Numerical Linear Algebra and Optimization 2021-08-25 Mathematics Linear Algebra Optimization Numerics Mathematician and Software Engineer Researcher, Engineer, Tinkerer, Scholar, and Philosopher. Related Conjugate Gradients Lanczos Algorithm Arnoldi Iterations Krylov Subspaces Designing Neural Networks in Mathematica Posts Conjugate Gradients 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. 2021-10-31 Mathematics Project Lanczos Algorithm 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. Last updated on 2021-08-22 Mathematics Project Arnoldi Iterations 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. Last updated on 2021-08-18 Mathematics Project Krylov Subspaces 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. Last updated on 2021-08-17 Mathematics Project Variations of the Bregman Algorithm (4/4) In the previous post in our series on the Bregman algorithm we discussed how to solve convex optimization problems. In this post we want to give reference to some variations and extensions of the Bregman algorithm. 2021-04-11 Mathematics Project Project The Bregman Algorithm (3/4) In a previous post we discussed how to solve constrained optimization problems by using the Bregman algorithm. Here we want to extend the approach unconstrained problems. Let’s start simple. Assume we want to minimize a convex and smooth function $f\colon\mathbb{R}^{n}\to\mathbb{R}$. 2021-03-21 Mathematics Project Project The Bregman Algorithm (2/4) In a previous post we discussed how to find a common point in a family of convex sets by using the Bregman algorithm. Actually the algorithm is capable of more. We can use it to solve constrained optimization problems. 2021-03-13 Mathematics Project Project The Bregman Algorithm (1/4) In the 1960s Lev Meerovich Bregman developed an optimization algorithm [1] which became rather popular beginning of 2000s. It’s not my intention to present the proofs for all the algorithmic finesse, but rather the general ideas why it is so appealing. 2021-03-06 Mathematics Project Project QR Decompositions and Rank Deficient Matrices We discuss the necessary changes to our QR decomposition algorithms to handle matrices which do not have full rank. 2020-10-04 Mathematics Project QR Comparison with other Implementations We developed a QR decomposition algorithm, based on the orthogonalisation process of Gram-Schmidt in a series of posts here, here, here, and here. Let’s have a look how good this algorithm performs against built-in implementations from julia and other programming languages. Last updated on 2020-12-20 Mathematics Project QR Decompositions with Reorthogonalisation Problem Formulation We already discussed QR decompositions and showed that using the modified formulation of Gram-Schmidt significantly improves the accuracy of the results. However, there is still an error of about $10^3 M_\varepsilon$ (where $M_\varepsilon$ is the machine epsilon) when using the modified Gram Schmidt as base algorithm for the orthogonalisation. Last updated on 2021-06-05 Mathematics Project QR Decompositions We consider the necessary changes to the Gram-Schmidt orthogonalisation to obtain a QR Decomposition Last updated on 2020-11-15 Mathematics Project Gram-Schmidt vs. Modified Gram-Schmidt We compare the accuracy of the classical Gram-Schmidt algorithm to the modified Gram-Schmidt algorithm. Last updated on 2021-08-17 Mathematics Project