Related papers: Characterizing and approximating eigenvalue sets o…
An a posteriori verification method is proposed for the generalized real-symmetric eigenvalue problem and is applied to densely clustered eigenvalue problems in large-scale electronic state calculations. The proposed method is realized by a…
The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we first give the error analysis for a single step or sweep of Jacobi's method in floating point arithmetic. Then we propose a mixed precision…
A novel method for approximating structured singular values (also known as mu-values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in…
Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially…
We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is…
The current work applies some recent combinatorial tools due to Jain to control the eigenvalue gaps of a matrix $M_n = M + N_n$ where $M$ is deterministic, symmetric with large operator norm and $N_n$ is a random symmetric matrix with…
An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive…
The computation of eigenvalues of large-scale matrices arising from finite element discretizations has gained significant interest in the last decade. Here we present a new algorithm based on slicing the spectrum that takes advantage of the…
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…
The Sinc quadrature and the Sinc indefinite integration are approximation formulas for definite integration and indefinite integration, respectively, which can be applied on any interval by using an appropriate variable transformation.…
We consider path-connected sets of matrices and the induced paths between eigenvalues. We discuss the equivalence relation generated by these paths, and how it relates to the presence of higher multiplicity eigenvalues realized by the set.…
Balancing a matrix is a preprocessing step while solving the nonsymmetric eigenvalue problem. Balancing a matrix reduces the norm of the matrix and hopefully this will improve the accuracy of the computation. Experiments have shown that…
We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schr\"odinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either…
Understanding how the optimal value of an optimisation problem changes when its input data is modified is an old question in mathematical optimisation. This paper investigates the computation of the optimal values of a family of (possibly…
We present a unified description of extremal metrics for the Laplace and Steklov eigenvalues on manifolds of arbitrary dimension using the notion of $n$-harmonic maps. Our approach extends the well-known results linking extremal metrics for…
We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions.…
We describe a subtle error which can appear in numerical calculations involving the spacing statistics of eigenvalues of random unitary matrices.
We prove conditions for equality between the extreme eigenvalues of a matrix and its quotient. In particular, we give a lower bound on the largest singular value of a matrix and generalize a result of Finck and Grohmann about the largest…
This paper is concerned with computations of a few smaller eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that smaller eigenvalues can be accurately computed for a diagonally dominant matrix or a…
Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a…