O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801-812.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Orthomin continues to be popular, since this variant can be easily truncated (Orthomin(s) in contrast to GMRES. The truncated and restarted versions of these algorithms are not necessarily mathematically equivalent. Methods that are mathematically equivalent to FOM are: Orthores [62] and GENCG [19, 106]. In these methods the approximate solutions are constructed such that they lead to orthogonal residuals (which form a basis for the Krylov subspace; analogously to the CG method) A good overview of all these methods and their relations is given in [83] The GMRES method and FOM are closely ....

O. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 15:801--812, 1978.

....definite, then the conjugate gradient algorithm is recovered and, moreover, r n 1 ; A r n 1 ) e n 1 ; Ae n 1 ) is minimized [8, p. 14] When the symmetric part (A A ) 2 of the real matrix A is positive definite, the conjugate gradient was generalized by Concus and Golub [38] and Widlund [131]. As proved by Eisenstat [48] the iterates constructed by this algorithm are the best approximants to the solution from certain affine subspaces (which are not the affine Krylov subspaces) When A = A u n = z n = r n the method of steepest descent is obtained. ffl u n = r n . This choice also ....

O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978) 801--812.

....the storage of previously calculated vectors. Therefore it is somewhat remarkable that preconditioning by the symmetric part (A A T ) 2 of the coefficient matrix A leads to a method that does not need this extended storage. Such a method was proposed by Concus and Golub [56] and Widlund [216]. However, solving a system with the symmetric part of a matrix may be no easier than solving a system with the full matrix. This problem may be tackled by imposing a nested iterative method, where a preconditioner based on the symmetric part is used. 54 CHAPTER 3. PRECONDITIONERS Vassilevski ....

O. Widlund, A Lanczos method for a class of non-symmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801--812.

....one has to pay is that SYMMLQ often takes more steps. A slight disadvantage is also that although the method minimizes the norm of the error, the value of this norm is not known and the only practical information one has is the norm of the residual. In 1976, Concus and Golub [43] and Widlund [190] came up with the idea of splitting a matrix into its symmetric and nonsymmetric parts and using the symmetric part as a preconditioner. With the proper inner product, the resulting algorithm corresponds to an iteration with a skew Hermitian matrix and therefore a three term recurrence CG like ....

O. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 15:801--812, 1978.

....the present paper, we extend this research by applying a multigrid method to the discretized integral equation. The aim of this adaption is to achieve rapid convergence for solving the discretized systems. This may appear rather unnecessary since the generalized conjugate gradient (CG) method of [3, 14] converges at respectable rates for these systems (see [12] But for large n, i.e. fine discretizations, it is faster to reconstruct the (complex valued) matrix of the discretized system for each iteration rather than storing it out of core memory. Therefore, each CG iteration requires about 10n ....

O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801--812.

....seems to be still popular, since this variant can be easily truncated (Orthomin(s) in contrast to GMRES. The truncated or restarted versions of these algorithms are not necessarily mathematically equivalent. Methods that are mathematically equivalent with FOM are: Orthores [47] and GENCG [14, 94]. In these methods the approximate solutions are constructed such that they lead to orthogonal residuals (which form a basis for the Krylov subspace; analogously to the CG method) A good overview of all these methods and their relations is given in [71] 26 4.3 Rank one updates for the Matrix ....

O. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 15:801--812, 1978.

....applicable for symmetric positive definite matrices. The second condition on the matrix of being positive definite can be relaxed by minor modifications. In contrast the symmetry assumption is essential in getting a minimizing scheme with low computational effort. A different attempt is made in [135], where the Lanczos recursion is applied for an anti symmetric system. Therefore a solver for the symmetric part has to be applied in every step, which is, in general, pretty expensive and the performance is usually not satisfactory. So far no optimal scheme has been found for this class of ....

O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801--812.

....polynomial p 2 P k , kr k k min p2Pk kp(A)k kr 0 k k p(A)k kr 0 k: 2. 1) Let A = S S Gamma1 , where is a Jordan canonical form of A, and j kSk kS Gamma1 k; ffl k j min p2Pk kp( k: Then kr k k ffl k kr 0 k: Upper bounds have been derived for ffl k when A is diagonalisable, e.g. [6, 14, 19, 20, 22], and when A is defective, e.g. 9, 10] In contrast, we construct polynomials p 2 P k that are related to the minimal polynomial of A to bound k p(A)k in (2.1) directly. Since our bounds hold for any matrix, they may not be as accurate as bounds expressly derived for specific classes of ....

O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801--812.

....price one has to pay is that SYMMLQ often takes more steps. A slight disadvantage is also that although the method minimizes the norm of the error, the value of this norm is not known and the only practical information one has is the norm of the residual. In 1976, Concus and Golub [38] and Widlund [181] came up with the idea of splitting a matrix into its symmetric and nonsymmetric parts and using the symmetric part as a preconditioner. With the proper inner product, the resulting algorithm corresponds to an iteration with a skew Hermitian matrix and therefore a three term recurrence CG like ....

O. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 15:801--812, 1978.

....Bielefeld, Germany where A 2 C N ThetaN is large, unsymmetric (non Hermitian) and nonsingular, and b an N Theta1 column vector. A major class of methods for solving (1) is Krylov subspace type or conjugate gradient type methods, and so far many researchers have made contributions to them [2, 9, 8, 6, 11, 22, 23, 24, 25, 31, 32]; see the survey papers [26, 12] There are some very successful schemes among them, e.g. GCR or ORTHOMIN [8, 9, 31] FOM or Arnoldi s method [22] GMRES [25] and QMR [11] to name only a few. We can observe that as far as the convergence analysis is concerned, many of the Krylov subspace methods ....

....may converge very slowly even though ffl (m) tends to zero quite rapidly. We can thus expect that all their restarted versions may not converge at all unless the steps m per restarting are taken sufficiently large. ffl When A is diagonalizable, there are many upper bounds for ffl (m) e.g. [14, 17, 1, 18, 19, 32, 22, 7, 3]. ffl For large unsymmetric eigenproblems, a much more complicated analog of ffl (m) has been established in [16] From Theorem 1, we now want to find a polynomial p 2 Qm such that it satisfies the condition p(0) 1, and its values of function and the jth order derivatives, j d i ; i = 1; 2; ....

O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978) 801--812.

....a matrix. Moreover M Gamma1 A is not evidently similar to a shifted and rotated hermitian matrix since M is indefinite. As a consequence, there does not exist a conjugate gradient method for these preconditioned systems ( 9] In particular the methods proposed by Concus and Golub [1] Widlund [18] and Rapoport [16] are unfortunately not applicable (see [8] Because of the projection property in the case C = 0, note also that M Gamma1 A has at most n Gamma m eigenvalues different from unity. Convergence (indeed termination) will thus occur after n Gamma m or possibly more iterations ....

O. Widlund, A Lanczos method for a class of non-symmetric systems of linear equations, SIAM J. Numer. Anal.,15, pp. 801-812, 1978.

....cannot be (or, at least, has not been) established and that the actual convergence behavior, although often good, is hard to predict. In the seventies it was also shown that in certain situations orthogonalization with respect to two previous iterates (like in standard CG) is still sufficient [7, 71]. The interesting reverse question of finding necessary and sufficient conditions for this to be true was then completely resolved in the 1984 paper of Faber and Manteuffel [13] In contrast to these orthogonalization methods the BO and the BCG algorithm produce biorthogonal sequences and, ....

O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801--811.

....Orthomin continues to be popular, since this variant can be easily truncated (Orthomin(s) in contrast to GMRES. The truncated and restarted versions of these algorithms are not necessarily mathematically equivalent. Methods that are mathematically equivalent to FOM are: Orthores [62] and GENCG [19, 106]. In these methods the approximate solutions are constructed such that they lead to orthogonal residuals (which form a basis for the Krylov subspace; analogously to the CG method) A good overview of all these methods and their relations is given in [83] The GMRES method and FOM are closely ....

O. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 15:801--812, 1978.

....the storage of previously calculated vectors. Therefore it is somewhat remarkable that preconditioning by the symmetric part (A A T ) 2 of the coefficient matrix A leads to a method that does not need this extended storage. Such a method was proposed by Concus and Golub [54] and Widlund [211]. However, solving a system with the symmetric part of a matrix may be no easier than solving a system with the full matrix. This problem may be tackled by imposing a nested iterative method, where a preconditioner based on the symmetric part is used. Vassilevski [207] proved that the efficiency ....

O. Widlund, A Lanczos method for a class of non-symmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801--812.

....seems to be still popular, since this variant can be easily truncated (Orthomin(s) in contrast to GMRES. The truncated or restarted versions of these algorithms are not necessarily mathematically equivalent. Methods that are mathematically equivalent with FOM are: Orthores [44] and GENCG [13, 93]. In these methods the approximate solutions are constructed such that they lead to orthogonal residuals (which form a basis for the Krylov subspace; analogously to the CG method) A good overview of all these methods and their relations is given in [71] 6.3 Rank one updates for the Matrix ....

O. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 15:801--812, 1978.

....problems by Jim Douglas and Todd Dupont [15] Two important extensions of the conjugate gradient algorithm were given during this period. Paul Concus, Gene Golub, and Olof Widlund solved problems in which the Hermitian part of the matrix was positive definite and could be used as a preconditioner [9, 54]. Chris Paige and Michael Saunders showed how to compute iterates in case a matrix was indefinite [44] resulting in SYMMLQ and related algorithms. Research on the eigenvalue problem also contributed to making the KMP algorithms more useful for solving linear systems. Chris Paige [43, 40, 41, 42] ....

O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801--812.

....(2:13) The class (2:13) consists of just the shifted and rotated Hermitian matrices. Note that the important subclass of real nonsymmetric matrices A = I Gamma S; where S = GammaS T is real; 2:14) is contained in (2:13) with e i = i, oe = Gammai, and T = iS. Concus and Golub (1976) and Widlund (1978) were the first to devise an implementation of a OR method for the family (2:14) The first MR algorithm for (2:14) was proposed by Rapoport (1978) and different implementations were given by Eisenstat et al. 1983) and Freund (1983) For a brief discussion of actual CG type algorithms for the ....

O. Widlund (1978), "A Lanczos method for a class of nonsymmetric systems of linear equations", SIAM J. Numer. Anal. 15, 801--812.

....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 = iS. Concus and Golub [6] and Widlund [56] were the first to devise a CG type algorithm for the family (1:5) Most of the non Hermitian Krylov subspace methods that have been proposed satisfy either (i) or (ii) Until recently, the emphasis was on requirement (i) and numerous algorithms with Advances in Lanczos Based Methods for Linear ....

O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 15 (1978), pp. 801--812.

....as a preconditioner and the problem is reduced to the 3 special nonsymmetric form. We use the splitting A = M Gamma N; 5) where M : A A T ) 2 (6) is the symmetric, positive definite and GammaN : A Gamma A T ) 2 is the skew symmetric part of A; see Concus and Golub [4] and Widlund [49]. The operator K : M Gamma1 N (7) plays a central role in the algorithm and in our analysis. It is easy to see that K is skew symmetric with respect to the M inner product defined by (u; v) M : u; Mv) u T Mv: 8) The results of the analysis of the algorithm obtained by using a Galerkin ....

....where v 1 = M Gamma1 r 0 = M Gamma1 b Gamma (I Gamma K)x 0 : 65) In this case, the Galerkin procedure works and gives invertible linear systems of equations and approximations for which error bounds can be derived. We first describe the method developed by Concus, Golub [4] and Widlund [49], which is commonly known as the CGW method. The Galerkin condition has the form (30) and, as in the symmetric, positive definite case, it is satisfied by x g k Gamma x 0 = V k y g k , where y g k is the solution of V T k AV k y g k = V T k r 0 : 66) From (22) 26) and (65) it follows ....

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Olof B. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM Journal on Numerical Analysis, 15:801--812, 1978.

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O. Widlund, A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), pp. 801-812.

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Widlund, O., A Lanczos method for a class of non-symmetric systems of linear equations, SIAM J. Numer. Anal., 15 (1978), 801-812. 20

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O. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 15:801--812, 1978.

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