Krylov subspace methods have had unparalleled success in solving real-life problems across disciplines ranging from computational fluid dynamics to statistics, machine learning, control theory, and chemistry, among many others. This article provides a brief history of these methods, discussing their origin, expansion, the lives people behind them.

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Algebraic solvers based on preconditioned Krylov subspace methods are among the most powerful tools for large scale numerical computations in applied mathematics, sciences, technology, as well as in emerging applications in social sciences. The study of mathematical properties of Krylov subspace methods, in both the cases of exact and inexact computations, is a very active area of research and ...

In this paper we consider the numerical solution of projected generalized continuous-time Lyapunov equations with low-rank right-hand sides. The interest in this problem stems from stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. Two projection methods are proposed for calculating low-rank approximate solutions. One is based ...

Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact in the non-Hermitian case introduces restrictions; e.g., initial residuals must be collinear and this collinearity must be maintained at restart. Thus we cannot simultaneously solve shifted systems with unrelated right-hand si...

This paper starts oo with studying simple extrapolation methods for the classical iteration schemes such as Richardson, Jacobi and Gauss-Seidel iteration. The extrapolation procedures can be interpreted as approximate minimal residual methods in a Krylov subspace. It seems therefore logical to consider, conversely, classical methods as pre-processors for Krylov subspace methods, as was done by ...

In this paper we describe and analyze Krylov subspace techniques for accelerating the convergence of waveform relaxation for solving time dependent problems. A new class of accelerated waveform methods, convolution Krylov subspace methods, is presented. In particular, we give convolution variants of the conjugate gradient algorithm and two convolution variants of the GMRES algorithm and analyze...