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Journal ArticleDOI

On the simplification of generalized conjugate-gradient methods for nonsymmetrizable linear systems

TL;DR: In this article, it was shown that the generalized conjugate-gradient (CC) can be simplified if a nonsingular matrix H is available such that HA = ATH.
About: This article is published in Linear Algebra and its Applications.The article was published on 1983-07-01 and is currently open access. It has received 73 citations till now. The article focuses on the topics: Conjugate gradient method & Identity matrix.
Citations
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01 Dec 1999
TL;DR: MT3DMS as discussed by the authors is the next generation of the modular three-dimensional transport model, with significantly expanded capabilities, including the addition of a third-order total-variation-diminishing (TVD) scheme for solving the advection term that is mass conservative but does not introduce excessive numerical dispersion and artificial oscillation.
Abstract: : This manual describes the next generation of the modular three-dimensional transport model, MT3D, with significantly expanded capabilities, including the addition of (a) a third-order total-variation-diminishing (TVD) scheme for solving the advection term that is mass conservative but does not introduce excessive numerical dispersion and artificial oscillation, (b) an efficient iterative solver based on generalized conjugate gradient methods and the Lanczos/ORTHOMIN acceleration scheme to remove stability constraints on the transport time-step size, (c) options for accommodating nonequilibrium sorption and dual-domain advection-diffusion mass transport, and (d) a multicomponent program structure that can accommodate add-on reaction packages for modeling general biological and geochemical reactions MT3DMS can be used to simulate changes in concentrations of miscible contaminants in groundwater considering advection, dispersion, diffusion, and some basic chemical reactions, with various types of boundary conditions and external sources or sinks The basic chemical reactions included in the model are equilibrium-controlled or rate-limited linear or nonlinear sorption and first-order irreversible or reversible kinetic reactions MT3DMS can accommodate very general spatial discretization schemes and transport boundary conditions, including: (a) confined, unconfined, or variably confined/unconfined aquifer layers, (b)inclined model layers and variable cell thickness within the same layer, (c) specified concentration or mass flux boundaries, and (d) the solute transport effects of external hydraulic sources and sinks such as wells, drains, rivers, areal recharge, and evapotranspiration

1,195 citations


Cites methods from "On the simplification of generalize..."

  • ...The basic iterative algorithm is accelerated by the Lanczos/ORTHOMIN acceleration scheme (Jea and Young 1983; Young and Mai 1988) if the coefficient matrix A is nonsymmetric, as in the case of fully implicit finite-difference formulation....

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Journal ArticleDOI
TL;DR: Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed.
Abstract: Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters. Copyright © 2006 John Wiley & Sons, Ltd.

408 citations

Journal ArticleDOI
TL;DR: It is shown that any CG method for $Ax = b$ is characterized by an hpd inner product matrix B and a left preconditioning matrix C and how eigenvalue estimates may be obtained from the iteration parameters, generalizing the well-known connection between CG and Lanczos.
Abstract: The conjugate gradient method of Hestenes and Stiefel is an effective method for solving large, sparse Hermitian positive definite (hpd) systems of linear equations, $Ax = b$. Generalizations to non-hpd matrices have long been sought. The recent theory of Faber and Manteuffel gives necessary and sufficient conditions for the existence of a CG method. This paper uses these conditions to develop and organize such methods. It is shown that any CG method for $Ax = b$ is characterized by an hpd inner product matrix B and a left preconditioning matrix C. At each step the method minimizes the B-norm of the error over a Krylov subspace. This characterization is then used to classify known and new methods. Finally, it is shown how eigenvalue estimates may be obtained from the iteration parameters, generalizing the well-known connection between CG and Lanczos. Such estimates allow implementation of a stopping criterion based more nearly on the true error.

251 citations

Journal ArticleDOI
TL;DR: It is shown how the cure for exact breakdown can be extended to near-breakdown in such a way that (in exact arithmetic) the well-conditioned formal orthogonal polynomials and the corresponding Krylov space vectors do not depend on the threshold specifying the near- breakdown.
Abstract: This paper is a continuation of Part I [M. H. Gutknecht, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 594--639], where the theory of the "unsymmetric" Lanczos biorthogonalization (BO) algorithm and the corresponding iterative method BIORES for non-Hermitian linear systems was extended to the nongeneric case. The analogous extension is obtained here for the biconjugate gradient (or BIOMIN) method and for the related BIODIR method. Here, too, the breakdowns of these methods can be cured. As a preparation, mixed recurrence formulas are derived for a pair of sequences of formal orthogonal polynomials belonging to two adjacent diagonals in a nonnormal Pade table, and a matrix interpretation of these recurrences is developed. This matrix interpretation leads directly to a completed formulation of the progressive qd algorithm, valid also in the case of a nonnormal Pade table. Finally, it is shown how the cure for exact breakdown can be extended to near-breakdown in such a way that (in exact arithmetic) the well-conditioned formal orthogonal polynomials and the corresponding Krylov space vectors do not depend on the threshold specifying the near-breakdown.

156 citations


Cites methods from "On the simplification of generalize..."

  • ...D Finally, we want to sketch the nongeneric generalization of yet another important algorithm, namely, of BIODIR [11] (or Lanczos/ORTHODIR [16])....

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  • ...From Algorithm 4 it is a small step to a nongeneric version of the BCG method, which also goes under the names Lanczos/ORTHOMIN [16] and BIOMIN [11]....

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Journal ArticleDOI
Martin H. Gutknecht1
TL;DR: This review article introduces the reader to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones.
Abstract: Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by look-ahead are also discussed.

121 citations

References
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Journal ArticleDOI
TL;DR: An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns and it is shown that this method is a special case of a very general method which also includes Gaussian elimination.
Abstract: An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns. The solution is given in n steps. It is shown that this method is a special case of a very general method which also includes Gaussian elimination. These general algorithms are essentially algorithms for finding an n dimensional ellipsoid. Connections are made with the theory of orthogonal polynomials and continued fractions.

7,598 citations

Journal ArticleDOI
TL;DR: In this article, a systematic method for finding the latent roots and principal axes of a matrix, without reducing the order of the matrix, has been proposed, which is characterized by a wide field of applicability and great accuracy, since the accumulation of rounding errors is avoided, through the process of minimized iterations.
Abstract: The present investigation designs a systematic method for finding the latent roots and the principal axes of a matrix, without reducing the order of the matrix. It is characterized by a wide field of applicability and great accuracy, since the accumulation of rounding errors is avoided, through the process of \"minimized iterations\". Moreover, the method leads to a well convergent successive approximation procedure by which the solution of integral equations of the Fredholm type and the solution of the eigenvalue problem of linear differential and integral operators may be accomplished.

3,947 citations

Journal ArticleDOI
Walter Arnoldi1
TL;DR: In this paper, an interpretation of Dr. Cornelius Lanczos' iteration method, which he has named ''minimized iterations'' is discussed, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and nonhomogeneous form.
Abstract: An interpretation of Dr. Cornelius Lanczos' iteration method, which he has named \"minimized iterations\", is discussed in this article, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and nonhomogeneous form. This interpretation leads to a variation of the Lanczos procedure which may frequently be advantageous by virtue of reducing the volume of numerical work in practical applications. Both methods employ essentially the same algorithm, requiring the generation of a series of orthogonal functions through which a simple matrix equation of reduced order is established. The reduced matrix equation may be solved directly in terms of certain polynomial functions obtained in conjunction with the generated orthogonal functions, and the convergence of the solution may be observed as the order of the reduced matrix is successively increased with the order of the original matrix as a limit. The method of minimized iterations is recommended as a rapid means for determining a small number of the larger eigenvalues and modal columns of a large matrix and as a desirable alternative for various series expansions of the Fredholm problem. 1. The conventional iterative procedures. It is frequently required that real latent roots, or eigenvalues, and modal columns be determined for a real numerical matrix, u, of order, n, in the characteristic homogeneous equation,*

1,826 citations

Journal ArticleDOI
TL;DR: The method of conjugate gradients for solving systems of linear equations with a symmetric positive definite matrix A is given as a logical development of the Lanczos algorithm for tridiagonalizing...
Abstract: The method of conjugate gradients for solving systems of linear equations with a symmetric positive definite matrix A is given as a logical development of the Lanczos algorithm for tridiagonalizing...

1,644 citations

Book
26 Jul 2004

1,106 citations