Numerical Methods for Large Eigenvalue Problems

By Y. Saad. 2011

1,374 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/66b4e2570fcf87566b49700f5c4ec4ff302bd5b1

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In Computational Theory and Mathematics

Citations in the corpus

Computing Personalized PageRank Quickly by Exploiting Graph Structures (2014, methodology): “We compute 200 “extreme” eigenvalues of the whole networks and of only their cores for the cases d = 10, 100, and 1000, respectively, by using the Arnoldi method [40].”
Model Reduction Using the Balanced Truncation Method and the Padé Approximation Method (2020, methodology): “For computing the Cholesky factor of a matrix, it should be a symmetric positive definite matrix [45].”
Trace optimization and eigenproblems in dimension reduction methods (2011, background): “While this result is seldom explicitly stated on its own in standard textbooks, it is an immediate consequence of the Courant-Fisher characterization, see, e.g., [25, 27 ].”
Model Order Reduction by Using the Balanced Truncation and Factor Division Methods (2018, background): “A symmetric positive definite matrix can be converted into a lower triangular matrix by using the Cholesky factorization [43].”
Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces (2013, positive): “Common to both of these ﬁelds are methods for matrix factor-izations and eigenvalue computations [35], which are replete with subspace oriented approaches.”
Block tensor conjugate gradient-type method for Rayleigh quotient minimization in two-dimensional case (2010, background): “defined on the square [0, 1] × [0, 1 ], the boundary conditions being the homogeneous Dirichlet condi� tions.”
Approximating Spectral Densities of Large Matrices (2013, positive): “Refined bounds which invoke the square of η ([46]) can also be used.”
Construction of eigenfunctions for scalar-type operators via Laplace averages with connections to the Koopman operator (2014, methodology): “e eigenvalues. Three other possible directions to pursue are the development of iterative methods using, for example, (i) Krylov subspace-type techniques and the methods of matrix-free linear algebra [36,26], (ii) using the conjugacy method to explicitly construct eigenfunctions and project an obser”
Knowledge Graph Completion via Complex Tensor Factorization (2017, neutral): “In practice, imposing orthonormality is essentially a numerical commodity for the decomposition of dense matrices, through iterative methods for example (Saad, 1992).”