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This paper is about GMRES algorithms for the solution of nonsingular linear systems.
The principle of minimized iterations in the solution of the matrix eigenvalue problem
W. E. Arnoldi · 1951
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An iterative least-square method suitable for solving large sparse matrices
I. M. Khabaza · 1963
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Solving linear least squares problems by Gram-Schmidt orthogonalization
Å. Björck · 1967
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Iterative refinement in floating point
C. B. Moler · 1967
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On the asymptotic directions of the s s -dimensional optimum gradient method
G. E. Forsythe · 1968
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Solution of sparse indefinite systems of linear equations
C. C. Paige and M. A. Saunders · 1975
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Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization
J. W. Daniel, W. B. Gragg, L. Kaufman, and G. W. Stewart · 1976
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Orthomin, an iterative method for solving sparse sets of simultaneous linear equations
P. K. W. Vinsome · 1976
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Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations
O. Axelsson · 1980
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Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices
Y. Saad · 1980
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Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods
D. M. Young and K. C. Jea · 1980
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Krylov subspace methods for solving large unsymmetric linear systems
Y. Saad · 1981
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Iterative Methods for Large, Sparse, Nonsymmetric Systems of Linear Equations
H. C. Elman · 1982
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Variational iterative methods for nonsymmetric systems of linear equations
S. C. Eisenstat, H. C. Elman, and M. H. Schultz · 1983
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On the simplification of generalized conjugate-gradient methods for nonsymmetrizable linear systems
K. C. Jea and D. M. Young · 1983
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GMRES acceleration of computational fluid dynamics codes
L. Wigton, N. Yu, and D. Young · 1985
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GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
Y. Saad and M. H. Schultz · 1986
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The rate of convergence of conjugate gradients
A. van der Sluis and H. A. van der Vorst · 1986
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Extrapolation methods for vector sequences
D. A. Smith, W. F. Ford, and A. Sidi · 1987
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Extrapolation vs. projection methods for linear systems of equations
A. Sidi · 1988
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Implementation of the GMRES method using Householder transformations
H. F. Walker · 1988
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Reduced storage matrix methods in stiff ODE systems
P. N. Brown and A. C. Hindmarsh · 1989
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Parallel conjugate gradient-like algorithms for solving sparse nonsymmetric linear systems on a vector multiprocessor
G. R. di Brozolo and Y. Robert · 1989
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Accelerating with rank-one updates
T. Eirola and O. Nevanlinna · 1989
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Implementations of the GMRES method
H. F. Walker · 1989
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s s -step Orthomin and GMRES implemented on parallel computers
A. T. Chronopoulos and S. K. Kim · 1990
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Iterative methods for nonsymmetric linear systems
W. D. Joubert and T. A. Manteuffel · 1990
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Approximation Theory and Numerical Linear Algebra
L. N. Trefethen · 1990
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Etude de Quelques Méthodes de Résolution de Problèmes Linéaires de Grande Taille sur Multiprocesseur
B. Vital · 1990
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A theoretical comparison of the Arnoldi and GMRES algorithms
P. N. Brown · 1991
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A parallel variant of GMRES( m m )
E. de Sturler · 1991
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Iterative solution of linear systems
R. W. Freund, G. H. Golub, and N. M. Nachtigal · 1992
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Parallelizable restarted iterative methods for nonsymmetric linear systems. Part I: Theory
W. D. Joubert and G. F. Carey · 1992
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Parallelizable restarted iterative methods for nonsymmetric linear systems. Part II: Parallel implementation
W. D. Joubert and G. F. Carey · 1992
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How fast are nonsymmetric matrix iterations?
N. M. Nachtigal, S. C. Reddy, and L. N. Trefethen · 1992
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A hybrid GMRES algorithm for nonsymmetric linear systems
N. M. Nachtigal, L. Reichel, and L. N. Trefethen · 1992
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Sparse iterative algorithm software for large-scale MIMD machines: An initial discussion and implementation
J. N. Shadid and R. S. Tuminaro · 1992
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Efficient high accuracy solutions with GMRES( m m )
K. Turner and H. F. Walker · 1992
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A comparison of some GMRES-like methods
C. Vuik and H. A. van der Vorst · 1992
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Parallel numerical linear algebra
J. W. Demmel, M. T. Heath, and H. A. van der Vorst · 1993
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Fields of values and iterative methods
M. Eiermann · 1993
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Convergence of Iterations for Linear Equations
O. Nevanlinna · 1993
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A flexible inner-outer preconditioned GMRES algorithm
Y. Saad · 1993
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A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations
G. Starke and R. S. Varga · 1993
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The superlinear convergence behaviour of GMRES
H. A. van der Vorst and C. Vuik · 1993
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A Newton basis GMRES implementation
Z. Bai, D. Hu, and L. Reichel · 1994
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A parallel implementation of the restarted GMRES iterative algorithm for nonsymmetric systems of linear equations
R. D. da Cunha and T. Hopkins · 1994
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Max-min properties of matrix factor norms
A. Greenbaum and L. Gurvits · 1994
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Matrices that generate the same Krylov residual spaces
A. Greenbaum and Z. Strakoš · 1994
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GMRES/CR and Arnoldi/Lanczos as matrix approximation problems
A. Greenbaum and L. N. Trefethen · 1994
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On the convergence behavior of the restarted GMRES algorithm for solving nonsymmetric linear systems
W. D. Joubert · 1994
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A robust GMRES-based adaptive polynomial preconditioning algorithm for nonsymmetric linear systems
W. D. Joubert · 1994
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GMRESR: A family of nested GMRES methods
H. A. van der Vorst and C. Vuik · 1994
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A simpler GMRES
H. F. Walker and L. Zhou · 1994
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Reducing the effect of global communication in GMRES( m m ) and CG on parallel distributed memory computers
E. de Sturler and H. A. van der Vorst · 1995
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Numerical stability of GMRES
J. Drkošová, A. Greenbaum, M. Rozložník, and Z. Strakoš · 1995
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A parallel GMRES version for general sparse matrices
J. Erhel · 1995
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Eigenvalue translation based preconditioners for the GMRES( k k ) method
S. A. Kharchenko and A. Y. Yeremin · 1995
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A restarted GMRES method augmented with eigenvectors
R. B. Morgan · 1995
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An iterative method for nonsymmetric systems with multiple right-hand sides
V. Simoncini and E. Gallopoulos · 1995
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A polynomial preconditioner for the GMRES algorithm
M. B. van Gijzen · 1995
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New insights in GMRES-like methods with variable preconditioners
C. Vuik · 1995
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GMRES and the minimal polynomial
S. L. Campbell, I. C. F. Ipsen, C. T. Kelley, and C. D. Meyer · 1996
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Relations between Galerkin and norm-minimizing iterative methods for solving linear systems
J. K. Cullum and A. Greenbaum · 1996
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Nested Krylov methods based on GCR
E. de Sturler · 1996
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Restarted GMRES preconditioned by deflation
J. Erhel, K. Burrage, and B. Pohl · 1996
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Minimal residual method stronger than polynomial preconditioning
V. Faber, W. D. Joubert, E. Knill, and T. A. Manteuffel · 1996
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Any nonincreasing convergence curve is possible for GMRES
A. Greenbaum, V. Pták, and Z. Strakoš · 1996
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Deflation techniques for an implicitly restarted Arnoldi iteration
R. B. Lehoucq and D. C. Sorensen · 1996
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DQGMRES: a direct quasi-minimal residual algorithm based on incomplete orthogonalization
Y. Saad and K. Wu · 1996
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Convergence properties of block GMRES and matrix polynomials
V. Simoncini and E. Gallopoulos · 1996
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GMRES on (nearly) singular systems
P. N. Brown and H. F. Walker · 1997
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Deflated and augmented Krylov subspace techniques
A. Chapman and Y. Saad · 1997
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Iterative Methods for Solving Linear Systems
A. Greenbaum · 1997
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Numerical behaviour of the modified Gram-Schmidt GMRES implementation
A. Greenbaum, M. Rozložník, and Z. Strakoš · 1997
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A note on the superlinear convergence of GMRES
I. Moret · 1997
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Analysis of augmented Krylov subspace methods
Y. Saad · 1997
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Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems
G. Starke · 1997
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GMRES vs. ideal GMRES
K.-C. Toh · 1997
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Krylov sequences of maximal length and convergence of GMRES
M. Arioli, V. Pták, and Z. Strakoš · 1998
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Adaptively preconditioned GMRES algorithms
J. Baglama, D. Calvetti, G. H. Golub, and L. Reichel · 1998
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On the performance of various adaptive preconditioned GMRES strategies
K. Burrage and J. Erhel · 1998
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Weighted FOM and GMRES for solving nonsymmetric linear systems
A. Essai · 1998
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Minimum residual methods for augmented systems
B. Fischer, A. Ramage, D. J. Silvester, and A. J. Wathen · 1998
Earlier work this paper cites.
Restarted GMRES for shifted linear systems
A. Frommer and U. Glässner · 1998
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Error analysis of Krylov methods in a nutshell
M. Hochbruck and C. Lubich · 1998
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The idea behind Krylov methods
I. C. F. Ipsen and C. D. Meyer · 1998
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Physics based GMRES preconditioner for compressible and incompressible navier-stokes equations
N. Nigro, M. Storti, S. Idelsohn, and T. Tezduyar · 1998
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A new adaptive GMRES algorithm for achieving high accuracy
M. Sosonkina, L. T. Watson, R. K. Kapania, and H. F. Walker · 1998
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Truncation strategies for optimal Krylov subspace methods
E. de Sturler · 1999
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How descriptive are GMRES convergence bounds?
M. Embree · 1999
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Ritz and harmonic Ritz values and the convergence of FOM and GMRES
S. Goossens and D. Roose · 1999
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Global FOM and GMRES algorithms for matrix equations
K. Jbilou, A. Messaoudi, and H. Sadok · 1999
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Implicitly restarted and deflated GMRES
C. Le Calvez and B. Molina · 1999
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CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm
H. Sadok · 1999
Earlier work this paper cites.
A relaxation strategy for inexact matrix-vector products for Krylov methods
A. Bouras and V. Frayssé · 2000
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GMRES-type methods for inconsistent systems
D. Calvetti, B. Lewis, and L. Reichel · 2000
Earlier work this paper cites.
Analysis of acceleration strategies for restarted minimal residual methods
M. Eiermann, O. G. Ernst, and O. Schneider · 2000
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Expressions and bounds for the GMRES residual
I. C. F. Ipsen · 2000
Cited alongside, same era.
Computable convergence bounds for GMRES
J. Liesen · 2000
Cited alongside, same era.
Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations
R. B. Morgan · 2000
Cited alongside, same era.
The DEFLATED-GMRES( m m , k k ) method with switching the restart frequency dynamically
K. Moriya and T. Nodera · 2000
Cited alongside, same era.
Further analysis of minimum residual iterations
Y. Saad · 2000
Cited alongside, same era.
Thick-restart Lanczos method for large symmetric eigenvalue problems
K. Wu and H. Simon · 2000
Cited alongside, same era.
GMRES implementations and residual smoothing techniques for solving ill-posed linear systems
M. Matinfar, H. Zareamoghaddam, M. Eslami, and M. Saeidy · 2012
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The complete stagnation of GMRES for n ≤ 4 n\leq 4
G. Meurant · 2012
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GMRES and the Arioli, Pták, and Strakoš parametrization
G. Meurant · 2012
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On the generation of Krylov subspace bases
B. Philippe and L. Reichel · 2012
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A new look at CMRH and its relation to GMRES
H. Sadok and D. B. Szyld · 2012
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A projection method to solve linear systems in tensor format
J. Ballani and L. Grasedyck · 2013
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Generalization of convergence conditions for a restarted GMRES
J. Zítko · 2000
Cited alongside, same era.
On the choice of subspace for iterative methods for linear discrete ill-posed problems
D. Calvetti, B. Lewis, and L. Reichel · 2001
Cited alongside, same era.
Geometric aspects of the theory of Krylov subspace methods
M. Eiermann and O. G. Ernst · 2001
Cited alongside, same era.
A block GMRES method augmented with eigenvectors
G.-D. Gu and Z.-H. Cao · 2001
Cited alongside, same era.
DGMRES: A GMRES-type algorithm for Drazin-inverse solution of singular non-symmetric linear systems
A. Sidi · 2001
Cited alongside, same era.
Preconditioning techniques for large linear systems: A survey
M. Benzi · 2002
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Reorthogonalized block classical Gram-Schmidt
J. L. Barlow and A. Smoktunowicz · 2013
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A modified block flexible GMRES method with deflation at each iteration for the solution of non-Hermitian linear systems with multiple right-hand sides
H. Calandra, S. Gratton, R. Lago, X. Vasseur, and L. M. Carvalho · 2013
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TT-GMRES: Solution to a linear system in the structured tensor format
S. V. Dolgov · 2013
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Properties of worst-case GMRES
V. Faber, J. Liesen, and P. Tichý · 2013
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A framework for deflated and augmented Krylov subspace methods
A. Gaul, M. H. Gutknecht, J. Liesen, and R. Nabben · 2013
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Hiding global communication latency in the GMRES algorithm on massively parallel machines
P. Ghysels, T. J. Ashby, K. Meerbergen, and W. Vanroose · 2013
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Matrix Computations
G. H. Golub and C. F. Van Loan · 2013
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Gram-Schmidt orthogonalization: 100 years and more
S. J. Leon, Å. Björck, and W. Gander · 2013
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Krylov Subspace Methods: Principles and Analysis
J. Liesen and Z. Strakoš · 2013
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On the choice of preconditioner for minimum residual methods for non-Hermitian matrices
J. Pestana and A. J. Wathen · 2013
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A flexible Krylov solver for shifted systems with application to oscillatory hydraulic tomography
A. K. Saibaba, T. Bakhos, and P. K. Kitanidis · 2013
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GMRES convergence for perturbed coefficient matrices, with application to approximate deflation preconditioning
J. A. Sifuentes, M. Embree, and R. B. Morgan · 2013
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Parallelism and robustness in GMRES with a newton basis and deflated restarting
D. N. Wakam and J. Erhel · 2013
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Block GMRES method with inexact breakdowns and deflated restarting
E. Agullo, L. Giraud, and Y.-F. Jing · 2014
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Communication lower bounds and optimal algorithms for numerical linear algebra
G. Ballard, E. C. Carson, J. W. Demmel, M. Hoemmen, N. Knight, and O. Schwartz · 2014
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Prescribing the behavior of early terminating GMRES and Arnoldi iterations
J. Duintjer Tebbens and G. Meurant · 2014
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On investigating GMRES convergence using unitary matrices
J. Duintjer Tebbens, G. Meurant, H. Sadok, and Z. Strakoš · 2014
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Some observations on weighted GMRES
S. Güttel and J. Pestana · 2014
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Hierarchical Krylov and nested Krylov methods for extreme-scale computing
L. C. McInnes, B. F. Smith, H. Zhang, and R. T. Mills · 2014
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Necessary and sufficient conditions for GMRES complete and partial stagnation
G. Meurant · 2014
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Krylov subspace recycling for sequences of shifted linear systems
K. M. Soodhalter, D. B. Szyld, and F. Xue · 2014
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GMRES convergence bounds that depend on the right-hand-side vector
D. Titley-Peloquin, J. Pestana, and A. J. Wathen · 2014
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Improving the performance of CA-GMRES on multicores with multiple GPUs
I. Yamazaki, H. Anzt, S. Tomov, M. Hoemmen, and J. Dongarra · 2014
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Deflation strategies to improve the convergence of communication-avoiding GMRES
I. Yamazaki, S. Tomov, and J. Dongarra · 2014
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Some properties of range restricted GMRES methods
M. Bellalij, L. Reichel, and H. Sadok · 2015
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Communication-Avoiding Krylov Subspace Methods in Theory and Practice
E. C. Carson · 2015
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Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: What is the largest shift for which wavenumber-independent convergence is guaranteed?
M. J. Gander, I. G. Graham, and E. A. Spence · 2015
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On the similarities between the quasi-Newton least squares method and GMRes
R. Haelterman, B. Lauwens, F. Van Utterbeeck, H. Bruyninckx, and J. Vierendeels · 2015
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Polynomial preconditioned GMRES and GMRES-DR
Q. Liu, R. B. Morgan, and W. Wilcox · 2015
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Preconditioning
A. J. Wathen · 2015
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Mixed-precision block Gram Schmidt orthogonalization
I. Yamazaki, S. Tomov, J. Kurzak, J. Dongarra, and J. L. Barlow · 2015
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Locally optimal and heavy ball GMRES methods
A. Imakura, R.-C. Li, and S.-L. Zhang · 2016
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Pseudoeigenvector bases and deflated GMRES for highly nonnormal matrices
R. B. Morgan, Z. Yang, and B. Zhong · 2016
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Pipelined, flexible Krylov subspace methods
P. Sanan, S. M. Schnepp, and D. A. May · 2016
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Computational methods for linear matrix equations
V. Simoncini · 2016
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Block Krylov subspace recycling for shifted systems with unrelated right-hand sides
K. M. Soodhalter · 2016
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Two recursive GMRES-type methods for shifted linear systems with general preconditioning
K. M. Soodhalter · 2016
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Multipreconditioned GMRES for shifted systems
T. Bakhos, P. K. Kitanidis, S. Ladenheim, A. K. Saibaba, and D. B. Szyld · 2017
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A new analysis of iterative refinement and its application to accurate solution of ill-conditioned sparse linear systems
E. C. Carson and N. J. Higham · 2017
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Any admissible harmonic Ritz value set is possible for GMRES
K. Du, J. Duintjer Tebbens, and G. Meurant · 2017
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Weighted inner products for GMRES and GMRES-DR
M. Embree, R. B. Morgan, and H. V. Nguyen · 2017
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GMRES with multiple preconditioners
C. Greif, T. Rees, and D. B. Szyld · 2017
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Varying the s s in your s s -step GMRES
D. Imberti and J. Erhel · 2017
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The coefficients of the FOM and GMRES residual polynomials
G. Meurant · 2017
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Improving performance of GMRES by reducing communication and pipelining global collectives
I. Yamazaki, M. Hoemmen, P. Luszczek, and J. Dongarra · 2017
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Shanks sequence transformations and Anderson acceleration
C. Brezinski, M. Redivo-Zaglia, and Y. Saad · 2018
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Accelerating the solution of linear systems by iterative refinement in three precisions
E. C. Carson and N. J. Higham · 2018
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On the convergence rate of DGMRES
A. Greenbaum, F. Kyanfar, and A. Salemi · 2018
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On GMRES for singular EP and GP systems
K. Morikuni and M. Rozložník · 2018
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A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right-hand sides
D.-L. Sun, T.-Z. Huang, Y.-F. Jing, and B. Carpentieri · 2018
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Enlarged GMRES for solving linear systems with one or multiple right-hand sides
H. Al Daas, L. Grigori, P. Hénon, and P. Ricoux · 2019
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Block modified Gram-Schmidt algorithms and their analysis
J. L. Barlow · 2019
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Scalable Non-blocking Krylov Solvers for Extreme-scale Computing
P. R. Eller · 2019
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Arnoldi decomposition, GMRES, and preconditioning for linear discrete ill-posed problems
S. Gazzola, S. Noschese, P. Novati, and L. Reichel · 2019
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A comparison of preconditioned Krylov subspace methods for large-scale nonsymmetric linear systems
A. Ghai, C. Lu, and X. Jiao · 2019
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Exploiting variable precision in GMRES, 2019
S. Gratton, E. Simon, D. Titley-Peloquin, and P. Toint · 2019
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Squeezing a matrix into half precision, with an application to solving linear systems
N. J. Higham, S. Pranesh, and M. Zounon · 2019
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A GMRES convergence analysis for localized invariant subspace ill-conditioning
G. Sacchi and V. Simoncini · 2019
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Weighted and deflated global GMRES algorithms for solving large Sylvester matrix equations
N. A. Zadeh, A. Tajaddini, and G. Wu · 2019
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Randomized Gram-Schmidt process with application to GMRES, 2020
O. Balabanov and L. Grigori · 2020
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On the cost of iterative computations
E. C. Carson and Z. Strakoš · 2020
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On the residual norms, the Ritz values and the harmonic Ritz values that can be generated by restarted GMRES
J. Duintjer Tebbens and G. Meurant · 2020
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Krylov type methods for linear systems exploiting properties of the quadratic numerical range
A. Frommer, B. Jacob, K. Kahl, C. Wyss, and I. Zwaan · 2020
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Block Krylov subspace methods for functions of matrices II: Modified block FOM
A. Frommer, K. Lund, and D. B. Szyld · 2020
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Admissible and attainable convergence behavior of block Arnoldi and GMRES
M. Kubínová and K. M. Soodhalter · 2020
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Improving the performance of the GMRES method using mixed-precision techniques
N. Lindquist, P. Luszczek, and J. Dongarra · 2020
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Polynomial preconditioned GMRES in Trilinos: Practical considerations for high-performance computing
J. A. Loe, H. K. Thornquist, and E. G. Boman · 2020
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Randomized numerical linear algebra: Foundations and algorithms
P.-G. Martinsson and J. A. Tropp · 2020
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Krylov Methods for Nonsymmetric Linear Systems: From Theory to Computations
G. Meurant and J. Duintjer Tebbens · 2020
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Preconditioners for Krylov subspace methods: An overview
J. W. Pearson and J. Pestana · 2020
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Iterative methods for linear systems of equations: A brief historical journey
Y. Saad · 2020
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A survey of subspace recycling iterative methods
K. M. Soodhalter, E. de Sturler, and M. E. Kilmer · 2020
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On fixed-point, Krylov, and 2 × 2 2\times 2 block preconditioners for nonsymmetric problems
B. S. Southworth, A. A. Sivas, and S. Rhebergen · 2020
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Low-synchronization orthogonalization schemes for s s -step and pipelined Krylov solvers in Trilinos
I. Yamazaki, S. Thomas, M. Hoemmen, E. G. Boman, K. Świrydowicz, and J. J. Elliott · 2020
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A survey of numerical linear algebra methods utilizing mixed-precision arithmetic
A. Abdelfattah, H. Anzt, E. G. Boman, E. C. Carson, T. Cojean, J. Dongarra, A. Fox, M. Gates, N. J. Higham, X. S. Li, J. Loe, P. Luszczek, S. Pranesh, S. Rajamanickam, T. Ribizel, B. F. Smith, K. Swirydowicz, S. Thomas, S. Tomov, Y. M. Tsai, and U. M. Yang · 2021
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Five-precision GMRES-based iterative refinement, 2021
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Some uses of the field of values in numerical analysis
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Low-synch Gram-Schmidt with delayed reorthogonalization for Krylov solvers, 2021
D. Bielich, J. Langou, S. Thomas, K. Swirydowicz, I. Yamazaki, and E. G. Boman · 2021
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Stability of linear GMRES convergence with respect to compact perturbations
J. Blechta · 2021
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The stability of block variants of classical Gram-Schmidt
E. C. Carson, K. Lund, and M. Rozložník · 2021
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On restarted and deflated block FOM and GMRES methods for sequences of shifted linear systems
L. Elbouyahyaoui, M. Heyouni, A. Tajaddini, and F. Saberi-Movahed · 2021
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Polynomial preconditioned Arnoldi with stability control
M. Embree, J. A. Loe, and R. B. Morgan · 2021
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Structure preserving quaternion generalized minimal residual method
Z. Jia and M. K. Ng · 2021
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Accelerating restarted GMRES with mixed precision arithmetic
N. Lindquist, P. Luszczek, and J. Dongarra · 2021
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Understanding performance variability in standard and pipelined parallel Krylov solvers
H. Morgan, P. Sanan, M. Knepley, and R. T. Mills · 2021
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Low synchronization Gram-Schmidt and generalized minimal residual algorithms
K. Swirydowicz, J. Langou, S. Ananthan, U. Yang, and S. Thomas · 2021
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Two new variants of the simpler block GMRES method with vector deflation and eigenvalue deflation for multiple linear systems
A. Tajaddini, G. Wu, F. Saberi-Movahed, and N. Azizizadeh · 2021
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Rounding error analysis of mixed precision block Householder QR algorithms
L. M. Yang, A. Fox, and G. Sanders · 2021
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Proxy-GMRES: Preconditioning via GMRES in polynomial space
X. Ye, Y. Xi, and Y. Saad · 2021
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Block Gram-Schmidt algorithms and their stability properties
E. C. Carson, K. Lund, M. Rozložník, and S. Thomas · 2022
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Toward efficient polynomial preconditioning for GMRES
J. A. Loe and R. B. Morgan · 2022
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