Gram‐Schmidt orthogonalization: 100 years and more
SUMMARY In 1907, Erhard Schmidt published a paper in which he introduced an
orthogonalization algorithm that has since become known as the classical Gram‐Schmidt …
orthogonalization algorithm that has since become known as the classical Gram‐Schmidt …
Mixed precision algorithms in numerical linear algebra
Today's floating-point arithmetic landscape is broader than ever. While scientific computing
has traditionally used single precision and double precision floating-point arithmetics, half …
has traditionally used single precision and double precision floating-point arithmetics, half …
Communication lower bounds and optimal algorithms for numerical linear algebra
The traditional metric for the efficiency of a numerical algorithm has been the number of
arithmetic operations it performs. Technological trends have long been reducing the time to …
arithmetic operations it performs. Technological trends have long been reducing the time to …
Communication-optimal parallel and sequential QR and LU factorizations
We present parallel and sequential dense QR factorization algorithms that are both optimal
(up to polylogarithmic factors) in the amount of communication they perform and just as …
(up to polylogarithmic factors) in the amount of communication they perform and just as …
[BOOK][B] Communication-avoiding Krylov subspace methods
M Hoemmen - 2010 - search.proquest.com
Krylov subspace methods (KSMs) are iterative algorithms for solving large, sparse linear
systems and eigenvalue problems. Current KSMs rely on sparse matrix-vector multiply …
systems and eigenvalue problems. Current KSMs rely on sparse matrix-vector multiply …
Less is more: Reweighting important spectral graph features for recommendation
As much as Graph Convolutional Networks (GCNs) have shown tremendous success in
recommender systems and collaborative filtering (CF), the mechanism of how they …
recommender systems and collaborative filtering (CF), the mechanism of how they …
Krylov methods for nonsymmetric linear systems
G Meurant, JD Tebbens - Cham: Springer, 2020 - Springer
Solving systems of algebraic linear equations is among the most frequent problems in
scientific computing. It appears in many areas like physics, engineering, chemistry, biology …
scientific computing. It appears in many areas like physics, engineering, chemistry, biology …
Amesos2 and Belos: Direct and iterative solvers for large sparse linear systems
E Bavier, M Hoemmen, S Rajamanickam… - Scientific …, 2012 - Wiley Online Library
Solvers for large sparse linear systems come in two categories: direct and iterative.
Amesos2, a package in the Trilinos software project, provides direct methods, and Belos …
Amesos2, a package in the Trilinos software project, provides direct methods, and Belos …
Reorthogonalized block classical Gram–Schmidt using two Cholesky-based TSQR algorithms
JL Barlow - SIAM Journal on Matrix Analysis and Applications, 2024 - SIAM
In [Numer. Math., 23 (2013), pp. 395–423], Barlow and Smoktunowicz propose the
reorthogonalized block classical Gram–Schmidt algorithm BCGS2. New conditions for the …
reorthogonalized block classical Gram–Schmidt algorithm BCGS2. New conditions for the …
Shifted Cholesky QR for computing the QR factorization of ill-conditioned matrices
The Cholesky QR algorithm is an efficient communication-minimizing algorithm for
computing the QR factorization of a tall-skinny matrix X∈R^m*n, where m≫n. Unfortunately …
computing the QR factorization of a tall-skinny matrix X∈R^m*n, where m≫n. Unfortunately …