Related papers: An algorithm for constructing doubly stochastic ma…
We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an…
We present here necessary and sufficient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse. The techniques we use are related with the…
The analysis of diagonalizable matrices in terms of their so-called isospectral reduction represents a versatile approach to the underlying eigenvalue problem. Starting from a symmetry of the isospectral reduction, we show in the present…
We first propose a concise singular value decomposition of dual matrices. Then, the randomized version of the decomposition is presented. It can significantly reduce the computational cost while maintaining the similar accuracy. We analyze…
Obtaining the inverse of a large symmetric positive definite matrix $\mathcal{A}\in\mathbb{R}^{p\times p}$ is a continual challenge across many mathematical disciplines. The computational complexity associated with direct methods can be…
In this study, an algorithm for computing the inverse of periodic k banded matrices, which are needed for solving the differential equations by using the finite differences, the solution of partial differential equations and the solution of…
We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner…
We study the eigenvalue problem for some special class of anti-triangular matrices. Though the eigenvalue problem is quite classical, as far as we know, almost nothing is known about properties of eigenvalues for anti-triangular matrices.…
In this note, we solve an inverse spectral problem for a class of finite band symmetric matrices. We provide necessary and sufficient conditions for a matrix valued function to be a spectral function of the operator corresponding to a…
Randomized iterative algorithms have attracted much attention in recent years because they can approximately solve large-scale linear systems of equations without accessing the entire coefficient matrix. In this paper, we propose two novel…
The dynamic matrix inverse problem is to maintain the inverse of a matrix undergoing element and column updates. It is the main subroutine behind the best algorithms for many dynamic problems whose complexity is not yet well-understood,…
In this paper a generalization of the Gram-Schmidt Algorithm is presented. Actually we provide an algorithm to construct a set of equiangular vectors with a given angle $\theta\in(0,\arccos(\frac{-1}{n-1}))$ using a set of input independent…
Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by…
We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient…
We investigate convex polytopes of doubly stochastic matrices having special structures: symmetric, Hankel symmetric, centrosymmetric, and both symmetric and Hankel symmetric. We determine dimensions of these polytopes and classify their…
Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to…
We introduce a new set of algorithms to compute Jacobi matrices associated with measures generated by infinite systems of iterated functions. We demonstrate their relevance in the study of theoretical problems, such as the continuity of…
An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is…
We derive an efficient stochastic algorithm for inverse problems that present an unknown linear forcing term and a set of nonlinear parameters to be recovered. It is assumed that the data is noisy and that the linear part of the problem is…