Related papers: Factored solution of nonlinear equation systems
The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products…
Numerically obtaining the inverse of a function is a common task for many scientific problems, often solved using a Newton iteration method. Here we describe an alternative scheme, based on switching variables followed by spline…
In this work we present an adaptive Newton-type method to solve nonlinear constrained optimization problems in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive…
The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of…
In this paper, a novel multigrid method based on Newton iteration is proposed to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue $\lambda$ and eigenfunction $u$ separately, we treat the eigenpair $(\lambda, u)$ as…
We describe an algorithm for the factorization of non-commutative polynomials over a field. The first sketch of this algorithm appeared in an unpublished manuscript (literally hand written notes) by James H. Davenport more than 20 years…
Solving semiparametric models can be computationally challenging because the dimension of parameter space may grow large with increasing sample size. Classical Newton's method becomes quite slow and unstable with intensive calculation of…
We present a new algorithm to decompose generic spinor polynomials into linear factors. Spinor polynomials are certain polynomials with coefficients in the geometric algebra of dimension three that parametrize rational conformal motions.…
New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…
A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely…
Nonnegative Matrix Factorization (NMF) is the problem of approximating a nonnegative matrix with the product of two low-rank nonnegative matrices and has been shown to be particularly useful in many applications, e.g., in text mining, image…
Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination,…
For the efficient compensation of fiber nonlinearity, one of the guiding principles appears to be: fewer steps are better and more efficient. We challenge this assumption and show that carefully designed multi-step approaches can lead to…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential…
This paper presents a methodology for constructing iterative schemes of any order of convergence for solving nonlinear systems of equations. It also provides formulas for the order of convergence of any iterative schemes constructed using…
We start with elementary algebraic theory of factorization of linear ordinary differential equations developed in the period 1880-1930. After exposing these classical results we sketch more sophisticated algorithmic approaches developed in…
In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with…
In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of…
Normals with unknown parameters (NUP) can be used to convert nontrivial model-based estimation problems into iterations of linear least-squares or Gaussian estimation problems. In this paper, we extend this approach by augmenting factor…