A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a Thick-Restart version of the Lanczos algorithm with deflation (`locking') and a new type of polynomial filters obtained from a least-squares technique. The resulting algorithm can be utilized in a `spectrum-slicing' approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different sub-intervals independently from one another.

[1]  Yousef Saad,et al.  Filtered Conjugate Residual-type Algorithms with Applications , 2006, SIAM J. Matrix Anal. Appl..

[2]  Lothar Reichel,et al.  Algorithm 827: irbleigs: A MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix , 2003, TOMS.

[3]  Y. Saad,et al.  Practical Use of Polynomial Preconditionings for the Conjugate Gradient Method , 1985 .

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

[5]  Kesheng Wu,et al.  Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods , 1998, SIAM J. Sci. Comput..

[6]  Eric Polizzi,et al.  A Density Matrix-based Algorithm for Solving Eigenvalue Problems , 2009, ArXiv.

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

[8]  Vicente Hernández,et al.  SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems , 2005, TOMS.

[9]  Yousef Saad,et al.  Self-consistent-field calculations using Chebyshev-filtered subspace iteration , 2006, J. Comput. Phys..

[10]  C. Paige.  Computational variants of the Lanczos method for the eigenproblem , 1972 .

[11]  Y. Saad,et al.  Electronic structure calculations for plane-wave codes without diagonalization , 1999 .

[12]  T. J. Rivlin.  An Introduction to the Approximation of Functions , 2003 .

[13]  T. Sakurai,et al.  CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems , 2007 .

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

[15]  M. Newman,et al.  Interpolation and approximation , 1965 .

[16]  Ping Tak Peter Tang,et al.  Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver , 2014, SIAM J. Sci. Comput..

[17]  M. Rozložník,et al.  The loss of orthogonality in the Gram-Schmidt orthogonalization process , 2005 .

[18]  C. Paige.  Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix , 1976 .

[19]  Richard B. Lehoucq,et al.  Anasazi software for the numerical solution of large-scale eigenvalue problems , 2009, TOMS.

[20]  Yousef Saad.  Analysis of Subspace Iteration for Eigenvalue Problems with Evolving Matrices , 2016, SIAM J. Matrix Anal. Appl..

[21]  Danny C. Sorensen,et al.  Deflation Techniques for an Implicitly Restarted Arnoldi Iteration , 1996, SIAM J. Matrix Anal. Appl..

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

[23]  A. Krall.  Applied Analysis , 1986 .

[24]  Y. Saad,et al.  PARSEC – the pseudopotential algorithm for real‐space electronic structure calculations: recent advances and novel applications to nano‐structures , 2006 .

[25]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[26]  W. Marsden.  I and J , 2012 .

[27]  T. Sakurai,et al.  A projection method for generalized eigenvalue problems using numerical integration , 2003 .

[28]  Yousef Saad,et al.  Numerical Methods for Electronic Structure Calculations of Materials , 2010, SIAM Rev..

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

[30]  Kesheng Wu,et al.  Adaptive Projection Subspace Dimension for the Thick-Restart Lanczos Method , 2010, ACM Trans. Math. Softw..

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

[32]  P. Revesz.  Interpolation and Approximation , 2010 .

[33]  T. Rose,et al.  Substructuring in MSC/NASTRAN for large scale parallel applications , 1991 .

[34]  Andreas Stathopoulos,et al.  PRIMME: preconditioned iterative multimethod eigensolver—methods and software description , 2010, TOMS.

[35]  Yousef Saad,et al.  A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems , 2012, SIAM J. Sci. Comput..

[36]  Yousef Saad,et al.  LEAST-SQUARES RATIONAL FILTERS FOR THE SOLUTION OF INTERIOR EIGENVALUE PROBLEMS , 2015 .

[37]  Chao Yang,et al.  A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix , 2014, J. Comput. Phys..

[38]  Yvan Notay,et al.  JADAMILU: a software code for computing selected eigenvalues of large sparse symmetric matrices , 2007, Comput. Phys. Commun..

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

[40]  G. Stewart,et al.  Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization , 1976 .