An invert-free Arnoldi method for computing interior eigenpairs of large matrices

References

• Bai , Z. , Barret , R. , Day , D. , Demmel , J. and Dongarra , J. 1997 . “ Test matrix collection for non-Hermitian eigenvalue problems ” . University of Tennessee . Technical Report CS-97-355 Available at http://math.nist.gov/MatrixMarket
   Google Scholar
• Morgan , R. B. and Zeng , M. 1998 . Harmonic projection methods for large nonsymmetric eigenvalue problems . Numerical Linear Algebra with Applications , 5 : 33 – 55 .
   Google Scholar
• Beattie , C. A. , Embree , M. and Rossi , J. 2004 . Convergence of restarted Krylov subspace to invariant subspaces . SIAM Journal on Matrix Analysis and Applications , 25 : 1074 – 1109 .
   Google Scholar
• Beattie , C. A. , Embree , M. and Sorensen , D. C. 2005 . Convergence of polynomial restarted Krylov methods for eigenvalue computations . SIAM Review , 47 : 492 – 515 .
   Google Scholar
• Saad , Y. 1992 . Numerical Methods for Large Eigenvalue Problems , Manchester : Manchester University Press .
   Google Scholar
• Morgan , R. B. 1991 . Computing interior eigenvalues of large matrices . Linear Algebra and its Applications , 154–156 : 289 – 309 .
   Google Scholar
• Sleijpen , G. L.G. and Van der Vorst , H. A. 1996 . A Jacobi–Davidson iteration method for linear eigenvalue problems . SIAM Journal on Matrix Analysis and Applications , 17 : 401 – 425 .
   Web of Science ®Google Scholar
• Scott , D. C. 1982 . The advantages of the inverted operators in Rayleigh–Ritz approximations . SIAM Journal on Scientific and Statistic Computation , 3 : 68 – 75 .
   Google Scholar
• Paige , C. C. , Parlett , B. N. and Van der Vorst , H. A. 1995 . Approximate solutions and eigenvalue bounds from Krylov subspace . Numerical Linear Algebra with Applications , 2 : 115 – 133 .
   Web of Science ®Google Scholar
• Hochstenbach , M. E. 2005 . Generalizations of harmonic and refined Rayleigh–Ritz . Electronic Transactions on Numerical Analysis , 20 : 235 – 252 .
   Google Scholar
• Jia , Z. 2002 . The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices . Applied Numerical Mathematics , 42 : 489 – 512 .
   Google Scholar
• Sleijpen , G. L.G. , Van der Vorst , H. A. and Meijerink , E. 1998 . Efficient expansion of subspaces in the Jacobi–Davidson method for standard and generalized eigenproblems . Electronic Transactions on Numerical Analysis , 7 : 75 – 89 .
   Google Scholar
• Arnoldi , W. E. 1951 . The principle of minimized iteration in the solution of the matrix eigenvalue problem . Quarterly of Applied Mathematics , 9 : 17 – 29 .
   Web of Science ®Google Scholar
• Walker , H. F. and Zhou , L. 1994 . A simpler GMRES . Numerical Linear Algebra with Applications , 1 : 571 – 581 .
   Web of Science ®Google Scholar
• Scott , J. A. 1995 . An Arnoldi code for computing selected eigenvalues of sparse, real, unsymmetric matrices . ACM Transactions on Mathematical Software , 214 : 432 – 475 .
   Google Scholar
• Saad , Y. 1980 . Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices . Linear Algebra and its Applications , 34 : 269 – 295 .
   Web of Science ®Google Scholar
• Sorensen , D. C. 1992 . Implicit application of polynomial filters in a k-step Arnoldi method . SIAM Journal on Matrix Analysis and Applications , 13 : 357 – 385 .
   Web of Science ®Google Scholar
• Lehoucq , R. B. , Sorensen , D. C. and Yang , C. 1998 . Arpack Users' Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods , Philadelphia : SIAM . Available at http://www.caam.rice.edu/software/ARPACK/
   Google Scholar
• Morgan , R. B. and Zeng , M. 2006 . A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity . Linear Algebra and its Applications , 415 : 96 – 113 .
   Google Scholar
• Wu , K. and Simon , H. D. 2000 . Thick-restart Lanczos method for large symmetric eigenvalue problems . SIAM Journal on Matrix Analysis and Applications , 22 : 602 – 616 .
   Web of Science ®Google Scholar
• Jia , Z. and Stewart , G. W. 2001 . An analysis of the Rayleigh–Ritz method for approximating eigenspaces . Mathematics of Computation , 70 : 637 – 647 .
   Google Scholar
• Elsner , L. 1985 . An optimal bound for the spectral variation of two matrices . Linear Algebra and its Applications , 71 : 77 – 80 .
   Google Scholar
• Jia , Z. 2005 . The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors . Mathematics of Computation , 74 : 1441 – 1456 .
   Google Scholar
• Duff , I. S. , Grimes , R. G. and Lewis , J. G. 1989 . Sparse matrix test problems . ACM Transactions on Mathematical Software , 15 : 1 – 14 .
   Web of Science ®Google Scholar
• Morgan , R. B. 2000 . Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations . SIAM Journal on Matrix Analysis and Applications , 21 : 1112 – 1135 .
   Web of Science ®Google Scholar