Numerical solution of the eigenvalue problem for symmetric rationally generated Toeplitz matrices
SIAM Journal on Matrix Analysis and Applications
Computational frameworks for the fast Fourier transform
Computational frameworks for the fast Fourier transform
A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems
SIAM Journal on Matrix Analysis and Applications
Using MPI: portable parallel programming with the message-passing interface
Using MPI: portable parallel programming with the message-passing interface
Fast Gaussian elimination with partial pivoting for matrices with displacement structure
Mathematics of Computation
ScaLAPACK user's guide
The symmetric eigenvalue problem
The symmetric eigenvalue problem
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
SIPs: Shift-and-invert parallel spectral transformations
ACM Transactions on Mathematical Software (TOMS)
Efficient buckling and free vibration analysis of cyclically repeated space truss structures
Finite Elements in Analysis and Design
Implementation and tuning of a parallel symmetric Toeplitz eigensolver
Journal of Parallel and Distributed Computing
Strategies for spectrum slicing based on restarted Lanczos methods
Numerical Algorithms
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This paper presents a new procedure to compute many or all of the eigenvalues and eigenvectors of symmetric Toeplitz matrices. The key to this algorithm is the use of the ''Shift-and-Invert'' technique applied with iterative methods, which allows the computation of the eigenvalues close to a given real number (the ''shift''). Given an interval containing all the desired eigenvalues, this large interval can be divided in small intervals. Then, the ''Shift-and-Invert'' version of an iterative method (Lanczos method, in this paper) can be applied to each subinterval. Since the extraction of the eigenvalues of each subinterval is independent from the other subintervals, this method is highly suitable for implementation in parallel computers. This technique has been adapted to symmetric Toeplitz problems, using the symmetry exploiting Lanczos process proposed by Voss [H. Voss, A symmetry exploiting Lanczos method for symmetric Toeplitz matrices, Numerical Algorithms 25 (2000) 377-385] and using fast solvers for the Toeplitz linear systems that must be solved in each Lanczos iteration. The method compares favourably with ScaLAPACK routines, specially when not all the spectrum must be computed.