Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp. 37 (1981), 105--126.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....efficient schemes for solving nonlinear problems. Key words. Nonlinear problems, Newton s method, Inexact Newton, Iterative methods. AMS subject classification. 65H10. 1. Introduction. Our goal in this paper is twofold. A number of iterative solvers for linear systems of equations, such as FOM [23], GMRES [26] GCR [31] Flexible GMRES [25] GMRESR [29] and GCRO [7] are in structure very similar to iterative methods for linear eigenproblems, like shift and invert Arnoldi [1, 24] Davidson [6, 24] and Jacobi Davidson [28] We will show that all these algorithms can be viewed as instances ....

....directions p j . Preferable updates p k : P jk fl j p j are those for which b Gamma Ax k 1 , where x k 1 = x k p k , is minimal in some sense: e.g. kb Gamma Ax k 1 k 2 is minimal, as in GMRES [26] and GCR [31] or b Gamma Ax k 1 is orthogonal to the p j for j k, as in FOM or GENCG [23], or b Gamma Ax k 1 is quasi minimal , as in Bi CG [17] and QMR [11] Of course the distinction between preconditioning and acceleration is not a clear one. Acceleration techniques with a limited number of steps can be seen as a kind of dynamic preconditioning as opposed to the static ....

Y. Saad, Krylov subspace method for solving large unsymmetric linear systems, Math. Comp., 37 (1981), pp. 105--126.

....CG method can be if combined with the ILU preconditioner. Pure generalizations of CG to solve non symmetric equations were introduced in the late seventies and early eighties. GMRES and variants use orthogonal basis. Algorithms of this type were proposed by Vinsome [41] Young and Jea [47] Saad [30], Elman [9] Axelsson [3] and others. The GMRES algorithm of Saad and Schultz [32] from 1986 seems to be the most 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 ....

....of this method is restricted to symmetric positive definite matrices, while for other symmetric matrices it is better to rely on the MINRES [27] algorithm, which is based on the MR approach. Likewise, in the unsymmetric case we obtain GMRES [32] by following the minimization approach, and FOM [30], if we follow the orthogonality approach. Depending on the particular construction of the minimal residual, or the orthogonal residual, we obtain differently named methods, but if they do not break down, then in exact arithmetic they produce the same results as either GMRES or FOM. For instance, ....

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Saad Y.: Krylov subspace method for solving large unsymmetric linear systems. Math. Comput., 37 (1981), pp. 105--126.

....Hence, we can hope that the convergence rate for the iteration for nite problem sizes will agree with the theory below for the asymptotic convergence rate. Next we de ne the asymptotic convergence factor by lim k 1 1=k (5.6) where k is de ned in (1. 3) Following e.g. Saad [6] we get an upper bound on which we now state and prove. Theorem 5.4 The asymptotic convergence factor de ned in (5.6) satis es b : 5.7) Proof Relation (5.7) follows immediately from Theorem 5.2 and [6] We also de ne the residual reduction after 20 iterations by ....

.... lim k 1 1=k (5.6) where k is de ned in (1.3) Following e.g. Saad [6] we get an upper bound on which we now state and prove. Theorem 5.4 The asymptotic convergence factor de ned in (5.6) satis es b : 5.7) Proof Relation (5.7) follows immediately from Theorem 5. 2 and [6]. We also de ne the residual reduction after 20 iterations by 1=20 (5.8) In Figure 5.3 we have plotted the upper bound on the asymptotic convergence factor de ned in (5.7) and the residual reduction , 5.8) 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 ....

Youcef Saad. Krylov subspace methods for solving large unsymmetric linear systems. Math. Comp., 37:105-126, 1981.

.... (CG) Conjugate Residuals (CR) Hestenes and Stiefel [25] for Hermitian positive definite systems, CG Minimal Residual Method (MINRES) Paige and Saunders [32] for Hermitian indefinite systems, Full Orthogonalization Method (FOM) Generalized Minimal Residual Method (GMRES) Saad and Schultz [37, 38] for the general non Hermitian case as well as Biconjugate Gradient Method (BCG) Quasi Minimal Residual Method (QMR) Lanczos [29] Freund and Nachtigal [14] and Conjugate Gradients Squared (CGS) Transpose Free QMR (TFQMR) Sonneveld [39] Freund [11] where the last two pairs of methods require ....

Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Math. Comp., 37:105--126, 1981.

....expression of q k is obtained from that of x k by replacing the first column in the numerator of (4. 8) by [r k ; Ar 0 ; Ar k ) Ar k Gamma1 ; Ar k ) Other iterative projection algorithms such as the orthogonal error method [7] the Orthodir [18] and the method of Arnoldi [15] could be obtained from the ORIA and the RIA. ....

Y. Saad , Krylov subspace methods for solving large unsymmetric linear systems , Math. Comput., 37 (1981) 105-126.

....(r 0 ; A) 2.7) for B(x 0 ) an HPD matrix. If B is independent of x 0 , we have a fixed metric conjugate gradient method, or just a conjugate gradient method. Otherwise, we have a variable metric conjugate gradient method. For more information about projection methods and their properties, see ([Sa81], Sa82] JoMa90] In Section 3 we will show how the biconjugate gradient (BCG) and the quasi minimal residual (QMR) methods fit into this framework as variable metric CG methods. Before demonstrating these specific cases, we first discuss some general relationships between variable metric methods ....

Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, J. Mathematics of Computation, Vol 37, pp. 105-126, (1981).

....authors are discussed in x3 and results and a complexity analysis of the parallel implementations are given in x4. Finally we give our conclusions in x5. 2. Description of GMRES(c) We first consider the non restarted GMRES algorithm. It is a modification of the Full Orthogonalisation method [16] which uses the Arnoldi process to compute an orthonormal basis fv 1 ; v 2 ; v k g of the Krylov subspace K k (A; v 1 ; k) The solution x (k) is taken as x (0) V y (k) where V is the matrix whose columns are the vectors v k computed by the Arnoldi process. The vector y (k) is ....

Y. SAAD, Krylov subspace methods for solving large unsymmetric systems, Mathematics of Computation, 37 (1981), pp. 105--126.

....on nonsymmetric system solvers has attracted much attention in recent years. Several effective Krylov subspace methods have been established to solve nonsymmetric systems, for example various generalization of the conjugate gradient method [1, 3, 5] GMRES [13] QMR [8] and some of their variants [9, 12, 15]. Generally, Krylov subspace methods look for some basis of the Krylov subspace such that the method still keeps the finite termination property or and short recurrence for nonsymmetric A. In fact, the conjugate gradient method finds a basis where the exact solution of the system can be determined ....

Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comput., 37(1981) 105-126.

....France. E mail: essai ano.univ lille1.fr 1 where A is a n Theta n nonsingular real matrix, b 2 IR n the right hand side, and the vector x is the solution of the linear system. In general, these systems are large, sparse and nonsymmetric. To solve such systems, Saad proposed the FOM method [13], and later Saad and Schultz proposed the GMRES method [14] Both methods use the Arnoldi process to construct an orthonormal basis Vm = v 1 ; vm ] of the Krylov subspace Km (A; r 0 ) span(r 0 ; Ar 0 ; A m Gamma1 r 0 ) where x 0 is an initial guess and r 0 = b Gamma Ax 0 is ....

Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comput., 37 (1981) 105--126.

....; A n Gamma1 bi such that the residual rn = b Gamma Axn has minimal norm over all x 2 Kn . This vector xn can be represented in the form xn = q(A)b for some polynomial q(z) of degree n Gamma 1, and (1) comes upon writing rn = p (A)b with p (z) 1 Gamma zq(z) 2 P n . The Arnoldi iteration [1,4,20,21,22] is an algorithm that solves the analogous problem involving P n instead of P n : Arnoldi approximation problem. Find p 2 P n such that (2) kp (A)bk = minimum : Nothing essential changes if we take N = 1 and let A be a bounded operator. y We have assumed that the initial guess for the ....

Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp., 37 (1981), pp. 105--126.

....2 A special multiple Arnoldi process For the computation of the approximate solution um 1 in (1.2) s systems of linear equations with the same coefficient matrix have to be solved. At the first stage, for the solution of (I Gamma hflA) k 1 = w 1 ; the classical Arnoldi method (cf. [12]) may be used to construct a sequence of orthonormal vectors q 1 ; q 2 ; q 1 ; that constitute a basis for the Krylov subspace K 1 = K(A; q 1 ; 1 ) spanfq 1 ; A 1 Gamma1 q 1 g: This process mixes Krylov steps Aq j with Gram Schmidt orthogonalization q j 1 = v kvk 2 ....

Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp. 37 (1981), 105--126.

.... (CG) Conjugate Residuals (CR) Hestenes and Stiefel [25] for Hermitian positive definite systems, CG Minimal Residual Method (MINRES) Paige and Saunders [32] for Hermitian indefinite systems, Full Orthogonalization Method (FOM) Generalized Minimal Residual Method (GMRES) Saad and Schultz [37, 38] for the general non Hermitian case as well as Biconjugate Gradient Method (BCG) Quasi Minimal Residual Method (QMR) Lanczos [29] Freund and Nachtigal [14] and Conjugate Gradients Squared (CGS) Transpose Free QMR (TFQMR) Sonneveld [39] Freund [11] where the last two pairs of methods require ....

Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Math. Comp., 37:105--126, 1981.

.... [36] Note that this method includes the popular GCR method (Generalized Conjugate Residual or restarted ORTHOMIN) 7] ORES (ORTHORES) This is another truncated restarted method for nonsymmetric systems [36] IOM (Incomplete Orthogonalization Method) This is a truncated method due to Saad [25, 26] which calculates the same iterates, in exact arithmetic, as ORTHORES. In the symmetric case, it runs the SYMMLQ algorithm of Paige and Saunders [21] Only left preconditioning is allowed. GMRES (Generalized Minimal Residual Method) This method is a truncated restarted method which, in the case ....

Saad, Y. "Krylov Subspace Methods for Solving Large Unsymmetric Linear Systems." Math. Comp., 37 (July 1981), pp. 105-126.

....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 ....

Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comput., 37 (1981), pp. 105--126.

....slow convergence. In some other applications it is a very attractive method though. For a discussion and for stable algorithms see Paige and Saunders [13] 2. Create explicitly an orthogonal basis for the Krylov subspace. This can be done by orthogonalizing the residual vectors, which leads to FOM [14] (or GENCG, ORTHORES, etc. The more popular approach is to construct approximations in the Krylov subspace for which the norm of the residual is minimal, which leads to GMRES [1] or GCR, GENCR, ORTHOMIN, ORTHODIR, etc. which comes down to finding an orthogonal basis for a suitable Krylov ....

....i in (3.1b) then together with (3. 1a) we get V T i (b Gamma V i H i;i y) 0: 3:1c) Since V T i V i is the unit matrix of dimension i, it follows that y is the solution of H i;i y = V T i b; and this is a small system that can be solved relatively easily (if not singular) The methods FOM [14], ORTHORES [15] and GENCG [16] are implementations of this approach. Another and more robust approach is to rewrite the orthogonality relations as AV i = V i 1 H i 1;i ; 3:1d) in which H i 1;i is an i 1 by i upper Hessenberg matrix. Now we try to find the x i in the Krylov 4 CFD Review ....

Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Math. Comput., 37:105-126, 1981.

....of robust schemes for solving nonlinear problems. Key words. Nonlinear Problems, Newton s method, Inexact Newton, Iterative Methods AMS subject classification. 65H10 1. Introduction. Our goal in this paper is twofold. A number of iterative solvers for linear systems of equations, such as FOM [23], GMRES [26] GCR [30] flexible GMRES [25] GMRESR [28] and GCRO [9] are in structure very similar to iterative methods for linear eigenproblems, like shift and invert Arnoldi [1, 24] Davidson [8, 24] and Jacobi Davidson [27] We will show that all these algorithms can be viewed as instances ....

....directions p j . Preferable updates p k : P jk fl j p j are those for which b Gamma Ax k 1 , where x k 1 = x k p k , is minimal in some sense: e.g. kb Gamma Ax k 1 k 2 is minimal, as in GMRES [26] and GCR [30] or b Gamma Ax k 1 is orthogonal to the p j for j k, as in FOM or GENCG [23], or b Gamma Ax k 1 is quasi minimal , as in Bi CG [17] and QMR [12] Of course the distinction between preconditioning and acceleration is not a clear one. Acceleration techniques with a limited number of steps can be seen as a kind of dynamic preconditioning as opposed to the static ....

Y. Saad, Krylov subspace method for solving large unsymmetric linear systems, Math. Comp., 37 (1981), pp. 105--126. 18 Diederik R. Fokkema, Gerard L. G. Sleijpen and Henk A. Van der Vorst

....references as well as in our own experiments with one of the classical low rank methods introduced in the sequel. The method we refer to as full orthogonalization method for Lyapunov equations (FOML) HR92, JK94, Saa90] could be considered as an extension of FOM for systems of linear equations [Saa81] to matrix equations. Note that this method is frequently called Arnoldi method or Galerkin method. FOM L is based on the Arnoldi process (if m = 1) or the block Arnoldi process (if m 1) applied to the matrices A T and B. The purpose of this process is to establish an orthonormal basis V k 2 ....

Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Math. Comp., 37:105--126, 1981.

....is preconditioned. The most common choices of Km and Lm are the following. 1. Lm = Km = Km (A; r 0 ) The conjugate gradient method is a particular instance of this method when the matrix is symmetric positive definite. Another method in this class is the Full Orthogonalization Method (FOM) [21] which is closely related to Arnoldi s method for solving eigenvalue problems [1] Also in this class is ORTHORES [14] a method that is mathematically equivalent to FOM. Axelsson [2] also derived a similar algorithm for general nonsymmetric matrices. As an example we outline here the FOM method ....

Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Mathematics of Computation, 37:105--126, 1981.

....known to be locally orthogonal to each other, in that (v i ; v j ) ffi ij for ji Gamma jj k In addition, the relations (4) 5) are still valid, but the matrix Hm now has a particular structure, namely, it is banded Hessenberg since h i;j = 0 for i j Gamma k. 3. 2 DIOM The IOM algorithm [5, 4] is defined similarly to the FOM algorithm except that the Arnoldi vectors obtained are not orthogonal but locally orthogonal. The Hessenberg matrix Hm obtained from the incomplete orthogonalization process has a band structure with a bandwidth of k 1. For example when k = 3 and m = 5 we obtain ....

Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Mathematics of Computation, 37:105--126, 1981.

....either another Krylov subspace or a related subspace. Specifically, the most popular choices of Km and Lm are the following. Orthogonal Projection Methods: Lm = Km = Km (A; r 0 ) This is the orthogonal projection or Galerkin case. A method in this class is the Full Orthogonalization Method (FOM) [18] which is closely related to Arnoldi s method for solving eigenvalue problems [1] Also in this class is ORTHORES [12] a method that is mathematically equivalent to FOM. Axelsson [2] also 4 Chapter 1 derived an algorithm of this class for general nonsymmetric matrices. When A is symmetric ....

Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Mathematics of Computation, 37:105--126, 1981.

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Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp. 37 (1981), 105--126.

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Y. SAAD, Krylov subspace methods for solving large unsymmetric systems, Mathematics of Computation, 37 (1981), pp. 105--126.

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Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comput., 37(1981) 105-126.

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Y. Saad. Krylov subspace methods for solving large unsymmetric linear systems. Mathematics of Computation, 37(155):105--126, July 1981.

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Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp., 37 (1981), pp. 105--126.

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