Related papers: Newton-Type Methods For Simultaneous Matrix Diagon…
It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In…
This article studies canonical forms derived from the finest simultaneous block diagonalization of a set of symmetric matrices via congruence. Our technique relies on Harrison's center theory, which is extended from a single higher degree…
Linear programming relaxations are central to {\sc map} inference in discrete Markov Random Fields. The ability to properly solve the Lagrangian dual is a critical component of such methods. In this paper, we study the benefit of using…
Considered herein is a modified Newton method for the numerical solution of nonlinear equations where the Jacobian is approximated using a complex-step derivative approximation. We show that this method converges for sufficiently small…
Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The…
This paper considers the problems of solving monotone variational inequalities with H\"older continuous Jacobians. By employing the knowledge of H\"older parameter $\nu$, we propose the $\nu$-regularized extra-Newton method within at most…
We expand the solutions of linearly coupled Mathieu equations in terms of infinite-continued matrix inversions, and use it to find the modes which diagonalize the dynamical problem. This allows obtaining explicitly the ('Floquet-Lyapunov')…
In this paper we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type $M(\lambda)v=0$, where $M:\mathbb{C}\rightarrow\mathbb{C}^{n\times n}$ is a holomorphic function. We investigate which types of approximations…
We apply matrix methods to arithmetic functions by associating matrices to the functions in a manner drawn from the theory of symmetric functions. Then we study the characteristic polynomials of the associated matrices.
We consider the problem of numerically computing a critical point of a functional $J\colon M\rightarrow R$ where $M$ is a Riemannian manifold. Due to local quadratic convergence a popular choice to solve this problem is the geometric Newton…
When a physical system is modeled by a nonlinear function, the unknown parameters can be estimated by fitting experimental observations by a least-squares approach. Newton's method and its variants are often used to solve problems of this…
We present a simple yet powerful technique for forming iterative methods of various convergence orders. Methods of various convergence orders (four, six, eight and ten) are formed through a modest modification of the classical Newton…
In order to avoid the evaluation of the Jacobian matrix and its inverse, the present author recently introduced the pseudo-Jacobian matrix with a general applicability of any nonlinear systems of equations. By using this concept, this paper…
This paper is concerned with the nonnegative inverse eigenvalue problem of finding a nonnegative matrix such that its spectrum is the prescribed self-conjugate set of complex numbers. We first reformulate the nonnegative inverse eigenvalue…
In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By…
Iterating Newton's method symbolically for the general quadratic yields a rational function, the numerator and denominator of which are polynomials with highly composite coefficients.
In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an adaptive…
In recent years, the proximal gradient method and its variants have been generalized to Riemannian manifolds for solving optimization problems with an additively separable structure, i.e., $f + h$, where $f$ is continuously differentiable,…
Newton's inequalities $c_n^2 \ge c_{n-1}c_{n+1}$ are shown to hold for the normalized coefficients $c_n$ of the characteristic polynomial of any $M$- or inverse $M$-matrix. They are derived by establishing first an auxiliary set of…
In this paper, we consider a family of Jacobi-type algorithms for simultaneous orthogonal diagonalization problem of symmetric tensors. For the Jacobi-based algorithm of [SIAM J. Matrix Anal. Appl., 2(34):651--672, 2013], we prove its…