相关论文: On two families of iterative methods without memor…
We describe four algorithms for neural network training, each adapted to different scalability constraints. These algorithms are mathematically principled and invariant under a number of transformations in data and network representation,…
Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerated versions. What are…
We analyse and compare several algorithms to compute numerically periodic solutions of high-dimensional dynamical systems and investigate their Floquet stability without building the monodromy matrix. The solution and its perturbation are…
The evolution, as functions of the "ticking time" $\ell =0,1,2,...$, of the solutions of the system of $N$ quadratic recursions \begin{eqnarray*} x_{n}\left( \ell +1\right) =c_{n}+\sum_{m=1}^{N}\left[ C_{nm}x_{m}\left( \ell \right) \right]…
In the present paper, we introduce a new family of $ \theta-$methods for solving delay differential equations. New methods are developed using a combination of decomposition technique viz. new iterative method proposed by Daftardar Gejji…
First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal…
This article generalizes a recently introduced procedure to solve nonlinear systems of equations, radically departing from the conventional Newton-Raphson scheme. The original nonlinear system is first unfolded into three simpler…
A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms builded up from Ostrowski's method for solving systems of nonlinear equations…
Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this…
We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction…
The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these…
A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that…
We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger…
This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the…
Inspired by Karr's algorithm, we consider the summations involving a sequence satisfying a recurrence of order two. The structure of such summations provides an algebraic framework for solving the difference equations of form…
We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion…
Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough…
A modification of Newton's method for solving systems of $n$ nonlinear equations is presented. The new matrix-free method relies on a given decomposition of the invertible Jacobian of the residual into invertible sparse local Jacobians…
Those lectures revolve around the following problem: given a system of n real polynomials in n variables, count the number of real roots. The first lecture is a course on Newton iteration and alpha-theory. The second describes an…
Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to enforce uniqueness and select a meaningful solution is to introduce a regularizer. While for most…