Krylov Subspace

This is the fourth post in our series on Krylov subspaces. The previous ones (i.e Arnoldi Iterations and Lanczos Algorithm were mostly focused on eigenvalue and eigenvector computations. In this post we will have a look at solving strategies for linear systems of equations. In particular, we will derive the famous conjugate gradients algorithm. The conjugate gradients method is one of the most popular solvers for linear systems of equations. There exist countless variations and extensions.

This is the third post in my series on Krylov subspaces. The first post is here and the second one is here. The Lanczos Algorithm In this post we cover the Lanczos algorithm that gives you eigenvalues and eigenvectors of symmetric matrices. The Lanczos algorithm is named after Cornelius Lanczos, who was quite influential with his research. He also proposed an interpolation method and a method to approximate the gamma function.

This is the second post in my series on Krylov subspaces. The first post is here and the third one is here. Arnoldi Iterations Arnoldi iterations is an algorithm to find eigenvalues and eigenvectors of general matrices. The algorithm constructs orthonormal bases of suitable Krylov subspaces and extracts the eigendata from the basis representation. The method is named after the American engineer Walter Edwin Arnoldi and was published in 1951 [1].

This is the first post in a (planned) series on Krylov subspaces, projection processes, and related algorithms. We already discussed projection processes when talking about the Bregman algorithm and we will see that the Krylov (sub-)spaces will be generated by a set of vectors that are not necessarily orthogonal. Getting an orthogonal basis for these spaces can for example be done by applying Gram-Schmidt. Forthcoming posts will discuss further interesting applications such as the Arnoldi iteration or the Lanczos method.