Related papers: How well-conditioned can the eigenvalue problem be…
It is well known the concept of the condition number $\kappa(A) = \|A\|\|A^{-1}\|$, where $A$ is a $n \times n$ real or complex matrix and the norm used is the spectral norm. Although it is very common to think in $\kappa(A)$ as "the"…
The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each…
A unitarily invariant projective framework is introduced to analyze the complexity of path-following methods for the eigenvalue problem. A condition number, and its relation to the distance to ill-posedness, is given. A Newton map…
In this paper, we study estimates for eigenvalues of the clamped plate problem. A sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in [10] is improved.
Analyses of accelerated (momentum-based) gradient descent usually assume bounded condition number to obtain exponential convergence rates. However, in many real problems, e.g., kernel methods or deep neural networks, the condition number,…
This paper investigates oscillation-free stability conditions of numerical methods for linear parabolic partial differential equations with some example extrapolations to nonlinear equations. Not clearly understood, numerical oscillations…
The problem of classifying all unitary R-matrices of arbitrary finite dimension that have precisely two distinct eigenvalues is described, working up to a natural equivalence relation given by the characters of their braid group…
The study of solving the inverse eigenvalue problem for nonnegative matrices has been around for decades. It is clear that an inverse eigenvalue problem is trivial if the desirable matrix is not restricted to a certain structure. Provided…
In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schr\"{o}dinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by…
We derive a hierarchy of continuous-variable multipartite entanglement conditions in terms of second-order moments of position and momentum operators that generalizes existing criteria. Each condition corresponds to a convex optimization…
The general expression with the physical significance and positive definite condition of the eigenvalues of $4\times 4$ Hermitian and trace-one matrix are obtained. This implies that the eigenvalue problem of the $4\times 4$ density matrix…
We study a broad class of numerical problems that can be defined as the solution of a system of (nonlinear) equations for a subset of the dependent variables. Given a system of the form $F(x,y,z) = c$ with multivariate input $x$ and…
Optimized, necessary and sufficient conditions for the identification of the Schmidt number will be derived in terms of general Hermitian operators. These conditions apply to arbitrary mixed quantum states. The optimization procedure…
Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of…
This paper concerns singular value decomposition (SVD)-based computable formulas and bounds for the condition number of the Total Least Squares (TLS) problem. For the TLS problem with the coefficient matrix $A$ and the right-hand side $b$,…
The problem of extracting a well conditioned submatrix from any rectangular matrix (with normalized columns) has been studied for some time in functional and harmonic analysis; see…
Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of…
This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on $\ell_2$, the space of square summable…
I revisit the condition number of computing left and right singular subspaces from [J.-G. Sun, Perturbation analysis of singular subspaces and deflating subspaces, Numer. Math. 73(2), pp. 235--263, 1996]. For real and complex matrices, I…
We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval. Such estimation is useful in a number of applications.In particular, it can be used to develop an efficient…