Towards understanding CG and GMRES through examples

[1]  Bjørn Fredrik Nielsen,et al.  A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators , 2024, SIAM Rev..

[2]  Richard J. Leute,et al.  An optimal preconditioned FFT-accelerated finite element solver for homogenization , 2023, Appl. Math. Comput..

[3]  B. F. Nielsen,et al.  Numerical approximation of the spectrum of self-adjoint operators in operator preconditioning , 2022, Numerical Algorithms.

[4]  Matthew J. Colbrook,et al.  SpecSolve: Spectral methods for spectral measures , 2022, ArXiv.

[5]  Jan Zeman,et al.  Elimination of ringing artifacts by finite-element projection in FFT-based homogenization , 2021, J. Comput. Phys..

[6]  Ivana Pultarová,et al.  Two‐sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems , 2021, Numer. Linear Algebra Appl..

[7]  Gérard Meurant,et al.  Accurate error estimation in CG , 2021, Numerical Algorithms.

[8]  John W. Pearson,et al.  Preconditioners for Krylov subspace methods: An overview , 2020, GAMM-Mitteilungen.

[9]  Matthew J. Colbrook,et al.  Computing spectral measures of self-adjoint operators , 2020, SIAM Rev..

[10]  Pseudospectra of matrices , 2020, Spectra and Pseudospectra.

[11]  Erin Carson,et al.  On the cost of iterative computations , 2020, Philosophical Transactions of the Royal Society A.

[12]  Jan Zeman,et al.  Guaranteed Two-Sided Bounds on All Eigenvalues of Preconditioned Diffusion and Elasticity Problems Solved by the Finite Element Method , 2020, Applications of Mathematics.

[13]  G. Meurant On prescribing the convergence behavior of the conjugate gradient algorithm , 2019, Numerical Algorithms.

[14]  Ivana Pultarová,et al.  Decomposition into subspaces preconditioning: abstract framework , 2019, Numerical Algorithms.

[15]  Miroslav Tuma,et al.  The Numerical Stability Analysis of Pipelined Conjugate Gradient Methods: Historical Context and Methodology , 2018, SIAM J. Sci. Comput..

[16]  Zdenek Strakos,et al.  Laplacian Preconditioning of Elliptic PDEs: Localization of the Eigenvalues of the Discretized Operator , 2018, SIAM J. Numer. Anal..

[17]  D. R. Lide Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators , 2018 .

[18]  Lloyd N. Trefethen,et al.  GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems , 2018, SIAM J. Sci. Comput..

[19]  Iveta Hnetynková,et al.  Relating Computed and Exact Entities in Methods Based on Lanczos Tridiagonalization , 2017, HPCSE.

[20]  Emmanuel Agullo,et al.  Analyzing the Effect of Local Rounding Error Propagation on the Maximal Attainable Accuracy of the Pipelined Conjugate Gradient Method , 2016, SIAM J. Matrix Anal. Appl..

[21]  A. Greenbaum,et al.  Matrices That Generate the Same Krylov Residual Spaces , 2015 .

[22]  Anne Greenbaum,et al.  Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations , 2015, SIAM J. Matrix Anal. Appl..

[23]  Zdenek Strakos,et al.  Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs , 2014, SIAM spotlights.

[24]  Gérard Meurant,et al.  On investigating GMRES convergence using unitary matrices , 2014 .

[25]  Howard C. Elman,et al.  IFISS: A Computational Laboratory for Investigating Incompressible Flow Problems , 2014, SIAM Rev..

[26]  Jörg Liesen,et al.  Properties of Worst-Case GMRES , 2013, SIAM J. Matrix Anal. Appl..

[27]  J. Liesen,et al.  Max-min and min-max approximation problems for normal matrices revisited , 2013, 1310.5880.

[28]  Tomáš Gergelits,et al.  Analysis of Krylov subspace methods , 2013 .

[29]  Y. Saad,et al.  Approximating Spectral Densities of Large Matrices , 2013, SIAM Rev..

[30]  Zdenek Strakos,et al.  Model reduction using the Vorobyev moment problem , 2009, Numerical Algorithms.

[31]  Aslak Tveito,et al.  Preconditioning by inverting the Laplacian: an analysis of the eigenvalues , 2008 .

[32]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[33]  G. Meurant The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations , 2006 .

[34]  G. Meurant,et al.  The Lanczos and conjugate gradient algorithms in finite precision arithmetic , 2006, Acta Numerica.

[35]  Miroslav Rozlozník,et al.  Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES , 2006, SIAM J. Matrix Anal. Appl..

[36]  Z. Strakos,et al.  Error Estimation in Preconditioned Conjugate Gradients , 2005 .

[37]  Julien Langou,et al.  Rounding error analysis of the classical Gram-Schmidt orthogonalization process , 2005, Numerische Mathematik.

[38]  Zdenek Strakos,et al.  GMRES Convergence Analysis for a Convection-Diffusion Model Problem , 2005, SIAM J. Sci. Comput..

[39]  Jörg Liesen,et al.  Convergence analysis of Krylov subspace methods , 2004 .

[40]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[41]  M. Benzi Preconditioning techniques for large linear systems: a survey , 2002 .

[42]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods , 2002, SIAM Rev..

[43]  Zdenek Strakos,et al.  Bounds for the least squares distance using scaled total least squares , 2002, Numerische Mathematik.

[44]  Zdenek Strakos,et al.  Residual and Backward Error Bounds in Minimum Residual Krylov Subspace Methods , 2001, SIAM J. Sci. Comput..

[45]  Arno B. J. Kuijlaars,et al.  Which Eigenvalues Are Found by the Lanczos Method? , 2000, SIAM J. Matrix Anal. Appl..

[46]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

[47]  M. Arioli,et al.  Krylov sequences of maximal length and convergence of GMRES , 1997 .

[48]  G. Golub,et al.  Matrices, moments and quadrature II; How to compute the norm of the error in iterative methods , 1997 .

[49]  A. Edelman The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law , 1997 .

[50]  Anne Greenbaum,et al.  Any Nonincreasing Convergence Curve is Possible for GMRES , 1996, SIAM J. Matrix Anal. Appl..

[51]  C. Brezinski The methods of Vorobyev and Lanczos , 1996 .

[52]  Vladimir Druskin,et al.  Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic , 1995, Numer. Linear Algebra Appl..

[53]  Gene H. Golub,et al.  Estimates in quadratic formulas , 1994, Numerical Algorithms.

[54]  Anne Greenbaum,et al.  Max-Min Properties of Matrix Factor Norms , 1994, SIAM J. Sci. Comput..

[55]  Wayne Joubert,et al.  A Robust GMRES-Based Adaptive Polynomial Preconditioning Algorithm for Nonsymmetric Linear Systems , 1994, SIAM J. Sci. Comput..

[56]  W. Hackbusch Iterative Solution of Large Sparse Systems of Equations , 1993 .

[57]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[58]  L. Trefethen,et al.  Eigenvalues and pseudo-eigenvalues of Toeplitz matrices , 1992 .

[59]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[60]  G. Golub,et al.  Iterative solution of linear systems , 1991, Acta Numerica.

[61]  Z. Strakos,et al.  On the real convergence rate of the conjugate gradient method , 1991 .

[62]  G. W. Stewart,et al.  Lanczos and linear systems , 1991 .

[63]  T. Manteuffel,et al.  A taxonomy for conjugate gradient methods , 1990 .

[64]  Thomas A. Manteuffel,et al.  On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations , 1990 .

[65]  Gene H. Golub,et al.  Some History of the Conjugate Gradient and Lanczos Algorithms: 1948-1976 , 1989, SIAM Rev..

[66]  A. Greenbaum Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences , 1989 .

[67]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[68]  H. V. D. Vorst,et al.  The rate of convergence of Conjugate Gradients , 1986 .

[69]  C. Paige Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem , 1980 .

[70]  A. Greenbaum Comparison of splittings used with the conjugate gradient algorithm , 1979 .

[71]  William P. Reinhardt,et al.  L2 discretization of atomic and molecular electronic continua: Moment, quadrature and J-matrix techniques , 1979 .

[72]  A. Jennings Influence of the Eigenvalue Spectrum on the Convergence Rate of the Conjugate Gradient Method , 1977 .

[73]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[74]  W. Gragg Matrix interpretations and applications of the continued fraction algorithm , 1974 .

[75]  G. Golub,et al.  Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. , 1972 .

[76]  Roy G. Gordon,et al.  Error Bounds in Equilibrium Statistical Mechanics , 1968 .

[77]  J. L. Rigal,et al.  On the Compatibility of a Given Solution With the Data of a Linear System , 1967, JACM.

[78]  J. H. Wilkinson The algebraic eigenvalue problem , 1966 .

[79]  George E. Forsythe,et al.  Solving linear algebraic equations can be interesting , 1953 .

[80]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[81]  C. Lanczos Chebyshev polynomials in the solution of large-scale linear systems , 1952, ACM '52.

[82]  C. Lanczos Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[83]  W. Karush Convergence of a method of solving linear problems , 1952 .

[84]  W. Arnoldi The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[85]  Gérard Meurant,et al.  Krylov Methods for Nonsymmetric Linear Systems , 2020 .

[86]  Erin Carson,et al.  Communication-Avoiding Krylov Subspace Methods in Theory and Practice , 2015 .

[87]  Numerische,et al.  On the convergence rate of the conjugate gradients in presence of rounding errors * , 2005 .

[88]  L. Trefethen,et al.  Spectra and pseudospectra : the behavior of nonnormal matrices and operators , 2005 .

[89]  Mark Embree,et al.  The Tortoise and the Hare Restart GMRES , 2003, SIAM Rev..

[90]  Z. Strakos,et al.  On error estimation in the conjugate gradient method and why it works in finite precision computations. , 2002 .

[91]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[92]  Gene H. Golub,et al.  Closer to the solutions: iterative linear solvers , 1997 .

[93]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[94]  Y. Saad Krylov subspace methods for solving large unsymmetric linear systems , 1981 .

[95]  P. Wesseling,et al.  Numerical experiments with a multiple grid and a preconditioned Lanczos type method , 1980 .

[96]  R. Fletcher Conjugate gradient methods for indefinite systems , 1976 .

[97]  Xiaomei Yang Rounding Errors in Algebraic Processes , 1964, Nature.

[98]  R. Varga,et al.  Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods , 1961 .