In this paper, we address a problem left open by Wilkinson concerning the computation of the shortest (least squares) distance of any matrix A to a matrix  with at least one repeated eigenvalue. We have found that by casting the problem in a more geometric light, we can find an answer. This paper presents two methods which can identify  for a given A based on a geometric interpretation of the level sets of the eig function (considered as a multivalued differentiable function on the set of diagonalizable matrices). One method will be based, ultimately, on the examination of the pseudospectra of A. Additionally, relations will be given between the conditioning of the problem of  and the conditioning of the algorithms for solving that problem.

We consider the existence of compactly supported distribution solutions ϕ:= (ϕ1,...,ϕr)T to matrix refinement equations of the form where a is a finitely supported sequence of r x r matrices called the refinement mask. Such multiple refinable distributions occur naturally in the study of multiple wavelets. A necessary condition for the existence of ϕ with https://www.w3.org/1998/Math/MathML" display="inline"> ϕ ^ ( 0 ) ≠ 0 https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math1127.tif"/> is that the matrix Σa(α)/2 has an eigenvalue 1. However, this condition is not sufficient, which is different from the scalar case. In this paper we provide a necessary and sufficient condition for the existence of a solution ϕ subject to https://www.w3.org/1998/Math/MathML" display="inline"> ϕ ^ ( 0 ) ≠ 0. https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003419839/4baaeab4-d6e3-45ed-9dec-1deb90c341dd/content/inline-math1128.tif"/> Such a solution can always be obtained through some infinite matrix product. Our condition is totally based on the refinement mask. A general factorization of the refinement mask is presented in the univariate case, which gives a way of constructing multiple refinable distributions.

Algorithms based on simple iteration combined with a checking step are suggested for two problems in control theory, computing the distance to instability of a stable matrix, and computing the H∞ norm of a transfer matrix. The checking step corresponds to one used in the bisection method, which is currently a popular approach to solving these problems. The new approach has potential efficiency gains.

We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method. Numerical examples are provided for the equation -Delta _{mathcal {M}} u + u = f on the 2- and 3-spheres, where Delta _{mathcal {M}} is the Laplace-Beltrami operator.

We show that the probability of the exceptional set decays exponentially for a broad class of randomized algorithms approximating solutions of ODEs, admitting a certain error decomposition. This class includes randomized explicit and implicit Euler schemes, and the randomized two-stage Runge-Kutta scheme (under inexact information). We design a confidence interval for the exact solution of an IVP and perform numerical experiments to illustrate the theoretical results.

In a recent work (Koc et al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the authors showed that including difference quotients (DQs) is necessary in order to prove optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order models of the heat equation. In this work, we introduce a new approach to including DQs in the POD procedure. Instead of computing the POD modes using all of the snapshot data and DQs, we only use the first snapshot along with all of the DQs and special POD weights. We show that this approach retains all of the numerical analysis benefits of the standard POD DQ approach, while using a POD data set that has approximately half the number of snapshots as the standard POD DQ approach, i.e., the new approach requires less computational effort. We illustrate our theoretical results with numerical experiments.