E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. SIAM J. Comput. Appl. Math., 86:387--414, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to 6.0, which has been determined through numerical experiments. Eq. 10 improves the diagonal dominance of the preconditioning matrix and reduces the work and storage requirements of the incomplete factorization. This approach is similar to the diagonal shift strategy suggested by Chow and Saad [4]. The present preconditioning matrix is a compromise between a preconditioner based on a first order upwind discretization of the flowfield equations and a preconditioner based on the actual second order discretization. This novel intermediate preconditioner is significantly more effective than ....

Chow E and Saad Y. Experimental study of ILU preconditioners for indefinite matrices. Journal of Computational and Applied Mathematics, Vol. 86, pp 387--414, 1997.

....is very sensitive to the condition number of the matrix. Since the Jacobian of the equations being solved is typically extremely ill conditioned, a good preconditioner is required to limit the number of inner iterations. Pueyo and Zingg [14] have shown that an incomplete LU preconditioner (ILU) [5] with two levels of fill minimizes solution time. They also found that a preconditioner based on a first order Jacobian is more efficient than the exact Jacobian, both in saving memory and CPU time. The first order Jacobian is formed by using only second difference dissipation. This reduces the ....

....the preconditioner settings. The preconditioner must be conditioned well enough to be stable, but remain close enough to the true Jacobian to provide adequate clustering of the eigenvalues. A useful tool in evaluating a preconditioner is its condition number estimate, as discussed by Chow and Saad [5]. This is simply the L 2 norm of the solution to LU c = 1. If a preconditioner is performing poorly, and has a high condition number estimate of the order of 10 7 or greater, more diagonal dominance is needed. It is usually better to err on the side of being too well conditioned. While this ....

Chow E and Saad Y. Experimental study of ILU preconditioners for indefinite matrices. J. of Comp. and Appl. Mathematics, Vol. 87, pp 387-- 414, 1997.

....considered. In addition, we will see that for problems that are strongly nonsymmetric and or are far from being diagonally dominant, the norm of the residual matrix R alone is usually not a reliable indicator of the quality of the corresponding preconditioner. It has been pointed out, e.g. in [11], that a more revealing measure of the quality of the preconditioner can be obtained by considering the Frobenius norm of the deviation of the preconditioned matrix from the identity, i.e. Note that this quantity is equal to F . Even if R is small in norm, it could happen that ( L ....

....is not surprising, considering that for general nonsymmetric problems it is not known how to predict the rate of convergence of iterative solvers. In practice, however, one can expect that very ill conditioned incomplete L and U factors will result in a poor preconditioner. As suggested in [11], an inexpensive way of detecting this illconditioning is by computing L U) 1 e# , where e denotes a vector of all ones. This is only a lower bound for ## but is quite useful in practice. The e#ects of permutations on preconditioned Krylov subspace methods for nonsymmetric problems ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387--414.

....if at least one level of fill in is retained. For higher values of density for the approximation of A, the factors may become very ill conditioned and consequently the preconditioner is very poor. This behaviour has been already observed on sparse real indefinite systems, see for instance [12]. As an attempt for a possible remedy, following [25, 26] we apply IC(#) to a perturbation of # A by a complex diagonal matrix, more specifically, we use # A # = # A i #h# r , 9) where # r = diag(Re(A) diag(Re( # A) and # stands for a nonnegative real parameter, while h = n 1 ....

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. Journal of Computational and Applied Mathematics, 86:387--414, 1997.

....assumption is not valid. In the case of nonsymmetric non diagonally dominant matrices such as the ones arising in our applications, the incomplete factors and can be more ill conditioned than the original matrix, and the long recurrences associated with backward and forward solves may be unstable [45, 46, 47]. Since diagonal dominance tends to alleviate this problem [48] we can benefit by using an approximate Jacobian matrix with reduced off diagonal dominance. This was shown in Ref. 18] The off diagonal dominance of the Jacobian matrix can be reduced by using only second difference dissipation in ....

Chow, E., and Saad, Y., "Experimental Study of ILU Preconditioners for Indefinite Matrices," Tech. Rep. UMSI 97/95, University of Minnesota Supercomputing Institute Research, June 1997.

....problem. An iterative solver that works well on one matrix may be ine#cient or even fail to converge on another. For example, Grote and Huckle [8] switch from right to left preconditioning in order to achieve convergence with a sparse approximate inverse preconditioner on pores2, and Chow and Saad [2] switch from row to column oriented factorization to achieve convergence with lhr01. Chow and Saad also use a variety of other techniques to solve other systems. There are no established criteria that can guide an automatic system as to which solver is appropriate for a given matrix. Therefore, it ....

Edmond Chow and Yousef Saad. Experimental study of ILU preconditioners for indefinite matrices. Technical Report UMSI 97/97, Supercomputer Institute, University of Minnesota, June 1997.

....this restriction. A disadvantage of the ILU(p) factorization is that the amount of fill and the amount of work needed to obtain the ILU(p) factorization is unpredictable for p 0. Guidelines for the use of the ILU(p) factorization and descriptions of several implementations can be found in [9, 11, 30, 45]. Drop tolerance strategies include nonzero elements in the incomplete factor if they are larger than some threshold parameter. For example, Munksgaard [32] drops a (k) ij during the kth step if a (k) ij # # # a (k) ii a (k) jj , where # is the drop tolerance. A disadvantage of ....

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387--414.

....this restriction. A disadvantage of the ILU(p) factorization is that the amount of fill and the amount of work needed to obtain the ILU(p) factorization is unpredictable for p 0. Guidelines for the use of the ILU(p) factorization, and descriptions of several implementations can be found in [9, 11, 30, 45]. Drop tolerance strategies include nonzero elements in the incomplete factor if they are larger than some threshold parameter. For example, Munksgaard [32] drops a (k) ij during the kth step if ja (k) ij j q a (k) ii a (k) jj ; where is the drop tolerance. A disadvantage of drop ....

E. Chow and Y. Saad, Experimental study of ilu preconditioners for indefinite matrices, Journal of Computational and Applied Mathematics, 86 (1997), pp. 387--414.

....Saad (1996) for an overview. Incomplete factorizations are used so that the resulting factors are more economical to store, to compute, and to solve with. One of the reasons why incomplete factorizations can behave poorly is that pivots can be arbitrarily small (Benzi, Szyld and van Duin 1997, Chow and Saad 1997). Pivots may even be zero in which case the incomplete factorization fails. Small pivots allow the numerical values of the entries in the incomplete factors to become very large, which leads to unstable and therefore inaccurate factorizations. In such cases, the norm of the residual matrix R = A ....

Chow, E. and Saad, Y. (1997), Experimental study of ILU preconditioners for indefinite matrices, Technical Report TR 97/95, Department of Computer Science, and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis.

....for solving many nonsymmetric and indefinite matrices, despite the fact that their existence in these applications is not guaranteed. However, their failure rates are still too high for them to be used as blackbox library software for solving general sparse matrices of practical interests [9, 25]. In fact, the lack of robustness of preconditioned iterative methods is currently the major impediment for them to gain acceptance in industrial applications, in spite of their intrinsic advantage for large scale problems. For indefinite matrices, there are at least two reasons that make ILU ....

....of preconditioned iterative methods is currently the major impediment for them to gain acceptance in industrial applications, in spite of their intrinsic advantage for large scale problems. For indefinite matrices, there are at least two reasons that make ILU factorization approaches problematic [9]. The first problem is due to small or zero pivots [26] Pivots in an indefinite matrix can be arbitrarily small. This may lead to unstable and inaccurate factorizations. In such cases, the size of the elements in the LU factors may be very large and these large size elements lead to inaccurate ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. J. Comput. Appl. Math., 86(2):387--414, 1997. 21

....in many cases. In addition, we will see that for problems that are strongly nonsymmetric and or far from being diagonally dominant, the norm of the residual matrix R alone is usually not a reliable indicator of the quality of the corresponding preconditioner. It has been pointed out, e.g. in [10], that a more revealing measure of the quality of the preconditioner can be obtained by considering the Frobenius norm of the deviation of the preconditioned matrix from the identity, i.e. #I A( L U) 1 # F . Note that this quantity is equal to #R( L U) 1 # F .EvenifR is small in ....

....This is not surprising, considering that for general nonsymmetric problems it is not known how to predict the rate of convergence of iterative solvers. In practice, however, one can expect that very ill conditioned incomplete L and U factors will result in a poor preconditioner. As suggested in [10], an inexpensive way of detecting this ill conditioning is by computing #( L U) 1 e# #,w heree denotes a vector of all ones. This is only a lower bound for #( L U) 1 # # , but is quite reliable in practice. The e#ects of permutations on preconditioned Krylov subspace methods for ....

[Article contains additional citation context not shown here]

Edmond Chow and Yousef Saad. Experimental study of ILU preconditioners for indefinite matrices. Minnesota Supercomputer Institute Technical Report TR 97/95, University of Minnesota, Minneapolis, June 1997. To appear in Journal of Computational and Applied Mathematics.

.... incomplete factorization methods (like Incomplete Cholesky or ILU and their variants) are widely popular and fairly robust techniques; see, e.g. 2] and [34] Yet, they are not always reliable, in that the incomplete factorization process may suffer from various types of instability [15], 18] Even in the SPD case, existence of the standard Incomplete Cholesky (IC) factorization [31] is guaranteed only for special classes of matrices, such as H matrices [30] For general SPD matrices, breakdown in the IC process may occur due to exceedingly small or negative pivots. For this ....

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387--414.

....under these circumstances, and typically deliver good rates of convergence. In contrast, such preconditioners are often unstable or may not even be defined when the coefficient matrix lacks diagonal dominance, has zeros on the main diagonal and or is highly nonsymmetric. We refer the reader to [16] for a study of the instabilities affecting ILU type techniques when applied to general sparse matrices. See also section 2. Furthermore, the presence of many eigenvalues with arbitrary real part (positive, negative and zero) causes serious difficulties for many Krylov subspace solvers. Matrices ....

....Frobenius norm minimization [19] 34] 17] 35] However, all these techniques are considerably more expensive than standard ones based on incomplete factorizations, and furthermore, as acknowledged in the papers just referenced, there are problems that cannot be solved even by such methods. In [16], which also addresses the problem of preconditioning general sparse matrices, several heuristics were tried in order to improve the reliability of ILU preconditioners. These include dynamic permutations (pivoting) reordering, scaling, perturbing diagonal elements, and blocking. Most of these ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387--414.

....An iterative solver that works well on one matrix may be inefficient or even fail to converge on another. For example, Grote and Huckle [8] switch from right to left preconditioning in order to achieve convergence with a sparse approximate inverse preconditioner on pores2, and Chow and Saad [2] switch from row to column oriented factorization to achieve convergence with lhr01. Chow and Saad also use a variety of other techniques to solve other systems. There are no established criteria that can guide an automatic system as to which solver is appropriate for a given matrix. Therefore, it ....

Edmond Chow and Yousef Saad. Experimental study of ILU preconditioners for indefinite matrices. Technical Report UMSI 97/97, Supercomputer Institute, University of Minnesota, June 1997.

....iterations and solution timings, but element by element comparisons between the factors themselves have apparently not been undertaken. The relationships between matrix orderings and ILU preconditioning performance is a complex issue which has, and continues to be, studied by numerous researchers [2, 5, 7, 8, 9]. At least four interrelated effects can be identified in parallel contexts. First, matrix ordering can be used to provide parallelism, i.e, as a partitioning method. Second, from a structural viewpoint, altering a matrix s ordering and ILU(k) or ILUT( p) parameters changes the amount and or ....

E. Chow and Y. Saad. Experimental study of ILU preconditioners of indefinite matrices. J. Comput. Appl. Math, 86:387--414, 1997.

....3 has 3; 864 unknowns and 238; 252 nonzeros. It comes from an Euler equation model and was supplied by L. Wigton from Boeing. It is solvable by ILUT with large values of p [14] This matrix was also used to compare BILUM with ILUT in [38] and to test point and block preconditioning techniques in [15, 16]. The test results in Table 2 show that BILUTM is the most robust preconditioner and ILUT is the least robust in solving the WIGTO966 matrix. In fact, for all the parameters tested, ILUT did not converge at all, even if it used more storage space than other preconditioners did in some cases. It is ....

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. J. Comput. Appl. Math., 86(2):387--414, 1997.

....for solving many nonsymmetric and indefinite matrices, despite the fact that their existence in these applications is not guaranteed. However, their failure rates are still too high for them to be used as blackbox library software for solving general sparse matrices of practical interests [9]. In fact, the lack of robustness of preconditioned iterative methods is currently the major impediment for them to gain acceptance in industrial applications, in spite of their intrinsic advantage for large scale problems. For indefinite matrices, there are at least two reasons that make ILU ....

....of preconditioned iterative methods is currently the major impediment for them to gain acceptance in industrial applications, in spite of their intrinsic advantage for large scale problems. For indefinite matrices, there are at least two reasons that make ILU factorization approaches problematic [9]. The first problem is due to small or zero pivots [23] Pivots in an indefinite matrix can be arbitrarily small. This may lead to unstable and inaccurate factorizations. In such cases, the size of the elements in the LU factors may be very large and these large size elements lead to inaccurate ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. J. Comput. Appl. Math., 86(2):387--414, 1997.

....known that incomplete factorization techniques can fail on matrices which are strongly nonsymmetric and or indefinite. The failure is usually due to some form of instability, either in the incomplete factorization itself (zero or very small pivots) or in the back substitution phase, or both; see [29]. Most approximate inverse techniques are largely immune from these problems, and therefore constitute an important complement to more standard preconditioning methods even on serial computers. We remark that approximate inverse techniques rely on the tacit assumption that for a given sparse ....

....have. These disadvantages all stem from the assumption that an ILU factorization has already been computed. This implies that these methods are not even applicable if an ILU factorization does not exist, or if it is unstable, as is sometimes the case for highly nonsymmetric, indefinite problems [29], 42] Clearly, this assumption also limits the parallel efficiency of this class of methods, since the preconditioner construction phase is not entirely parallelizable (computing an ILU factorization is a highly sequential process) Another disadvantage is that the computation of the ....

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. Journal of Computational and Applied Mathematics, 86:387--414, 1997.

....[2] and Saad [19] among others. Duff and Meurant have studied the impact of reorderings on incomplete factorization preconditioning for SPD problems. The sources of inaccuracy in incomplete factorizations and the effect of reorderings on accuracy and stability have been analyzed by Chow and Saad [4] and Benzi et al. 2] Let A = L U . The residual matrix R = A Gamma A measures the accuracy of the incom M. Benzi, W. Joubert, and G. Mateescu 10003 plete factorization. Let k Delta kF denote the Frobenius norm of a matrix. Chow and Saad [4] and Benzi et al. 2] have shown that ....

....have been analyzed by Chow and Saad [4] and Benzi et al. 2] Let A = L U . The residual matrix R = A Gamma A measures the accuracy of the incom M. Benzi, W. Joubert, and G. Mateescu 10003 plete factorization. Let k Delta kF denote the Frobenius norm of a matrix. Chow and Saad [4] and Benzi et al. 2] have shown that for nonsymmetric problems kRkF may be an insufficient characterization of the convergence of the preconditioned iterative methods. This is in contrast with symmetric positive definite problems where R provides a good measure of convergence. These authors have ....

E. CHOW AND Y. SAAD, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387--414.

....results, there seems to be no comparison of preconditioned Krylov subspace methods involving a more powerful general purpose preconditioner and more realistic nonsymmetric problems from CFD applications. High accuracy preconditioners are relatively new and they have been mostly tested with GMRES [6, 4]. A comprehensive comparison of Krylov subspace methods with modern preconditioning techniques will undoubtedly offer information that may help researchers and application engineers choose suitable iterative schemes for their own problems. 3 Multi Level Block ILU Preconditioner Multi level block ....

.... matrix, which was constructed using a fully coupled solution method (mixed u p formulation) The penalty method gives very ill conditioned matrices, whereas mixed u p gives indefinite, larger systems (they include pressure variables) Several of these matrices contain small or zero diagonal values [6]. 3 The zero diagonals are due to the incompressibility condition of the Navier Stokes equations [6] The substantial amount of zero diagonals makes these matrices indefinite. It is remarked in [4] that the FIDAP matrices are difficult 2 All FIDAP matrices are available from the MatrixMarket ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. J. Comput. Appl. Math., 86(2):387--414, 1997.

....indicator of the quality of the corresponding preconditioner. A more revealing measure of the quality of the preconditioner can be obtained by considering the Frobenius norm of the deviation of the preconditioned matrix from the identity, i.e. kI Gamma A( L U) Gamma1 kF ; see, e.g. [7]. This quantity is equal to kR( L U) Gamma1 kF . Even if R is small in norm, it could happen that ( L U) Gamma1 has very large entries, resulting in a large deviation of the preconditioned matrix from the identity. As a result, the preconditioned iteration fails to converge. A ....

....kRkF need not be very large, but kI Gamma A( L U) Gamma1 kF = kR( L U) Gamma1 kF will be. Again, this is a common situation when the coefficient matrix is far from being diagonally dominant. Of course, both types of instabilities can simultaneously occur for a given problem: see [7] for an extensive experimental study of the causes of failure of incomplete factorizations. In [7] however, only scant attention is devoted to the effect of reorderings. In order to gain some insight about the effect of reorderings, we computed the Frobenius norms of R and R( L U) Gamma1 ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. To appear in J. Comput. Appl. Math..

....ff are proposed in subsection 3.3 for an exact factorization of A ffI used with the CG algorithm. Here, for a test problem, we showthe dependences of the convergence rate and the stability of the preconditioner on ff and outline an automatic process of arriving at an appropriate shift value. In [4], a strong correlation between stability of the preconditioner and the size of E = log (k(LU) 1 k inf )was shown and was suggested as a practical means of evaluating the quality of a preconditioner. We can inexpensively compute E ff as E ff = log (k(LU) 1 ek 1 )# where e is a vector ....

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, Journal of Computational and Applied Mathematics, 87 (1997), pp. 387--414.

....be factored using Gaussian elimination and the factors will then be kept instead of the explicit inverses. If explicit inversion is employed, it is best to utilize a thresholded pseudo inverse employing a singular value decomposition as is often done in block ILU factorizations, for details, see [7, 8]. The above reduction process can be continued recursively with the matrix A j being replaced by A j 1 , until we reach a Schur complement matrix A nlev which is small enough to be solved by a direct or a preconditioned iterative method. The multi level block factorization described above becomes ....

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math. (to appear).

....ff are proposed in subsection 3.3 for an exact factorization of A ffI used with the CG algorithm. Here, for a test problem, we show the dependences of the convergence rate and the stability of the preconditioner on ff and outline an automatic process of arriving at an appropriate shift value. In [4], a strong correlation between stability of the preconditioner and the size of E = log (k(LU) Gamma1 k inf ) was shown and was suggested as a practical means of evaluating the quality of a preconditioner. We can inexpensively compute E ff as E ff = log (k(LU) Gamma1 ek 1 ) where e is a ....

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, Journal of Computational and Applied Mathematics, 87 (1997), pp. 387--414.

....be factored using Gaussian elimination and the factors will then be kept instead of the explicit inverses. If explicit inversion is employed, it is best to utilize a thresholded pseudo inverse employing a singular value decomposition as is often done in block ILU factorizations, for details, see [10, 11]. The multi level block factorization described above becomes more expensive as the number of levels increases. This is because of the fill ins introduced by the elimination process. The matrices E j D Gamma1 j and A j 1 as in (4.2) become denser as the factorization proceeds. A general ....

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387--414.

....be factored using Gaussian elimination and the factors will then be kept instead of the explicit inverses. If explicit inversion is employed, it is best to utilize a thresholded pseudo inverse employing a singular value decomposition as is often done in block ILU factorizations, for details, see [10, 11]. The multi level block factorization described above becomes more expensive as the number of levels increases. This is because of the fill ins introduced by the elimination process. The matrices E j D Gamma1 j and A j 1 as in (7) become denser as the factorization proceeds. A general practice ....

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math. 86, 387--414 (1997).

....with very small or zero diagonal elements. Matrices with these features are usually difficult to solve by standard ILU factorization techniques because small pivots yield large L and U factors that may cause stability problems during the forward elimination and the backward substitution processes [5, 9]. Liquid Flow surface Wall Wall Air Figure 1: Air penetrating the liquid flowing between two walls (PDF 1 matrix) Roll Air Air Blade Liquid Pan Figure 2: The knife coating model, a blade is wiping away the liquid from a rotating roll (BKNIFE 01 matrix) Multi Level Preconditioner for Coating ....

Chow, E., and Saad, Y., Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), 387--414.

....1998. y E mail: saad cs.umn.edu. URL: http: www.cs.umn.edu saad. z E mail: jzhang cs.umn.edu. URL: http: www.cs.umn.edu jzhang. Current affiliation: Department of Computer Science, University of Kentucky, 773 Anderson Hall, Lexington, KY 40506 0046. preconditioners has been conducted in [21] and a number of ILU type preconditioners have been tested for solving some difficult problems from computational fluid dynamics in [18, 19] The Incomplete LU factorization without fill ins (ILU(0) is probably the best known general purpose preconditioner [38] However, this preconditioner is ....

....4 has 3; 864 unknowns and 238; 252 nonzeros. It comes from an Euler equation model and was supplied by L. Wigton from Boeing. It is solvable by ILUT with large values of p [18] This matrix was also used to compare BILUM with ILUT in [52] and to test point and block preconditioning techniques in [19, 21]. BILUM (with GMRES(10) was shown to be 6 times faster than ILUT with only one third of the memory required by ILUT [52] In our current tests, we chose several values for and p for BILUTM and ILUT, and the size of the blocks in case of BILUTM. 4 The WIGTO966 matrix is available from the ....

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. J. Comput. Appl. Math., 86(2):387--414, 1997.

....such their efficiency is somewhat limited to those applications. The main trade offs when comparing preconditioners are in their intrinsic efficiency, their parallelism, and their robustness. An experimental study on robustness of some of these general purpose preconditioners has been conducted in [14]. For an introduction to Krylov subspace methods and various preconditioning techniques, see [30] The multi elimination ILU preconditioner (ILUM) introduced in [29] is based on exploiting the idea of successive independent set orderings and several heuristic algorithms have been suggested to ....

....the adjacency graph of the matrices will cause nodes with small or zero diagonal values to be part of the independent set. As a result, the inverse of the diagonal matrix with such diagonal entries will contain very large elements and this may cause the resulting preconditioner to be unstable [14]. Although techniques may be used to invert a near singular diagonal matrix approximately, the resulting preconditioner is sometimes less efficient. A common method for assessing the quality of a preconditioner is to estimate the condition number of the preconditioned system. These estimates are ....

[Article contains additional citation context not shown here]

E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math. 86 (2), 387--414 (1997).

No context found.

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. SIAM J. Comput. Appl. Math., 86:387--414, 1997.

No context found.

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. J. Comput. Appl. Math., 86(2):387--414, 1997.

No context found.

E. Chow and Y. Saad. Experimental study of ILU preconditioners for indefinite matrices. J. Comput. Appl. Math., 86(2):387--414, 1997.

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