On the simplification of generalized conjugate-gradient methods for nonsymmetrizable linear systems
TLDR
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.read more
Citations
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MT3DMS: A Modular Three-Dimensional Multispecies Transport Model for Simulation of Advection, Dispersion, and Chemical Reactions of Contaminants in Groundwater Systems; Documentation and User's Guide
Chunmiao Zheng,P P Wang +1 more
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.
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Recent computational developments in krylov subspace methods for linear systems
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.
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A taxonomy for conjugate gradient methods
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.
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A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms, Part II
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.
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Lanczos-type Solvers for Nonsymmetric Linear Systems of Equations
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.
References
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The Use of Pre-conditioning in Iterative Methods for Solving Linear Equations with Symmetric Positive Definite Matrices
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Practical Use of Some Krylov Subspace Methods for Solving Indefinite and Nonsymmetric Linear Systems
TL;DR: It is shown how an algorithm similar to the SYMMLQ can be derived for nonsymmetric problems and a more economical algorithm based upon the $LU$ factorization with partial pivoting is described.
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The Use of Conjugate Gradients for Systems of Linear Equations Possessing “Property A”
TL;DR: It is shown that for systems possessing Young’s “Property A” the work in applying the conjugate gradients algorithm may be approximately halved and a vector of storage may be saved by using a technique analogous to that of Golub and Varga for Chebyshev semi-iteration.
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The Conjugate Residual Method for Constrained Minimization Problems
TL;DR: The conjugate gradient method can be viewed as a descent procedure for general unconstrained minimization problems as discussed by the authors, and through this viewpoint it can be generalized to general problems.
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