Eigenvalue Algorithms for Symmetric Hierarchical Matrices

[1]  Claude Le Bris,et al.  Computational chemistry from the perspective of numerical analysis , 2005, Acta Numerica.

[2]  Wolfgang Hackbusch,et al.  A projection method for the computation of inner eigenvalues using high degree rational operators , 2007, Computing.

[3]  David S. Watkins,et al.  Convergence of algorithms of decomposition type for the eigenvalue problem , 1991 .

[4]  Shivkumar Chandrasekaran,et al.  A fast and stable solver for recursively semi-separable systems of linear equations , 2001 .

[5]  Tri Nguyen,et al.  The Interplay of Ranks of Submatrices , 2004, SIAM Rev..

[6]  W. Hackbusch,et al.  Introduction to Hierarchical Matrices with Applications , 2003 .

[7]  Peter Benner,et al.  An Implicitly Restarted Symplectic Lanczos Method for the Symplectic Eigenvalue Problem , 2000, SIAM J. Matrix Anal. Appl..

[8]  Steffen Börm,et al.  Data-sparse approximation of non-local operators by H2-matrices , 2007 .

[9]  Michael Lintner.  Lösung der 2D Wellengleichung mittels hierarchischer Matrizen , 2002 .

[10]  Ulrike Baur,et al.  Control-Oriented Model Reduction for Parabolic Systems , 2008 .

[11]  Lars Grasedyck,et al.  Theorie und Anwendungen Hierarchischer Matrizen , 2006 .

[12]  Boris N. Khoromskij,et al.  Numerical Solution of the Hartree - Fock Equation in Multilevel Tensor-Structured Format , 2011, SIAM J. Sci. Comput..

[13]  W. Hackbusch.  A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices , 1999, Computing.

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

[15]  M. Barel,et al.  TRANSFORMING A HIERARCHICAL INTO A UNITARY-WEIGHT REPRESENTATION , 2008 .

[16]  Steffen Börm,et al.  Hybrid cross approximation of integral operators , 2005, Numerische Mathematik.

[17]  S. Delvaux,et al.  Transforming a hierarchical matrix into a unitary-weight representation , 2008 .

[18]  Paul Willems,et al.  On MR3-type Algorithms for the Tridiagonal Symmetric Eigenproblem and the Bidiagonal SVD , 2018 .

[19]  Boris N. Khoromskij,et al.  Solution of Large Scale Algebraic Matrix Riccati Equations by Use of Hierarchical Matrices , 2003, Computing.

[20]  B. Parlett,et al.  From qd to LR, or, how were the qd and LR algorithms discovered? , 2011 .

[21]  Rajeev Motwani,et al.  The PageRank Citation Ranking : Bringing Order to the Web , 1999, WWW 1999.

[22]  Kai Sundmacher,et al.  H-matrix methods for linear and quasi-linear integral operators appearing in population balances , 2007, Comput. Chem. Eng..

[23]  M. SIAMJ..  THE RELATION BETWEEN THE QR AND LR ALGORITHMS∗ , 1998 .

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

[25]  Lin-Wang Wang,et al.  Electronic Structure Pseudopotential Calculations of Large (-1000 Atoms ) Si Quantum Dots , 2001 .

[26]  J. Bunch,et al.  Some stable methods for calculating inertia and solving symmetric linear systems , 1977 .

[27]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[28]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[29]  Lin-Wang Wang,et al.  Electronic Structure Pseudopotential Calculations of Large (.apprx.1000 Atoms) Si Quantum Dots , 1994 .

[30]  Ronald Kriemann,et al.  Hierarchical Matrices Based on a Weak Admissibility Criterion , 2004, Computing.

[31]  Marc Van Barel,et al.  A Cholesky LR algorithm for the positive definite symmetric diagonal-plus-semiseparable eigenproblem , 2008 .

[32]  Robert A. van de Geijn,et al.  Parallelizing the QR Algorithm for the Unsymmetric Algebraic Eigenvalue Problem: Myths and Reality , 1996, SIAM J. Sci. Comput..

[33]  L. Grasedyck,et al.  Domain decomposition based $${\mathcal H}$$ -LU preconditioning , 2009, Numerische Mathematik.

[34]  Structures preserved by the QR-algorithm , 2006 .

[35]  G. Golub,et al.  Eigenvalue computation in the 20th century , 2000 .

[36]  Shivkumar Chandrasekaran,et al.  A Fast ULV Decomposition Solver for Hierarchically Semiseparable Representations , 2006, SIAM J. Matrix Anal. Appl..

[37]  T. Chan.  Rank revealing QR factorizations , 1987 .

[38]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .

[39]  Andrew Knyazev,et al.  Preconditioned Eigensolvers - an Oxymoron? , 1998 .

[40]  E. Tyrtyshnikov.  Mosaic-Skeleton approximations , 1996 .

[41]  J. G. F. Francis,et al.  The QR Transformation A Unitary Analogue to the LR Transformation - Part 1 , 1961, Comput. J..

[42]  N. Higham.  Computing the polar decomposition with applications , 1986 .

[43]  K. Neymeyr.  A geometric theory for preconditioned inverse iteration. I : Extrema of the Rayleigh quotient , 2001 .

[44]  Gene H. Golub,et al.  Matrix computations , 1983 .

[45]  Wolfgang Hackbusch,et al.  Construction and Arithmetics of H-Matrices , 2003, Computing.

[46]  S Sauter,et al.  Boundary elements methods. Analysis, numerics and implementation of fast algorithms. (Randelementmethoden. Analyse, Numerik und Implementierung schneller Algorithmen.) , 2004 .

[47]  Eugene E. Tyrtyshnikov,et al.  Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions , 2009, SIAM J. Sci. Comput..

[48]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[49]  Gene H. Golub,et al.  Calculating the singular values and pseudo-inverse of a matrix , 2007, Milestones in Matrix Computation.

[50]  G. W. Stewart,et al.  Four algorithms for the the efficient computation of truncated pivoted QR approximations to a sparse matrix , 1999, Numerische Mathematik.

[51]  B. Parlett,et al.  Accurate singular values and differential qd algorithms , 1994 .

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

[53]  G. Schulz.  Iterative Berechung der reziproken Matrix , 1933 .

[54]  S. Chandrasekaran,et al.  A fast adaptive solver for hierarchically semiseparable representations , 2005 .

[55]  W. Hackbusch.  Integral Equations: Theory and Numerical Treatment , 1995 .

[56]  Peter Benner,et al.  The Symplectic Eigenvalue Problem, the Butterfly Form, the SR Algorithm, and the Lanczos Method , 1998 .

[57]  R. Vandebril,et al.  An Implicit Q Theorem for Hessenberg-like Matrices , 2005 .

[58]  Boris N. Khoromskij,et al.  Use of tensor formats in elliptic eigenvalue problems , 2012, Numer. Linear Algebra Appl..

[59]  J. G. F. Francis,et al.  The QR Transformation - Part 2 , 1962, Comput. J..

[60]  Marc Van Barel,et al.  Divide and conquer algorithms for computing the eigendecomposition of symmetric diagonal-plus-semiseparable matrices , 2005, Numerical Algorithms.

[61]  Peter Benner,et al.  Two connections between the SR and HR eigenvalue algorithms , 1998 .

[62]  A. Knyazev,et al.  A Geometric Theory for Preconditioned Inverse Iteration. III:A Short and Sharp Convergence Estimate for Generalized EigenvalueProblems. , 2001 .

[63]  Jianlin Xia,et al.  Fast algorithms for hierarchically semiseparable matrices , 2010, Numer. Linear Algebra Appl..

[64]  Timothy A. Davis,et al.  The university of Florida sparse matrix collection , 2011, TOMS.

[65]  J. Cuppen.  A divide and conquer method for the symmetric tridiagonal eigenproblem , 1980 .

[66]  Robert E. Mahony,et al.  A Grassmann-Rayleigh Quotient Iteration for Computing Invariant Subspaces , 2002, SIAM Rev..

[67]  Peter Benner,et al.  On the QR decomposition of $${\fancyscript {H}}$$ -matrices , 2010, Computing.

[68]  John R. Gilbert,et al.  Sparse Matrices in MATLAB: Design and Implementation , 1992, SIAM J. Matrix Anal. Appl..

[69]  Ilse C. F. Ipsen.  A history of inverse iteration , 1994 .

[70]  G. Golub,et al.  A bibliography on semiseparable matrices* , 2005 .

[71]  Dario Fasino.  Rational Krylov matrices and QR steps on Hermitian diagonal-plus-semiseparable matrices , 2005, Numer. Linear Algebra Appl..

[72]  David S. Watkins.  QR -like algorithms for eigenvalue problems , 2000 .

[73]  Mario Bebendorf,et al.  Stabilized rounded addition of hierarchical matrices , 2007, Numer. Linear Algebra Appl..

[74]  Heinz Rutishauser,et al.  Bestimmung der Eigenwerte und Eigenvektoren einer Matrix mit Hilfe des Quotienten-Differenzen-Algorithmus , 1955 .

[75]  Christian H. Bischof,et al.  Algorithm 782: codes for rank-revealing QR factorizations of dense matrices , 1998, TOMS.

[76]  Mario Bebendorf,et al.  Mathematik in den Naturwissenschaften Leipzig Existence of H-Matrix Approximants to the Inverse FE-Matrix of Elliptic Operators with L ∞-Coefficients , 2003 .

[77]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[78]  Shivkumar Chandrasekaran,et al.  A Hierarchical Semi-separable Moore-Penrose Equation Solver , 2006 .

[79]  Sabine Le Borne,et al.  Numerische Mathematik Domain decomposition based H-LU preconditioning , 2009 .

[80]  Mario Bebendorf,et al.  Approximation of boundary element matrices , 2000, Numerische Mathematik.

[81]  T. Mach.  Lösung von Randintegralgleichungen zur Bestimmung derKapazitätsmatrix von Elektrodenanordnungen mittels H -Arithmetik , 2008 .