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Localization in Matrix Computations: Theory and Applications

  • Michele BenziEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2173)

Abstract

Many important problems in mathematics and physics lead to (non-sparse) functions, vectors, or matrices in which the fraction of nonnegligible entries is vanishingly small compared the total number of entries as the size of the system tends to infinity. In other words, the nonnegligible entries tend to be localized, or concentrated, around a small region within the computational domain, with rapid decay away from this region (uniformly as the system size grows). When present, localization opens up the possibility of developing fast approximation algorithms, the complexity of which scales linearly in the size of the problem. While localization already plays an important role in various areas of quantum physics and chemistry, it has received until recently relatively little attention by researchers in numerical linear algebra. In this chapter we survey localization phenomena arising in various fields, and we provide unified theoretical explanations for such phenomena using general results on the decay behavior of matrix functions. We also discuss computational implications for a range of applications.

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Notes

Acknowledgements

I would like to express my sincere gratitude to several friends and collaborators without whose contributions these lecture notes would not have been written, namely, Paola Boito, Matt Challacombe, Nader Razouk, Valeria Simoncini, and the late Gene Golub. I am also grateful for financial support to the Fondazione CIME and to the US National Science Foundation (grants DMS-0810862, DMS-1115692 and DMS-1418889).

References

  1. 1.
    M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, NY, 1965)zbMATHGoogle Scholar
  2. 2.
    S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N -Body Schrödinger Operators. Mathematical Notes, vol. 29 (Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982)Google Scholar
  3. 3.
    G. Alléon, M. Benzi, L. Giraud, Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics. Numer. Algorithms 16, 1–15 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    P.W. Anderson, Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)CrossRefGoogle Scholar
  5. 5.
    M. Arioli, M. Benzi, A finite element method for quantum graphs. Math/CS Technical Report TR-2015-009, Emory University, Oct 2015Google Scholar
  6. 6.
    E. Aune, D.P. Simpson, J. Eidsvik, Parameter estimation in high dimensional Gaussian distributions. Stat. Comput. 24, 247–263 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    O. Axelsson, Iterative Solution Methods (Cambridge University Press, Cambridge, 1994)zbMATHCrossRefGoogle Scholar
  8. 8.
    O. Axelsson, B. Polman, On approximate factorization methods for block matrices suitable for vector and parallel processors. Linear Algebra Appl. 77, 3–26 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    R. Baer, M. Head-Gordon, Sparsity of the density matrix in Kohn–Sham density functional theory and an assessment of linear system-size scaling methods. Phys. Rev. Lett. 79, 3962–3965 (1997)CrossRefGoogle Scholar
  10. 10.
    R. Baer, M. Head-Gordon, Chebyshev expansion methods for electronic structure calculations on large molecular systems. J. Chem. Phys. 107, 10003–10013 (1997)CrossRefGoogle Scholar
  11. 11.
    H. Bağci, J.E. Pasciak, K.Y. Sirenko, A convergence analysis for a sweeping preconditioner for block tridiagonal systems of linear equations. Numer. Linear Algebra Appl. 22, 371–392 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    A.G. Baskakov, Wiener’s theorem and the asymptotic estimates of the elements of inverse matrices. Funct. Anal. Appl. 24, 222–224 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    A.G. Baskakov, Estimates for the entries of inverse matrices and the spectral analysis of linear operators. Izv. Math. 61, 1113–1135 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    R. Bellman, Introduction to Matrix Analysis, 2nd edn. (McGraw-Hill, New York, NY, 1970)zbMATHGoogle Scholar
  15. 15.
    C.M. Bender, S. Boettcher, P.N. Meisinger, PT-symmetric quantum mechanics. J. Math. Phys. 40, 2201–2229 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    C.M. Bender, D.C. Brody, H.F. Jones, Must a Hamiltonian be Hermitian? Am. J. Phys. 71, 1095–1102 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    M. Benzi, Preconditioning techniques for large linear systems: a survey. J. Comp. Phys. 182, 418–477 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    M. Benzi, P. Boito, Quadrature rule-based bounds for functions of adjacency matrices. Linear Algebra Appl. 433, 637–652 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    M. Benzi, P. Boito, Decay properties for functions of matrices over C -algebras. Linear Algebra Appl. 456, 174–198 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    M. Benzi, G.H. Golub, Bounds for the entries of matrix functions with applications to preconditioning. BIT Numer. Math. 39, 417–438 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    M. Benzi, N. Razouk, Decay bounds and O(n) algorithms for approximating functions of sparse matrices. Electron. Trans. Numer. Anal. 28, 16–39 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    M. Benzi, V. Simoncini, Decay bounds for functions of Hermitian matrices with banded or Kronecker structure. SIAM J. Matrix Anal. Appl. 36, 1263–1282 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    M. Benzi, M. T˚uma, A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19, 968–994 (1998)Google Scholar
  24. 24.
    M. Benzi, M. T˚uma, Orderings for factorized approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1851–1868 (2000)Google Scholar
  25. 25.
    M. Benzi, C.D. Meyer, M. T˚uma, A sparse approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput. 17, 1135–1149 (1996)Google Scholar
  26. 26.
    M. Benzi, P. Boito, N. Razouk, Decay properties of spectral projectors with applications to electronic structure. SIAM Rev. 55, 3–64 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M. Benzi, T. Evans, S. Hamilton, M. Lupo Pasini, S. Slattery, Analysis of Monte Carlo accelerated iterative methods for sparse linear systems. Math/CS Technical Report TR-2016-002, Emory University. Numer. Linear Algebra Appl. 2017, to appearGoogle Scholar
  28. 28.
    S.K. Berberian, G.H. Orland, On the closure of the numerical range of an operator. Proc. Am. Math. Soc. 18, 499–503 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    L. Bergamaschi, M. Vianello, Efficient computation of the exponential operator for large, sparse, symmetric matrices. Numer. Linear Algebra Appl. 7, 27–45 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    L. Bergamaschi, M. Caliari, M. Vianello, Efficient approximation of the exponential operator for discrete 2D advection-diffusion problems. Numer. Linear Algebra Appl. 10, 271–289 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    D.A. Bini, G. Latouche, B. Meini, Numerical Methods for Structured Markov Chains (Oxford University Press, Oxford, 2005)zbMATHCrossRefGoogle Scholar
  32. 32.
    D.A. Bini, S. Dendievel, G. Latouche, B. Meini, Computing the exponential of large block-triangular block-Toeplitz matrices encountered in fluid queues. Linear Algebra Appl. 502, 387–419 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    I.A. Blatov, Incomplete factorization methods for systems with sparse matrices. Comput. Math. Math. Phys. 33, 727–741 (1993)MathSciNetzbMATHGoogle Scholar
  34. 34.
    I.A. Blatov, On algebras and applications of operators with pseudosparse matrices. Siber. Math. J. 37, 32–52 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    I.A. Blatov, A.A. Terteryan, Estimates of the elements of the inverse matrices and pivotal condensation methods of incomplete block factorization. Comput. Math. Math. Phys. 32, 1509–1522 (1992)MathSciNetzbMATHGoogle Scholar
  36. 36.
    N. Bock, M. Challacombe, An optimized sparse approximate matrix multiply for matrices with decay. SIAM J. Sci. Comput. 35, C72–C98 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    N. Bock, M. Challacombe, L.V. Kalé, Solvers for \(\mathcal{O}(N)\) electronic structure in the strong scaling limit. SIAM J. Sci. Comput. 38, C1–C21 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    L. Bonaventura, Local exponential methods: a domain decomposition approach to exponential time integration of PDEs. arXiv:1505.02248v1, May 2015Google Scholar
  39. 39.
    F. Bonchi, P. Esfandiar, D.F. Gleich, C. Greif, L.V.S. Lakshmanan, Fast matrix computations for pair-wise and column-wise commute times and Katz scores. Internet Math. 8, 73–112 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    A. Böttcher, S.M. Grudsky, Spectral Properties of Banded Toeplitz Matrices (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2005)zbMATHCrossRefGoogle Scholar
  41. 41.
    A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices (Springer, New York, NY, 1998)zbMATHGoogle Scholar
  42. 42.
    D.R. Bowler, T. Miyazaki, O(N) methods in electronic structure calculations. Rep. Prog. Phys. 75, 036503 (2012)CrossRefGoogle Scholar
  43. 43.
    S. Brooks, E. Lindenstrauss, Non-localization of eigenfunctions on large regular graphs. Isr. J. Math. 193, 1–14 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    C. Brouder, G. Panati, M. Calandra, C. Mourougane, N. Marzari, Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007)CrossRefGoogle Scholar
  45. 45.
    S. Bruciapaglia, S. Micheletti, S. Perotto, Compressed solving: a numerical approximation technique for elliptic PDEs based on compressed sensing. Comput. Math. Appl. 70, 1306–1335 (2015)MathSciNetCrossRefGoogle Scholar
  46. 46.
    K. Bryan, T. Lee, Making do with less: an introduction to compressed sensing. SIAM Rev. 55, 547–566 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    C. Canuto, V. Simoncini, M. Verani, On the decay of the inverse of matrices that are sum of Kronecker products. Linear Algebra Appl. 452, 21–39 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    C. Canuto, V. Simoncini, M. Verani, Contraction and optimality properties of an adaptive Legendre–Galerkin method: the multi-dimensional case. J. Sci. Comput. 63, 769–798 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    M. Challacombe, A simplified density matrix minimization for linear scaling self-consistent field theory. J. Chem. Phys. 110, 2332–2342 (1999)CrossRefGoogle Scholar
  50. 50.
    T. Chan, W.-P. Tang, J. Wan, Wavelet sparse approximate inverse preconditioners. BIT Numer. Math. 37, 644–660 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    J. Chandrasekar, D.S. Bernstein, Correlation bounds for discrete-time systems with banded dynamics. Syst. Control Lett. 56, 83–86 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    E. Chow, A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1804–1822 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    J.-M. Combes, L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys. 34, 251–270 (1973)zbMATHCrossRefGoogle Scholar
  54. 54.
    P. Concus, G.H. Golub, G. Meurant, Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comput. 6, 220–252 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    M. Cramer, J. Eisert, Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices. New J. Phys. 8, 71 (2006)MathSciNetCrossRefGoogle Scholar
  56. 56.
    M. Cramer, J. Eisert, M.B. Plenio, J. Dreissig, Entanglement-area law for general Bosonic harmonic lattice systems. Phys. Rev. A 73, 012309 (2006)CrossRefGoogle Scholar
  57. 57.
    M. Crouzeix, Numerical range and functional calculus in Hilbert space. J. Funct. Anal. 244, 668–690 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    C.K. Chui, M. Hasson, Degree of uniform approximation on disjoint intervals. Pac. J. Math. 105, 291–297 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    J.J.M. Cuppen, A divide and conquer method for the symmetric tridiagonal eigenproblem. Numer. Math. 36, 177–195 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    S. Dahlke, M. Fornasier, K. Gröchenig, Optimal adaptive computations in the Jaffard algebra and localized frames. J. Approx. Theory 162, 153–185 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    A. Damle, L. Lin, L. Ying, Compressed representations of Kohn–Sham orbitals via selected columns of the density matrix. J. Chem. Theory Comput. 11, 1463–1469 (2015)CrossRefGoogle Scholar
  62. 62.
    A. Damle, L. Lin, L. Ying, Accelerating selected columns of the density matrix computations via approximate column selection. arXiv:1604.06830v1, April 2016Google Scholar
  63. 63.
    P.J. Davis, Circulant Matrices (Wiley, New York, 1979)zbMATHGoogle Scholar
  64. 64.
    T.A. Davis, Y. Hu, The University of Florida sparse matrix collection. ACM Trans. Math. Softw. 38, 1–25 (2011)MathSciNetGoogle Scholar
  65. 65.
    Y. Dekel, J.R. Lee, N. Linial, Eigenvectors of random graphs: nodal domains. Random Struct. Algorithm 39, 39–58 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    N. Del Buono, L. Lopez, R. Peluso, Computation of the exponential of large sparse skew-symmetric matrices. SIAM J. Sci. Comput. 27, 278–293 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    S. Demko, Inverses of band matrices and local convergence of spline projections. SIAM J. Numer. Anal. 14, 616–619 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    S. Demko, W.F. Moss, P.W. Smith, Decay rates for inverses of band matrices. Math. Comput. 43, 491–499 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    J. des Cloizeaux, Energy bands and projection operators in a crystal: analytic and asymptotic properties. Phys. Rev. 135, A685–A697 (1964)Google Scholar
  70. 70.
    I.S. Dhillon, B.S. Parlett, C. Vömel, The design and implementation of the MRRR algorithm. ACM Trans. Math. Softw. 32, 533–560 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    R. Diestel, Graph Theory (Springer, Berlin, 2000)zbMATHGoogle Scholar
  72. 72.
    I.S. Duff, A.M. Erisman, J.K. Reid, Direct Methods for Sparse Matrices (Oxford University Press, Oxford, 1986)zbMATHGoogle Scholar
  73. 73.
    I. Dumitriu, S. Pal, Sparse regular random graphs: spectral density and eigenvectors. Ann. Prob. 40, 2197–2235 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    W.E, J. Lu, The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy–Born rule. Arch. Ration. Mech. Anal. 199, 407–433 (2011)Google Scholar
  75. 75.
    V. Eijkhout, B. Polman, Decay rates of inverses of banded M-matrices that are near to Toeplitz matrices. Linear Algebra Appl. 109, 247–277 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    J. Eisert, M. Cramer, M.B. Plenio, Colloquium: area laws for the entanglement entropy. Rev. Modern Phys. 82, 277–306 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    S.W. Ellacott, Computation of Faber series with application to numerical polynomial approximation in the complex plane. Math. Comput. 40, 575–587 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    E. Estrada, The Structure of Complex Networks: Theory and Applications (Oxford University Press, Oxford, 2012)zbMATHGoogle Scholar
  79. 79.
    E. Estrada, N. Hatano, Communicability in complex networks. Phys. Rev. E 77, 036111 (2008)MathSciNetCrossRefGoogle Scholar
  80. 80.
    E. Estrada, D.J. Higham, Network properties revealed by matrix functions. SIAM Rev. 52, 696–714 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    E. Estrada, N. Hatano, M. Benzi, The physics of communicability in complex networks. Phys. Rep. 514, 89–119 (2012)MathSciNetCrossRefGoogle Scholar
  82. 82.
    I. Faria, Permanental roots and the star degree of a graph. Linear Algebra Appl. 64, 255–265 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    N.J. Ford, D.V. Savostyanov, N.L. Zamarashkin, On the decay of the elements of inverse triangular Toeplitz matrices. SIAM J. Matrix Anal. Appl. 35, 1288–1302 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    R. Freund, On polynomial approximations to f a(z) = (za)−1 with complex a and some applications to certain non-Hermitian matrices. Approx. Theory Appl. 5, 15–31 (1989)MathSciNetzbMATHGoogle Scholar
  85. 85.
    I.M. Gelfand, Normierte Ringe. Mat. Sb. 9, 3–23 (1941)MathSciNetGoogle Scholar
  86. 86.
    I.M. Gelfand, M.A. Neumark, On the imbedding of normed rings in the ring of operators in Hilbert space. Mat. Sb. 12, 197–213 (1943)MathSciNetzbMATHGoogle Scholar
  87. 87.
    I.M. Gelfand, D.A. Raikov, G.E. Shilov, Commutative Normed Rings (Chelsea Publishing Co., Bronx/New York, 1964)Google Scholar
  88. 88.
    P.-L. Giscard, K. Lui, S.J. Thwaite, D. Jaksch, An exact formulation of the time-ordered exponential using path-sums. J. Math. Phys. 56, 053503 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    D.F. Gleich, PageRank beyond the Web. SIAM Rev. 57, 321–363 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    D.F. Gleich, K. Kloster, Sublinear column-wise actions of the matrix exponential on social networks. Internet Math. 11, 352–384 (2015)MathSciNetCrossRefGoogle Scholar
  91. 91.
    S. Goedecker, Linear scaling electronic structure methods. Rev. Mod. Phys. 71, 1085–1123 (1999)CrossRefGoogle Scholar
  92. 92.
    S. Goedecker, O.V. Ivanov, Frequency localization properties of the density matrix and its resulting hypersparsity in a wavelet representation. Phys. Rev. B 59, 7270–7273 (1999)CrossRefGoogle Scholar
  93. 93.
    K.-I. Goh, B. Khang, D. Kim, Spectra and eigenvectors of scale-free networks. Phys. Rev. E 64, 051903 (2001)CrossRefGoogle Scholar
  94. 94.
    G.H. Golub, G. Meurant, Matrices, Moments and Quadrature with Applications (Princeton University Press, Princeton, NJ, 2010)zbMATHGoogle Scholar
  95. 95.
    G.H. Golub, C.F. Van Loan, Matrix Computations, 4th edn. (Johns Hopkins University Press, Baltimore/London, 2013)zbMATHGoogle Scholar
  96. 96.
    K. Gröchenig, A. Klotz, Noncommutative approximation: inverse-closed subalgebras and off-diagonal decay of matrices. Constr. Approx. 32, 429–466 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    K. Gröchenig, M. Leinert, Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices. Trans. Am. Math. Soc. 358, 2695–2711 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    K. Gröchenig, Z. Rzeszotnik, T. Strohmer, Convergence analysis of the finite section method and Banach algebras of matrices. Integr. Equ. Oper. Theory 67, 183–202 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    M. Grote, T. Huckle, Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18, 838–853 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    J. Gutiérrez-Gutiérrez, P.M. Crespo, A. Böttcher, Functions of the banded Hermitian block Toeplitz matrices in signal processing. Linear Algebra Appl. 422, 788–807 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    S. Güttel, L. Knizhnerman, A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions. BIT Numer. Math. 53, 595–616 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    A. Haber, M. Verhaegen, Subspace identification of large-scale interconnected systems. IEEE Trans. Automat. Control 59, 2754–2759 (2014)MathSciNetCrossRefGoogle Scholar
  103. 103.
    A. Haber, M. Verhaegen, Sparse solution of the Lyapunov equation for large-scale interconnected systems. Automatica 73, 256–268 (2016)MathSciNetCrossRefGoogle Scholar
  104. 104.
    M. Hasson, The degree of approximation by polynomials on some disjoint intervals in the complex plane. J. Approx. Theory 144, 119–132 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  105. 105.
    L. He, D. Vanderbilt, Exponential decay properties of Wannier functions and related quantities. Phys. Rev. Lett. 86, 5341–5344 (2001)CrossRefGoogle Scholar
  106. 106.
    V.E. Henson, G. Sanders, Locally supported eigenvectors of matrices associated with connected and unweighted power-law graphs. Electron. Trans. Numer. Anal. 39, 353–378 (2012)MathSciNetzbMATHGoogle Scholar
  107. 107.
    N.J. Higham, Matrix Functions. Theory and Computation (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008)Google Scholar
  108. 108.
    N.J. Higham, D.S. Mackey, N. Mackey, F. Tisseur, Functions preserving matrix groups and iterations for the matrix square root. SIAM J. Matrix Anal. Appl. 26, 1178–1192 (2005)MathSciNetzbMATHGoogle Scholar
  109. 109.
    M. Hochbruck, Ch. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    P. Hohenberg, W. Kohn, Inhomogeneous electron gas. Phys. Rev. 136, B864–871 (1964)MathSciNetCrossRefGoogle Scholar
  111. 111.
    R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1994)zbMATHGoogle Scholar
  112. 112.
    R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edn. (Cambridge University Press, Cambridge, 2013)zbMATHGoogle Scholar
  113. 113.
    T. Huckle, Approximate sparsity patterns for the inverse of a matrix and preconditioning. Appl. Numer. Math. 30, 291–303 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    M. Hutchinson, A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat. Simul. Comput. 18, 1059–1076 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    A. Iserles,How large is the exponential of a banded matrix? N. Z. J. Math. 29, 177–192 (2000)MathSciNetzbMATHGoogle Scholar
  116. 116.
    S. Ismail-Beigi, T.A. Arias, Locality of the density matrix in metals, semiconductors, and insulators. Phys. Rev. Lett. 82, 2127–2130 (1999)CrossRefGoogle Scholar
  117. 117.
    S. Jaffard, Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications. Ann. Inst. Henri Poincarè 7, 461–476 (1990)MathSciNetzbMATHGoogle Scholar
  118. 118.
    J. Janas, S. Naboko, G. Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Intern. Math. Res. Notices 4, 736–764 (2009)MathSciNetzbMATHGoogle Scholar
  119. 119.
    R. Kadison, Diagonalizing matrices. Am. J. Math. 106, 1451–1468 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    R. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras. Elementary Theory, vol. I (Academic Press, Orlando, FL, 1983)Google Scholar
  121. 121.
    W. Kohn, Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, 809–821 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  122. 122.
    W. Kohn, Density functional and density matrix method scaling linearly with the number of atoms. Phys. Rev. Lett. 76, 3168–3171 (1996)CrossRefGoogle Scholar
  123. 123.
    W. Kohn, Nobel lecture: electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999)CrossRefGoogle Scholar
  124. 124.
    W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. Lett. 140, A1133–1138 (1965)MathSciNetGoogle Scholar
  125. 125.
    L.Y. Kolotilina, A.Y. Yeremin, Factorized sparse approximate inverse preconditioning I. Theory. SIAM J. Matrix Anal. Appl. 14, 45–58 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  126. 126.
    A. Koskela, E. Jarlebring, The infinite Arnoldi exponential integrator for linear inhomogeneous ODEs. arXiv:1502.01613v2, Feb 2015Google Scholar
  127. 127.
    I. Kryshtal, T. Strohmer, T. Wertz, Localization of matrix factorizations. Found. Comput. Math. 15, 931–951 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    R. Lai, J. Lu, Localized density matrix minimization and linear-scaling algorithms. J. Comput. Phys. 315, 194–210 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    C.S. Lam, Decomposition of time-ordered products and path-ordered exponentials. J. Math. Phys. 39, 5543–5558 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  130. 130.
    A.N. Langville, C.D. Meyer Google’s PageRank and Beyond: The Science of Search Engine Rankings (Princeton University Press, Princeton, NJ, 2006)Google Scholar
  131. 131.
    A.J. Laub, Matrix Analysis for Scientists and Engineers (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2005)zbMATHCrossRefGoogle Scholar
  132. 132.
    C. Le Bris, Computational chemistry from the perspective of numerical analysis. Acta Numer. 14, 363–444 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    X.-P. Li, R.W. Nunes, D. Vanderbilt, Density-matrix electronic structure method with linear system-size scaling. Phys. Rev. B 47, 10891–10894 (1993)CrossRefGoogle Scholar
  134. 134.
    W. Liang, C. Saravanan, Y. Shao, R. Baer, A. T. Bell, M. Head-Gordon, Improved Fermi operator expansion methods for fast electronic structure calculations. J. Chem. Phys. 119, 4117–4124 (2003)CrossRefGoogle Scholar
  135. 135.
    L. Lin, Localized spectrum slicing. Math. Comput. (2016, to appear). DOI:10.1090/mcom/3166Google Scholar
  136. 136.
    L. Lin, J. Lu, Sharp decay estimates of discretized Green’s functions for Schrödinger type operators. Sci. China Math. 59, 1561–1578 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  137. 137.
    F.-R. Lin, M.K. Ng, W.-K. Ching, Factorized banded inverse preconditioners for matrices with Toeplitz structure. SIAM J. Sci. Comput. 26, 1852–1870 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  138. 138.
    L. Lin, J. Lu, L. Ying, R. Car, E. Weinan, Multipole representation of the Fermi operator with application to the electronic structure analysis of metallic systems. Phys. Rev. B 79, 115133 (2009)CrossRefGoogle Scholar
  139. 139.
    M. Lindner, Infinite Matrices and Their Finite Sections (Birkhäuser, Basel, 2006)zbMATHGoogle Scholar
  140. 140.
    X. Liu, G. Strang, S. Ott, Localized eigenvectors from widely spaced matrix modifications. SIAM J. Discrete Math. 16, 479–498 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 141.
    L. Lopez, A. Pugliese, Decay behaviour of functions of skew-symmetric matrices, in Proceedings of HERCMA 2005, 7th Hellenic-European Conference on Computer Mathematics and Applications, 22–24 Sept 2005, Athens, ed. By E.A. Lipitakis, Electronic Editions (LEA, Athens, 2005)Google Scholar
  142. 142.
    T. Malas, L. Gürel, Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel multipole algorithm. SIAM J. Sci. Comput. 33, 2440–2467 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  143. 143.
    A.I. Markushevich, Theory of Functions of a Complex Variable, vol. III (Prentice-Hall, Englewood Cliffs, NJ, 1967)zbMATHGoogle Scholar
  144. 144.
    O.A. Marques, B.N. Parlett, C. Vömel, Computation of eigenpair subsets with the MRRR algorithm. Numer. Linear Algebra Appl. 13, 643–653 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  145. 145.
    R.M. Martin, Electronic Structure. Basic Theory and Practical Methods (Cambridge University Press, Cambridge, 2004)Google Scholar
  146. 146.
    P.E. Maslen, C. Ochsenfeld, C.A. White, M.S. Lee, M. Head-Gordon, Locality and sparsity of ab initio one-particle density matrices and localized orbitals. J. Phys. Chem. A 102, 2215–2222 (1998)CrossRefGoogle Scholar
  147. 147.
    N. Mastronardi, M.K. Ng, E.E. Tyrtyshnikov, Decay in functions of multi-band matrices. SIAM J. Matrix Anal. Appl. 31, 2721–2737 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  148. 148.
    G. Meinardus, Approximation of Functions: Theory and Numerical Methods. Springer Tracts in Natural Philosophy, vol. 13 (Springer, New York, 1967)Google Scholar
  149. 149.
    P.N. McGraw, M. Menzinger, Laplacian spectra as a diagnostic tool for network structure and dynamics. Phys. Rev. E 77, 031102 (2008)CrossRefGoogle Scholar
  150. 150.
    N. Merkle, Completely monotone functions—a digest. arXiv:1211.0900v1, Nov 2012Google Scholar
  151. 151.
    G. Meurant, A review of the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM J. Matrix Anal. Appl. 13, 707–728 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  152. 152.
    N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University Press, Cambridge, 2011)zbMATHCrossRefGoogle Scholar
  153. 153.
    L. Molinari, Identities and exponential bounds for transfer matrices. J. Phys. A: Math. Theor. 46, 254004 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  154. 154.
    R. Nabben, Decay rates of the inverse of nonsymmetric tridiagonal and band matrices. SIAM J. Matrix Anal. Appl. 20, 820–837 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  155. 155.
    Y. Nakatsukasa, Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices. Appl. Numer. Math. 62, 67–78 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  156. 156.
    Y. Nakatsukasa, N. Saito, E. Woei, Mysteries around the graph Laplacian eigenvalue 4. Linear Algebra Appl. 438, 3231–3246 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  157. 157.
    H. Nassar, K. Kloster, D.F. Gleich, Strong localization in personalized PageRank vectors, in Algorithms and Models for the Web Graph, ed. by D.F. Gleich et al. Lecture Notes in Computer Science, vol. 9479 (Springer, New York, 2015), pp. 190–202Google Scholar
  158. 158.
    G. Nenciu, Existence of the exponentially localised Wannier functions. Commun. Math. Phys. 91, 81–85 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  159. 159.
    A.M.N. Niklasson, Density matrix methods in linear scaling electronic structure theory, in Linear-Scaling Techniques in Computational Chemistry and Physics, ed. by R. Zaleśny et al. (Springer, New York, 2011), pp. 439–473CrossRefGoogle Scholar
  160. 160.
    J. Pan, R. Ke, M.K. Ng, H.-W. Sun, Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations. SIAM. J. Sci. Comput. 36, A2698–A2719 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  161. 161.
    B.N. Parlett, Invariant subspaces for tightly clustered eigenvalues of tridiagonals. BIT Numer. Math. 36, 542–562 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  162. 162.
    B.N. Parlett, A result complementary to Geršgorin’s circle theorem. Linear Algebra Appl. 432, 20–27 (2009)zbMATHCrossRefGoogle Scholar
  163. 163.
    B.N. Parlett, I.S. Dhillon, Relatively robust representations of symmetric tridiagonals. Linear Algebra Appl. 309, 121–151 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  164. 164.
    M.S. Paterson, L.J. Stockmeyer, On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM J. Comput. 2, 60–66 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  165. 165.
    E. Prodan, Nearsightedness of electronic matter in one dimension. Phys. Rev. B 73, 085108 (2006)CrossRefGoogle Scholar
  166. 166.
    E. Prodan, W. Kohn, Nearsightedness of electronic matter. Proc. Nat. Acad. Sci., 102, 11635–11638 (2005)CrossRefGoogle Scholar
  167. 167.
    E. Prodan, S.R. Garcia, M. Putinar, Norm estimates of complex symmetric operators applied to quantum systems. J. Phys. A: Math. Gen. 39, 389–400 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  168. 168.
    N. Razouk, Localization phenomena in matrix functions: theory and algorithms, Ph.D. Thesis, Emory University, 2008Google Scholar
  169. 169.
    L. Reichel, G. Rodriguez, T. Tang, New block quadrature rules for the approximation of matrix functions. Linear Algebra Appl. 502, 299–326 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  170. 170.
    S. Roch, Finite Sections of Band-Dominated Operators, vol. 191, no. 895 (Memoirs of the American Mathematical Society, Providence, RI, 2008)zbMATHGoogle Scholar
  171. 171.
    G. Rodriguez, S. Seatzu, D. Theis, An algorithm for solving Toeplitz systems by embedding in infinite systems. Oper. Theory Adv. Appl. 160, 383–401 (2005)MathSciNetzbMATHGoogle Scholar
  172. 172.
    E.H. Rubensson, E. Rudberg, P. Salek, Methods for Hartree–Fock and density functional theory electronic structure calculations with linearly scaling processor time and memory usage, in Linear-Scaling Techniques in Computational Chemistry and Physics, ed. by R. Zaleśny et al. (Springer, New York, NY, 2011), pp. 269–300Google Scholar
  173. 173.
    W. Rudin, Functional Analysis (McGraw-Hill, New York, NY, 1973)zbMATHGoogle Scholar
  174. 174.
    Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edn. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003)zbMATHCrossRefGoogle Scholar
  175. 175.
    Y. Saad, J.R. Chelikowsky, S.M. Shontz, Numerical methods for electronic structure calculations of materials. SIAM Rev. 52, 3–54 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  176. 176.
    N. Schuch, J.I. Cirac, M.M. Wolf, Quantum states on harmonic lattices. Commun. Math. Phys. 267, 65–92 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  177. 177.
    M. Shao, On the finite section method for computing exponentials of doubly-infinite skew-Hermitian matrices. Linear Algebra Appl. 451, 65–96 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  178. 178.
    D.I. Shuman, B. Ricaud, P. Vandergheynst, Vertex-frequency analysis on graphs. Appl. Comput. Harmon. Anal. 40, 260–291 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  179. 179.
    C. Siefert, E. de Sturler, Probing methods for saddle-point problems. Electron. Trans. Numer. Anal. 22, 163–183 (2006)MathSciNetzbMATHGoogle Scholar
  180. 180.
    B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A 38, 295–308 (1983)zbMATHGoogle Scholar
  181. 181.
    V. Simoncini, Computational methods for linear matrix equations. SIAM Rev. 58, 377–441 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  182. 182.
    D.T. Smith, Exponential decay of resolvents and discrete eigenfunctions of banded infinite matrices. J. Approx. Theory 66, 83–97 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  183. 183.
    G. Stolz, An introduction to the mathematics of Anderson localization, in Entropy and the Quantum II, ed. by R. Sims, D. Ueltschi. Contemporary Mathematics, vol. 552 (American Mathematical Society, Providence, RI, 2011), pp. 71–108Google Scholar
  184. 184.
    G. Strang, S. MacNamara, Functions of difference matrices are Toeplitz plus Hankel. SIAM Rev. 56, 525–546 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  185. 185.
    T. Strohmer, Four short stories about Toeplitz matrix calculations. Linear Algebra Appl. 343/344, 321–344 (2002)Google Scholar
  186. 186.
    Q. Sun, Wiener’s lemma for infinite matrices with polynomial off-diagonal decay. C. R. Acad. Sci. Paris Ser. I 340, 567–570 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  187. 187.
    P. Suryanarayana, On spectral quadrature for linear-scaling density functional theory. Chem. Phys. Lett. 584, 182–187 (2013)CrossRefGoogle Scholar
  188. 188.
    H. Tal-Ezer, Polynomial approximation of functions of matrices and applications. J. Sci. Comput. 4, 25–60 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  189. 189.
    L.V. Tran, V.H. Vu, K. Wang, Sparse random graphs: eigenvalues and eigenvectors. Random Struct. Algorithm. 42, 110–134 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  190. 190.
    L.N. Trefethen, Numerical computation of the Schwarz–Christoffel transformation. SIAM J. Sci. Stat. Comput. 1, 82–102 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  191. 191.
    L.N. Trefethen, D. Bau, Numerical Linear Algebra (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997)zbMATHCrossRefGoogle Scholar
  192. 192.
    L.N. Trefethen, M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators (Princeton University Press, Princeton, NJ, 2005)Google Scholar
  193. 193.
    L.N. Trefethen, M. Contedini, M. Embree, Spectra, pseudospectra, and localization for random bidiagonal matrices. Commun. Pure Appl. Math. 54, 595–623 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  194. 194.
    C.V.M. van der Mee, G. Rodriguez, S. Seatzu, LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices. Calcolo 33, 307–335 (1998)MathSciNetzbMATHGoogle Scholar
  195. 195.
    C.V.M. van der Mee, G. Rodriguez, S. Seatzu, Block Cholesky factorization of infinite matrices and orthonormalization of vectors of functions, in Advances in Computational Mathematics (Guangzhou, 1997). Lecture Notes in Pure and Applied Mathematics (Dekker, New York, 1999), pp. 423–455Google Scholar
  196. 196.
    R.S. Varga, Nonnegatively posed problems and completely monotonic functions. Linear Algebra Appl. 1, 329–347 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  197. 197.
    P.S. Vassilevski, On some ways of approximating inverses of band matrices in connection with deriving preconditioners based on incomplete block factorizations. Computing 43, 277–296 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  198. 198.
    C. Vömel, B. N. Parlett, Detecting localization in an invariant subspace. SIAM J. Sci. Comput. 33, 3447–3467 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  199. 199.
    H. Wang, Q. Ye, Error bounds for the Krylov subspace methods for computations of matrix exponentials. Tech. Rep., Department of Mathematics, University of Kentucky, Lexington, KY, 2016Google Scholar
  200. 200.
    H.F. Weinberger, A First Course in Partial Differential Equations (Wiley, New York, 1965)zbMATHGoogle Scholar
  201. 201.
    D.V. Widder, The Laplace Transform (Princeton University Press, Princeton, 1946)zbMATHGoogle Scholar
  202. 202.
    W. Yang, Direct calculation of electron density in density-functional theory. Phys. Rev. Lett. 66, 1438–1441 (1991)CrossRefGoogle Scholar
  203. 203.
    Q. Ye, Error bounds for the Lanczos method for approximating matrix exponentials. SIAM J. Numer. Anal. 51, 68–87 (2013)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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