7,338 research outputs found
On the Number of Places of Convergence for Newton's Method over Number Fields
Let f be a polynomial of degree at least 2 with coefficients in a number
field K, let x_0 be a sufficiently general element of K, and let alpha be a
root of f. We give precise conditions under which Newton iteration, started at
the point x_0, converges v-adically to the root alpha for infinitely many
places v of K. As a corollary we show that if f is irreducible over K of degree
at least 3, then Newton iteration converges v-adically to any given root of f
for infinitely many places v. We also conjecture that the set of places for
which Newton iteration diverges has full density and give some heuristic and
numerical evidence.Comment: 9 pages; minor changes from the previous version; to appear in
Journal de Th\'eorie des Nombres de Bordeau
Computational aspects of helicopter trim analysis and damping levels from Floquet theory
Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated
The use of approximate factorization in stiff ODE solvers
AbstractWe consider implicit integration methods for the numerical solution of stiff initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonstiff or mildly stiff with respect to the stepsize. In these nonstiff subintervals, we do not need the (expensive) Newton iteration process. This motivated us to look for an iteration process that converges in mildly stiff situations and is less costly than Newton iteration. The process we have in mind uses modified Newton iteration as the outer iteration process and a linear solver for solving the linear Newton systems as an inner iteration process. This linear solver is based on an approximate factorization of the Newton system matrix by splitting this matrix into its lower and upper triangular part. The purpose of this paper is to combine fixed point iteration, approximate factorization iteration and Newton iteration into one iteration process for use in initial-value problems where the degree of stiffness is changing during the integration
A Framework for Generalising the Newton Method and Other Iterative Methods from Euclidean Space to Manifolds
The Newton iteration is a popular method for minimising a cost function on
Euclidean space. Various generalisations to cost functions defined on manifolds
appear in the literature. In each case, the convergence rate of the generalised
Newton iteration needed establishing from first principles. The present paper
presents a framework for generalising iterative methods from Euclidean space to
manifolds that ensures local convergence rates are preserved. It applies to any
(memoryless) iterative method computing a coordinate independent property of a
function (such as a zero or a local minimum). All possible Newton methods on
manifolds are believed to come under this framework. Changes of coordinates,
and not any Riemannian structure, are shown to play a natural role in lifting
the Newton method to a manifold. The framework also gives new insight into the
design of Newton methods in general.Comment: 36 page
A new modified Newton iteration for computing nonnegative Z-eigenpairs of nonnegative tensors
We propose a new modification of Newton iteration for finding some
nonnegative Z-eigenpairs of a nonnegative tensor. The method has local
quadratic convergence to a nonnegative eigenpair of a nonnegative tensor, under
the usual assumption guaranteeing the local quadratic convergence of the
original Newton iteration
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