On Some Applications to the Newton--Raphson Method

Treanungkur Mal

Home/Archive/arXiv

arXiv · 2211.07433

On Some Applications to the Newton--Raphson Method

Treanungkur Mal

View original source ↗Read source document ↗Export metadata ↓

Abstract

Newton's method is classically viewed as an iterative procedure for approximating a zero of a nonlinear function. In this paper, we consider the Newton iterates from a quadrature perspective: the iterates generate a partition of the interval between the initial point and the root, which in turn induces a composite trapezoidal rule. We establish an exact decomposition of the integral associated with this Newton-generated quadrature and characterize the resulting quadrature error. We also obtain an explicit bound for this error in terms of the initial Newton decrement and give a condition under which it is smaller than the error of the trapezoidal rule on the original interval. These results provide a connection between Newton's method and the quadrature rule induced by its iterates.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Treanungkur Mal. 2026-09-16. On Some Applications to the Newton--Raphson Method. https://arxiv.org/abs/2211.07433

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quelques remarques sur les vari{é}t{é}s, fonctions de Green et formule de Stokes

We give some remarks on some manifolds K3 surfaces, Complex projective spaces, real projective space and Torus and the classification of two dimensional Riemannian surfaces, Green functions and the Stokes formula. We also, talk about traces of Sobolev spaces, the distance function, the notion of degree and a duality theorem, the variational formulation and conformal map in dimension 2, the metric on the boundary of a Lipschitz domain and polar geodesic coordinates and the Gauss-Bonnet formula and the positive mass theorem in dimension $ \geq 3 $ and in the flat and non flat case. And the Ricci flow. And fields and their relation to the equations.And obstructions in astronomy. And on strings, superstrings and D-branes. And topological solutions in the negative case, critical, supercritical and superstrings and symmetry. And geometrization. And Decision problem, SAT problem and p=np problem.

math.GM↗

arXiv2026

Counting Truchet Tile Balls

A formula is established that counts the number of different balls that can be made by decorating the pentagons and hexagons of a classic football with Truchet-like patterns.

math.GM↗

arXiv2026

A Theory of Scales and Orbit Covers

This paper develops a formal theory of musical scales and their harmonic coverings and introduces orbit covers: coverings obtained by translating a fixed subset across a scale via a group action. Orbit covers generalize familiar constructions, such as the covering of the diatonic scale by tertian triads, and are motivated by the search for a generalized harmonic framework extending common-practice tonality. We model modes as group structures associated with pitch-class sets and scales as torsors, introducing scale covers and, in particular, orbit covers. To each orbit cover we associate a nerve complex encoding its intersection structure and associated topological invariants. We classify triadic orbit covers of heptatonic scales up to affine symmetry and nerve isomorphism. These results support a broader theory of harmonic organization with analytical and compositional applications.

math.GM↗