Solution of Systems of Linear Equations by Minimized Iterations1

By C. Lanczos. 1952

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Citations in the corpus

Optimal combination forecasts for hierarchical time series (2011, methodology): “For very large hierarchies, an iterative approach can be used based on the method of Lanczos (1950, 1952).”
Efficient Perturbation Methods for Solving Regime-Switching DSGE Models (2015, unknown): “3) (c) k=k+1 The BCG method, due to Lanczos (1952), extends the classical conjugate gradient algorithm (CG) to non-Hermitian matrices.”
Iterative methods for sparse linear systems (2003, methodology): “Lanczos himself certainly used this viewpoint when he wrote his breakthrough papers [140, 142] in the early 1950s.”
Iterative Krylov Methods for Large Linear Systems (2003, methodology): “The resulting method is known as the Bi-Lanczos method [130].”
Vector extrapolation methods. applications and numerical comparison (2000, methodology): “It was shown in [34] that these methods are equivalent to the method of Arnoldi [26], the generalized minimal residual method (GMRES) [27] and the method of Lanczos [21], respectively.”
Methods of conjugate gradients for solving linear systems (1952, background): “The papers referred to in this summary are those of Stiefel [3], Hestenes [4], and Lanczos [5,6], the last of which is discussed elsewhere in this volume.”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “Solvers based on Krylov subspace methods [121, 9, 122, 184] and on Anderson relaxation [5, 211, 202] perform intermediate computations in low-dimensional subspaces that are updated as the computation proceeds.”
A two-directional Arnoldi process and its application to parametric model order reduction (2009, unknown): “…Arnoldi process play an important role in modern computational techniques for large-scale matrix computation problems, such as solving linear systems of equations [13] [15] [23] [24], computing a few selected eigenpairs [3] [25] [21] and reduced-order modeling of dynamical systems [18] [2] [16].”
The block grade of a block Krylov space (2009, methodology): “The nonsymmetric Lanczos process has also been implemented in these two flavors; see, e.g. [1,3] or Sections 4.6, 7.9, and 7.10 of [4].”
Contraction integral equation method in three‐dimensional electromagnetic modeling (2002, methodology): “[31] This method was introduced as a generalization of CG for nonhermitian systems [Lanczos, 1952; Fletcher, 1976].”