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Open AccessJournal ArticleDOI

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.

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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, +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.
Journal ArticleDOI

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.
Journal ArticleDOI

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.
Journal ArticleDOI

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.
Journal ArticleDOI

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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Journal ArticleDOI

Krylov subspace methods for solving large unsymmetric linear systems

TL;DR: Some algorithms based upon a projection process onto the Krylov subspace K/sub m/ = Span(r/sub 0/,..., A/sup m-1/r/ sub 0/) are developed, generalizing the method of conjugate gradients to unsymmetric systems, extensions of Arnoldi's algorithm for solving eigenvalue problems.
Journal ArticleDOI

The Lanczos algorithm with selective orthogonalization

TL;DR: It is shown how a modification called selective orthogonalization stifles the formation of duplicate eigenvectors without increasing the cost of a Lanczos step signifi'cantly.
Book ChapterDOI

A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations

TL;DR: A generalized conjugate gradient method for solving sparse, symmetric, positive-definite systems of linear equations, principally those arising from the discretization of boundary value problems for elliptic partial differential equations is considered.
Journal ArticleDOI

Some stable methods for calculating inertia and solving symmetric linear systems

TL;DR: Several decompositions of symmetry matrices for calculating inertia and solving systems of linear equations are discussed and new partial pivoting strategies for decomposing symmetric matrices are introduced and analyzed.
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