The Symmetric Eigenvalue Problem

By Beresford N. Parlett. 1981

Subfields: Statistical Learning Theory, Numerical Methods & Scientific Computing

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Citations in the corpus

User-Friendly Tail Bounds for Sums of Random Matrices (2010, methodology): “We adopt Parlett’s convention [41] that bold capital letters symmetric about the vertical axis (A, .”
Structured inverse eigenvalue problems (2002, unknown): “One instance, according to Parlett (1998), is that ‘Vibrations are everywhere, and so too are the eigenvalues associated with them.’”
Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems (2000, methodology): “When the matrix is symmetric, the Lanczos method (see Algorithm 1, [11, 16, 20]) is the most commonly used method.”
Communication-optimal Parallel and Sequential QR and LU Factorizations (2008, background): “This is the source of Kahan’s maxim, “Twice is enough” [48]: the accuracy reaches its theoretical best after one reorthogonalization pass (see also [1]), and further reorthogonalizations do not improve orthogonality.”
Multisurface proximal support vector machine classification via generalized eigenvalues (2006, methodology): “SUPPORT vector machines (SVMs) [23], [4], [27] constitute the method of choice for classification problems while the generalized eigenvalue problem [22], [5] is a simple problem of classical linear algebra solvable by a single command of MATLAB [17] or Scilab [24] or by using standard linear algebra”
Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra (2019, methodology): “y reproduced in this notation; much as in [Pai1971], these identities are then used to analyze various iterative methods for computing eigenvectors. The proof of the theorem is left as an exercise in [Par1980], with the adjugate method given as a hint. Essentially the same result is also stated in t”
Numerical Methods for Electronic Structure Calculations of Materials (2010, background): “In practice, orthogonality of the La nczos vectors, which is guaranteed in theory, is lost as soon as one of the eigenvectors starts to converge [123].”
Recent progress and applications in group FFTs (2002, methodology): “This is an algorithm that may be used to efficiently compute the eigenspace projections of a real symmetric matrix when, as in all of our examples, it has relatively few eigenspaces and when it may be applied efficiently to arbitrary vectors, either directly or through a given subroutine (see, e.g.,”
Testing that distributions are close (2000, methodology): “However, for a sparse n × n symmetric matrix with m nonzero entries, n ≤ m , “Lanczos algorithms” [20] accomplish the same task in Θ( n [ m +log n ]) ﬂops, consuming Θ( n + m ) storage.”