The computation of eigenvalues and eigenvectors of very large sparse matrices

By C. Paige. 1971

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

Frank-Wolfe Splitting via Augmented Lagrangian Method (2018, methodology): “(34) can be computed efficiently by the Lanczos algorithm (Paige, 1971; Kuczyński and Woźniakowski, 1992) whereas the forward backward splitting which is the standard splitting method to solve (33) needs to compute projections over the trace norm ball via a complete diagonalization which is O(d(3)).”
Numerical methods for large eigenvalue problems (2002, unknown): “In the Hermitian case, Paige (1971) showed that the loss of orthogonality occurs precisely when an eigenvalue of Hj is close to an eigenvalue of A.”
On error estimation in the conjugate gradient method and why it works in finite precision computations. (2002, methodology): “Estimation of the A-norm of the error in CG computations 67 see [33], [34], [35], [36], the review paper [44, Section 3.1] and the works quoted there (in particular [38], [39], [32] and [45]).”
The Quadratic Eigenvalue Problem (2001, background): “The new vector is generally a linear combination of the eigenvectors of interest and can be defined either explicitly [112], [125] or implicitly [94].”
Symplectic Methods for the Symplectic Eigenproblem (2002, background): “10) are sufficient to guarantee It is known [108] that bi-orthogonality will in fact be lost when at least one of the eigenvalues of converges to an eigenvalue of A.”
On the Insufficiency of Existing Momentum Schemes for Stochastic Optimization (2018, background): “…given the instability of fast gradient methods in dealing with noise, as evidenced by several negative results which consider statis- tical (Proakis, 1974; Polyak, 1987; Roy and Shynk, 1990), numerical (Paige, 1971; Greenbaum, 1989) and adversarial errors (d’Aspremont, 2008; Devolder et al., 2014).”
Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start (1992, methodology): “The analysis of convergence of the Lanczos algorithm is more complex and some of it may be found in e.g., Wilkinson 65], Kaniel 66], Paige 71,72], Kahan and Parlett 76], Scott 78], Parlett 80] and Saad 80].”
A Krylov–Schur approach to the truncated SVD (2012, background): “It is well known that a loss of orthogonality can occur when the algorithm progresses, see [24, 22].”
Sourcebook of parallel computing (2003, methodology): “Explicit restarting schemes have a history going back to the original Lanczos method [237, 387, 536, 733].”
Accelerating Stochastic Gradient Descent (2017, methodology): “In essence, this paper presents the first known provable analysis of the claim that fast gradient methods are stable when dealing with statistical errors, in contrast to negative results in statistical and non-statistical settings (Paige, 1971; Proakis, 1974; Polyak, 1987; Greenbaum, 1989; Roy and S”