references nor to describe its connections with other questions. A quite complete account of the history of the subject and an annotated bibliography can be found in [28] See also [31] ....

.... It is essentially due to Vinsome [53] Another implementation, corresponding to a different choice of the auxiliary polynomials U k , is the biconjugate gradient algorithm (BCG) due to L anczos [39, 40] but which only became known after having being put under a more algorithmic form by Fletcher [24]. We shall now explain how to avoid breakdowns in the recursive algorithms for the L anczos method. 3. Avoiding true breakdowns The treatment of a true breakdown consists in the following operations 1. to be able to recognize the occurrence of such a breakdown, that is that the next orthogonal ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, G.A. Watson ed., LNM vol. 506, Springer--Verlag, Berlin, 1976, pp. 73--89.

.... k (A)p k ) ff k 1 = Gamma(y; AU k (A)r k 1 ) y; AU k (A)p k ) This algorithm is due to Vinsome [31] For the choice U k = V k = P k , it is called Lanczos Orthomin and is equivalent to the bi conjugate gradient (BCG) of Lanczos [22, 23] which was written under an algorithmic form by Fletcher [18]. 4 Transpose free algorithms We shall now discuss how to avoid the use of A in the computation of the coefficients of the recurrence relationships for orthogonal polynomials given in the preceding Section. The coefficients of all the preceding recurrence relationships can be, in fact, ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, Dundee 1975.

....generated by the columns of the matrices V; AV; A V . 2 The block BCG algorithm The block Biconjugate gradient (Bl BCG) algorithm was first proposed by O Leary [16] for solving the problem (1. 1) This algorithm is a generalization of the one right hand side well known BCG algorithm [7]. Bl BCG computes two sets of direction matrices fP 0 ; P k g and f P k g that span the block Krylov subspaces K k 1 (A; R 0 ) and K k 1 (A ; R 0 ) where R 0 = B Gamma AX 0 and R 0 is an arbitrary N Theta s matrix. The algorithm is summarized as follows [16] ALGORITHM 1: ....

R. Fletcher, Conjugate gradient methods for indefinite systems, In G. A. Watson, editor, Proceedings of the Dundee Biennal Conference on Numerical Analysis

....minimal residual, they feature short recurrences (that is, three term or coupled two term recurrences) for generating the approximations (or iterates) x and the corresponding residuals r : b Axe. However, the classical biconjugate gradient (BICG) method of Lanczos [9] reformulated by Fletcher [2]) has also a number of shortcomings: i) BICG may break down (even for well conditioned problems) Preprint submitted to Elsevier Science 3 April 2001 (ii) BICG requires matrix vector products with the transpose of the matrix, iii) BICG needs two matrix vector products to gain one dimension in ....

....a version that is easier to analyze and can be shown to have high ultimate accuracy at the same cost as GPBICG. 2 Lanczos type product methods based on two pairs of coupled two term recurrences Starting point for all LTPMs is the classical method BICG of Lanczos [9] reformulated by Fletcher [2]. It is given as Algorithm 1. Algorithm 1 (BICG) For computing Ax = b choose an initial approximation Xe and let Ve : re : b Axe. Choose Ye such that de : Ye, re) 0 and 5 : o, Avo) 0. Then, compute for n = O, 1, The basic idea behind BICG is to generate residuals r and a ....

R. Fletcher. Conjugate gradient methods for indefinite systems. In G. A. Watson, editor, Numerical Analysis, Dundee, 1975.

....The other approach involves using the unsymmetric Lanczos process to generate two sequence of vectors that satisfy a bi orthogonality condition. The method of Bi Conjugate Gradients (BiCG) is based on using these vectors to obtain iterates by enforcing a Galerkintype condition on the residuals [32]. The advantage of such an approach is that, unlike GMRES, only a limited amount of work and storage is needed per iteration. However, the residuals of BiCG iterates display irregular convergence and the method can sometimes experience numerical breakdowns [32] Recently, Freund and Nachtigal [39] ....

....condition on the residuals [32] The advantage of such an approach is that, unlike GMRES, only a limited amount of work and storage is needed per iteration. However, the residuals of BiCG iterates display irregular convergence and the method can sometimes experience numerical breakdowns [32]. Recently, Freund and Nachtigal [39] have proposed a new approach, the Quasi Minimal Residual method (QMR) which applies a least squares solve and update to the BiCG residuals, thereby smoothing out the irregular convergence behavior of BiCG. A special implementation of QMR (with look ahead ) ....

R. Fletcher. Conjugate gradient methods for indefinite systems. In G. A. Watson, editor, Proc. Dundee Conference on Numerical Analysis, 1975, Lecture Notes in Mathematics 506, pages

....vector distribution must be chosen the same. This extra constraint makes it more di#cult to balance the communication, and some times it may even lead to an increase in communication volume. First, we consider a square nonsymmetric matrix. Iterative algorithms such as GMRES [36] QMR [18] BiCG [16], and Bi CGSTAB [37] target this type of matrix. These algorithms are most conveniently carried out in parallel if all vectors involved are distributed in the same way, to facilitate vector operations such as norm and inner product computations and DAXPYs. The matrix partitioning can be done as ....

R. Fletcher, Conjugate gradient methods for indefinite systems, in Proceedings of the Dundee Biennial Conference on Numerical Analysis, G. A. Watson, ed., vol. 506 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1976, pp. 73--89.

....For the gen eral Lanczos method it gives the minimization of the inner product sr, but this implies no minimization for either the r or the s vectors. This minimiza tion property led Lanczos [13, 14] to name this method minimized iterations . Another name is the biconjugate gradient method [7]. 5.2 Symmetric CG: Minimization in the A norm In addition to theorem 3 we can prove a further minimization property for the conjugate gradient method for spd systems. Theorem 40rthogonalizing the residuals ri for the conjugate gradient method applied to symmetric positive definite systems A ....

.... rh rn lhn ln = Ar. Traditionally this was the way Lanczos [13] derived his bi orthogonalization method. The conjugate gradient method, however, was presented by its dis coverers Hestenes and Stiefel [11] as two coupled two term recurrences. Such a formulation was later given by Fletcher [7] for the Lanczos method, who called it the biconjugate gradient method . In this section we will show how the two formulations are equivalent, and how the view as coupled two term recurrences arises from factoring the Hessenberg matrix. 6.1 Search directions and the solution of linear systems ....

R. Fletcher. Conjugate gradient methods for indefinite systems. In G.A. Watson, editor, Numerical Analysis Dundee 1975.

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R. Fletcher, Conjugate gradient methods for indefinite systems, Proc. Dundee Conference Numer. Anal., ed. G. A. Watson, Springer-Verlag, New York, 1976, 73-89.

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R. Fletcher. Conjugate Gradient Methods for Indefinite Systems, volume 506 of Lecture Notes in Mathematics, pages 73--89. Spring-Verlag, Heidelberg, 1976.

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R. Fletcher. Conjugate gradient methods for indefinite systems, volume 506 of Lecture Notes Math., pages 73--89. Springer-Verlag, Berlin-- Heidelberg--New York, 1976.

No context found.

R. Fletcher. Conjugate Gradient Methods for Indefinite Systems, volume 506 of Lecture Notes in Mathematics, pages 73--89. Springer-Verlag, Heidelberg, 1976.

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