Related papers: Fast Symbolic Integer-Linear Spectra
This paper proposes hybrid high-order eigensolvers for the computation of guaranteed lower eigenvalue bounds. These bounds display higher order convergence rates and are accessible to adaptive mesh-refining algorithms. The involved…
Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a…
We present an iteration for the computation of simple eigenvalues using a pseudospectrum approach. The most appealing characteristic of the proposed iteration is that it reduces the computation of a single eigenvalue to a small number of…
Recently a novel family of eigensolvers, called spectral indicator methods (SIMs), was proposed. Given a region on the complex plane, SIMs first compute an indicator by the spectral projection. The indicator is used to test if the region…
We describe algorithms for computing eigenpairs (eigenvalue--eigenvector) of a complex $n\times n$ matrix $A$. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
Exact analytic expression is derived for the matrix elements of the Coulomb interaction in two dimensions in the form of a closed finite sum expression. The orthonormal complete set of eigenfunctions of the harmonic oscillator is used as…
This paper presents a fast, randomized divide-and-conquer algorithm for the definite generalized eigenvalue problem, which corresponds to pencils $(A,B)$ in which $A$ and $B$ are Hermitian and the Crawford number $\gamma(A,B) =…
In this paper, we study the hardness of solving graph-structured linear systems with coefficients over a finite field $\mathbb{Z}_p$ and over a polynomial ring $\mathbb{F}[x_1,\ldots,x_t]$. We reduce solving general linear systems in…
This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the…
Many real-world problems can be naturally described by mathematical formulas. The task of finding formulas from a set of observed inputs and outputs is called symbolic regression. Recently, neural networks have been applied to symbolic…
Counting the number of perfect matchings in bipartite graphs, or equivalently computing the permanent of 0-1 matrices, is an important combinatorial problem that has been extensively studied by theoreticians and practitioners alike. The…
We give formulae for first and second derivatives of generalized eigenvalues/eigenvectors of symmetric matrices and generalized singular values/singular vectors of rectangular matrices when the matrices are linear or nonlinear functions of…
Effective matrix methods for solving standard linear algebra problems in a commutative domains are discussed. Two of them are new. There are a methods for computing adjoined matrices and solving system of linear equations in a commutative…
Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to…
We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…
The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none of which…
This paper discusses a new method to solve definite integrals using artificial neural networks. The objective is to build a neural network that would be a novel alternative to pre-established numerical methods and with the help of a…
Estimating the diagonal entries of a matrix, that is not directly accessible but only available as a linear operator in the form of a computer routine, is a common necessity in many computational applications, especially in image…
We investigate the question which Q-valued characters and characters of Q-representations of finite groups are Z-linear combinations of permutation characters. This question is known to reduce to that for quasi-elementary groups, and we…