Live briefing
Everything about Lanczos algorithm
Numerical eigenvalue calculation
The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the "most useful" eigenvalues and eigenvectors of an Hermitian matrix, where is often but not necessarily much smaller than. Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Lanczos algorithmA 5-minute orientation
- Skim a recent paper: On Krylov complexity in open systems: an approach via bi-Lanczos algorithmInfluential work since 2022
- Pick up a classic book: Strongly Correlated SystemsThe most-reprinted book on the subject
- Read the landmark paper: Generalized Lanczos algorithm for variational quantum Monte CarloCited 339 times
- Go sideways: Arrowhead matrixA closely related topic
- Go sideways: Power iterationA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🏆 Landmark papers (most cited)
Generalized Lanczos algorithm for variational quantum Monte Carlo
The Lanczos algorithm with selective orthogonalization
The Lanczos algorithm with partial reorthogonalization
A rational Lanczos algorithm for model reduction
On estimating the largest eigenvalue with the Lanczos algorithm
The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem
The Lanczos-Arnoldi algorithm and controllability
A Look-Ahead Lanczos Algorithm for Unsymmetric Matrices
🆕 Influential since 2022
On Krylov complexity in open systems: an approach via bi-Lanczos algorithm
Lanczos Algorithm, the Transfer Matrix, and the Signal-to-Noise Problem
Block Lanczos algorithm for lattice QCD spectroscopy and matrix elements
Lanczos algorithm for lattice QCD matrix elements
Smooth particle mesh Ewald-integrated stochastic Lanczos many-body dispersion algorithm
📚 Books worth reading
Strongly Correlated SystemsAdolfo Avella · 2011
Electron transfer- from isolated molecules to biomoleculesJoshua Jortner, Stuart Alan Rice · 1999
Lanczos algorithms for large symmetric eigenvalue computationsJane K. Cullum · 1985- 📕An implementation of the look-ahead Lanczos algorithm for non-Hermitian matricesRoland W. Freund · 1990
- 📕A look-ahead variant of the Lanczos algorithm and its application to the quasi-minimal residual method for non-Hermitian linear systemsNoël Maurice Nachtigal · 1991
The Lanczos and Conjugate Gradient AlgorithmsGérard Meurant · 2006
The Lanczos methodLouis Komzsik · 1987
Total least squares and errors-in-variables modelingSabine van Huffel · 2002
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Arrowhead matrix In the mathematical field of linear algebra, an arrowhead matrix is a square matrix containing zeros in all entries except for the first row, first column…
- Power iteration In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A {\displaystyle A} , the algorithm…
- Nonlinear eigenproblem In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations…
- Rayleigh quotient In mathematics, the Rayleigh quotient (/ˈreɪ.li/) for a given complex Hermitian matrix M {\displaystyle M} and nonzero vector x {\displaystyle x} is defined…
- Divide-and-conquer eigenvalue algorithm Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s)…
- Expander mixing lemma The expander mixing lemma intuitively states that the edges of certain d {\displaystyle d} -regular graphs are evenly distributed throughout the graph…
- Perron–Frobenius theorem In matrix theory, the Perron–Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by Georg Frobenius (1912), asserts that a…
- Segmentation-based object categorization The image segmentation problem is concerned with partitioning an image into multiple regions according to some homogeneity criterion. This article is primarily…