An iterative method for the solution of the eigenvalue problem of linear differential and integral

1950

Citations in the corpus

• The Full Spectrum of Deep Net Hessians At Scale: Dynamics with Sample Size (2018, methodology): “The Lanczos algorithm [8] computes the spectrum of a symmetric matrix H ∈ Rp×p by first reducing it to a tridiagonal form Tp ∈ Rp×p and then computing the spectrum of that matrix instead.”
• A Krylov-Schur Algorithm for Large Eigenproblems (2001, background): “In 1950, Lanczos [6] showed that if A is Hermitian then the vectors ui satisfy a threeterm recurrence of the form”
• The Full Spectrum of Deepnet Hessians at Scale: Dynamics with SGD Training and Sample Size. (2018, methodology): “The Lanczos algorithm [Lanczos, 1950] computes the spectrum of a symmetric matrix A ∈ Rp×p by first reducing it to a tridiagonal form Tp ∈ Rp×p and then computing the spectrum of that (a) Outlier attribution toG.”
• Sketching Curvature for Efficient Out-of-Distribution Detection for Deep Neural Networks (2021, background): “Moreover, stochastic versions of Lanczos algorithm [Lanczos, 1950] or power iteration are difficult to tune.”
• Neon2: Finding Local Minima via First-Order Oracles (2017, methodology): “Most notably, power method uses Õ(n/δ) computations of Hessian-vector products, and Lanscoz method [18] uses Õ(n/ √ δ) computations.”
• The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems Part I - System overview and formulation (2011, unknown): “Following Lanczos (1950), a reduced rank factorization of the symmetric, positive definite matrix A can be found which satisfies the recurrence relation given by: AVm ¼ VmTm þ cmqmþ1eTm ð14Þ where Vm = (. . .,qi, . . .), i = 1, . . . ,m, is a matrix composed of the orthonormal vectors qi, referred…”
• Model reduction of large-scale systems by least squares (2006, methodology): “However, the methods are numerically reliable and can be implemented iteratively; see, for example, [17, 23 ,8,12,13] for efficient implementations of the Krylov based methods.”
• Symplectic Methods for the Symplectic Eigenproblem (2002, methodology): “4 LANCZOS ALGORITHM In 1950, Lanczos [84] proposed a method for the successive reduction of a given, in general non-Hermitian, matrix A to tridiagonal form.”
• Inexact Inverse Iteration for Generalized Eigenvalue Problems (2000, methodology): “It is usually used in combination with an iterative method such as the power method (inverse iteration), subspace iteration [19], the Lanczos algorithm [ 7 ] and the Arnoldi algorithm [1] (see also [4, 13]).”