Deflated and Restarted Symmetric Lanczos Methods for Eigenvalues and Linear Equations with Multiple Right-Hand Sides

A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is necessary to control roundoff error, and several approaches are discussed. The eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides. Experiments are given with large matrices from quantum chromodynamics that have many right-hand sides.

[1]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

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

[3]  Christopher C. Paige,et al.  The computation of eigenvalues and eigenvectors of very large sparse matrices , 1971 .

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

[5]  B. Parlett,et al.  The Lanczos algorithm with selective orthogonalization , 1979 .

[6]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[7]  B. Parlett.  A new look at the Lanczos algorithm for solving symmetric systems of linear equations , 1980 .

[8]  D. O’Leary.  The block conjugate gradient algorithm and related methods , 1980 .

[9]  J. Grcar.  Analyses of the lanczos algorithm and of the approximation problem in richardson's method , 1981 .

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

[11]  Y. Saad,et al.  Practical Use of Some Krylov Subspace Methods for Solving Indefinite and Nonsymmetric Linear Systems , 1984 .

[12]  H. Simon.  Analysis of the symmetric Lanczos algorithm with reorthogonalization methods , 1984 .

[13]  H. Simon.  The Lanczos algorithm with partial reorthogonalization , 1984 .

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

[15]  Y. Saad,et al.  On the Lánczos method for solving symmetric linear systems with several right-hand sides , 1987 .

[16]  H. V. D. Vorst,et al.  An iterative solution method for solving f ( A ) x = b , using Krylov subspace information obtained for the symmetric positive definite matrix A , 1987 .

[17]  R. Nicolaides.  Deflation of conjugate gradients with applications to boundary value problems , 1987 .

[18]  R. Mittra,et al.  A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields , 1989 .

[19]  R. Morgan.  Computing Interior Eigenvalues of Large Matrices , 1991 .

[20]  R. Freund.  Quasi-kernel polynomials and their use in non-Hermitian matrix iterations , 1992 .

[21]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[22]  D. Calvetti,et al.  AN IMPLICITLY RESTARTED LANCZOS METHOD FOR LARGE SYMMETRIC EIGENVALUE PROBLEMS , 1994 .

[23]  Ronald B. Morgan,et al.  A Restarted GMRES Method Augmented with Eigenvectors , 1995, SIAM J. Matrix Anal. Appl..

[24]  P. Forcrand.  Progress on lattice QCD algorithms , 1995, hep-lat/9509082.

[25]  Andy A. Nikishin,et al.  Variable Block CG Algorithms for Solving Large Sparse Symmetric Positive Definite Linear Systems on Parallel Computers, I: General Iterative Scheme , 1995, SIAM J. Matrix Anal. Appl..

[26]  Efstratios Gallopoulos,et al.  An Iterative Method for Nonsymmetric Systems with Multiple Right-Hand Sides , 1995, SIAM J. Sci. Comput..

[27]  S. A. Kharchenko,et al.  Eigenvalue translation based preconditioners for the GMRES(k) method , 1995, Numer. Linear Algebra Appl..

[28]  Henk A. van der Vorst,et al.  Approximate solutions and eigenvalue bounds from Krylov subspaces , 1995, Numer. Linear Algebra Appl..

[29]  D. Calvetti,et al.  Iterative methods for the computation of a few eigenvalues of a large symmetric matrix , 1996 .

[30]  K. Burrage,et al.  Restarted GMRES preconditioned by deflation , 1996 .

[31]  V. Simoncini,et al.  A hybrid block GMRES method for nonsymmetric systems with multiple right-hand sides , 1996 .

[32]  Ronald B. Morgan,et al.  On restarting the Arnoldi method for large nonsymmetric eigenvalue problems , 1996, Math. Comput..

[33]  R. Freund,et al.  A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides , 1997 .

[34]  Tony F. Chan,et al.  Analysis of Projection Methods for Solving Linear Systems with Multiple Right-Hand Sides , 1997, SIAM J. Sci. Comput..

[35]  A. Frommer.  Linear systems solvers — recent developments and implications for lattice computations , 1996, hep-lat/9608074.

[36]  Yousef Saad,et al.  Deflated and Augmented Krylov Subspace Techniques , 1997, Numer. Linear Algebra Appl..

[37]  Y. Saad.  Analysis of Augmented Krylov Subspace Methods , 1997, SIAM J. Matrix Anal. Appl..

[38]  Gene H. Golub,et al.  Adaptively Preconditioned GMRES Algorithms , 1998, SIAM J. Sci. Comput..

[39]  Kesheng Wu,et al.  Thick-Restart Lanczos Method for Symmetric Eigenvalue Problems , 1998, IRREGULAR.

[40]  R. Morgan,et al.  Harmonic projection methods for large non-symmetric eigenvalue problems , 1998 .

[41]  Min Zeng,et al.  Harmonic projection methods for large non-symmetric eigenvalue problems , 1998, Numer. Linear Algebra Appl..

[42]  K. Burrage,et al.  On the Performance of Various Adaptive Preconditioned GMRES Strategies , 1998 .

[43]  U. Heller,et al.  Study of chiral symmetry in quenched QCD using the overlap Dirac operator , 1998, hep-lat/9811030.

[44]  E. Sturler,et al.  Truncation Strategies for Optimal Krylov Subspace Methods , 1999 .

[45]  Kesheng Wu,et al.  Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems , 2000, SIAM J. Matrix Anal. Appl..

[46]  R. Narayanan,et al.  Alternative to domain wall fermions , 2000, hep-lat/0005004.

[47]  Frédéric Guyomarc'h,et al.  A Deflated Version of the Conjugate Gradient Algorithm , 1999, SIAM J. Sci. Comput..

[48]  Frédéric Guyomarc'h,et al.  An Augmented Conjugate Gradient Method for Solving Consecutive Symmetric Positive Definite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

[49]  Ronald B. Morgan,et al.  Implicitly Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of Equations , 2000, SIAM J. Matrix Anal. Appl..

[50]  Liu,et al.  Chiral symmetry, quark mass, and scaling of the overlap fermions , 2000, Physical review letters.

[51]  Eric L. Miller,et al.  QMR-Based Projection Techniques for the Solution of Non-Hermitian Systems with Multiple Right-Hand Sides , 2001, SIAM J. Sci. Comput..

[52]  T. Lippert,et al.  Low fermionic eigenmode dominance in QCD on the lattice , 2001, hep-lat/0106016.

[53]  H. V. D. Vorst,et al.  Numerical methods for the QCDd overlap operator. I. Sign-function and error bounds , 2002, hep-lat/0202025.

[54]  Ronald B. Morgan,et al.  GMRES WITH DEFLATED , 2008 .

[55]  G. W. Stewart,et al.  A Krylov-Schur Algorithm for Large Eigenproblems , 2001, SIAM J. Matrix Anal. Appl..

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

[57]  M. Gutknecht,et al.  Block Krylov methods for Hermitian linear systems , 2004 .

[58]  R. Morgan,et al.  Deflation of eigenvalues for iterative methods in lattice QCD , 2003, hep-lat/0309068.

[59]  R. Morgan,et al.  Deflated Iterative Methods for Linear Equations with Multiple Right-Hand Sides , 2004, math-ph/0405053.

[60]  C. Le Calvez,et al.  Implicitly restarted and deflated GMRES , 1999, Numerical Algorithms.

[61]  M. Gutknecht.  BLOCK KRYLOV SPACE METHODS FOR LINEAR SYSTEMS WITH MULTIPLE RIGHT-HAND SIDES : AN , 2005 .

[62]  R. Morgan.  Restarted block-GMRES with deflation of eigenvalues , 2005 .

[63]  R. Morgan,et al.  A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity , 2006 .

[64]  Eric de Sturler,et al.  Recycling Krylov Subspaces for Sequences of Linear Systems , 2006, SIAM J. Sci. Comput..

[65]  Wolfgang Fichtner,et al.  Improving the Accuracy of GMRes with Deflated Restarting , 2007, SIAM J. Sci. Comput..

[66]  M. Lüscher.  Local coherence and deflation of the low quark modes in lattice QCD , 2007 .

[67]  M. Luscher.  Local coherence and deflation of the low quark modes in lattice QCD , 2007, 0706.2298.

[68]  J. Baglama.  Augmented Block Householder Arnoldi Method , 2008 .

[69]  Yousef Saad,et al.  Block Krylov–Schur method for large symmetric eigenvalue problems , 2008, Numerical Algorithms.

[70]  A. Stathopoulos,et al.  Computing and deflating eigenvalues while solving multiple right hand side linear systems in Quantum Chromodynamics , 2008 .

[71]  Andreas Stathopoulos,et al.  Computing and Deflating Eigenvalues While Solving Multiple Right-Hand Side Linear Systems with an Application to Quantum Chromodynamics , 2007, SIAM J. Sci. Comput..

[72]  G. Golub,et al.  Gmres: a Generalized Minimum Residual Algorithm for Solving , 2022 .