**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Newton polynomial

Summary

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method.
Given a set of k + 1 data points
where no two xj are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials
with the Newton basis polynomials defined as
for j > 0 and .
The coefficients are defined as
where
is the notation for divided differences.
Thus the Newton polynomial can be written as
The Newton polynomial can be expressed in a simplified form when
are arranged consecutively with equal spacing.
Introducing the notation
for each
and ,
the difference can be written as .
So the Newton polynomial becomes
This is called the Newton forward divided difference formula.
If the nodes are reordered as , the Newton polynomial becomes
If are equally spaced with and for i = 0, 1, ..., k, then,
is called the Newton backward divided difference formula.
Newton's formula is of interest because it is the straightforward and natural differences-version of Taylor's polynomial. Taylor's polynomial tells where a function will go, based on its y value, and its derivatives (its rate of change, and the rate of change of its rate of change, etc.) at one particular x value. Newton's formula is Taylor's polynomial based on finite differences instead of instantaneous rates of change.
As with other difference formulas, the degree of a Newton interpolating polynomial can be increased by adding more terms and points without discarding existing ones. Newton's form has the simplicity that the new points are always added at one end: Newton's forward formula can add new points to the right, and Newton's backward formula can add new points to the left.
The accuracy of polynomial interpolation depends on how close the interpolated point is to the middle of the x values of the set of points used.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (61)

Related concepts (8)

Related courses (9)

Related MOOCs (4)

Related people (15)

Related lectures (115)

Related units (3)

Lagrange polynomial

In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs with the are called nodes and the are called values. The Lagrange polynomial has degree and assumes each value at the corresponding node, Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler.

Divided differences

In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation. Divided differences is a recursive division process. Given a sequence of data points , the method calculates the coefficients of the interpolation polynomial of these points in the Newton form.

Polynomial interpolation

In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of n + 1 data points , with no two the same, a polynomial function is said to interpolate the data if for each . There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.

MATH-212: Analyse numérique et optimisation

L'étudiant apprendra à résoudre numériquement divers problèmes mathématiques. Les propriétés théoriques de ces
méthodes seront discutées.

MATH-451: Numerical approximation of PDEs

The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimens

MATH-251(a): Numerical analysis

This course presents numerical methods for the solution of mathematical problems such as systems of linear and non-linear equations, functions approximation, integration and differentiation, and diffe

Digital Signal Processing I

Basic signal processing concepts, Fourier analysis and filters. This module can
be used as a starting point or a basic refresher in elementary DSP

Digital Signal Processing II

Adaptive signal processing, A/D and D/A. This module provides the basic
tools for adaptive filtering and a solid mathematical framework for sampling and
quantization

Digital Signal Processing III

Advanced topics: this module covers real-time audio processing (with
examples on a hardware board), image processing and communication system design.

Explores Taylor polynomials of orders 1 and 2 for linear function approximation around a point.

Explores the intuition behind transforms of the place and addresses audience questions on integral calculations and function choices.

Explores Gauss-Legendre quadrature formulas using Legendre polynomials for accurate function approximation.

Mathieu Salzmann, Alexandre Massoud Alahi, Megh Hiren Shukla

Deep heteroscedastic regression involves jointly optimizing the mean and covariance of the predicted distribution using the negative log-likelihood. However, recent works show that this may result in sub-optimal convergence due to the challenges associated ...

20242D cardiac MR cine images provide data with a high signalto-noise ratio for the segmentation and reconstruction of the heart. These images are frequently used in clinical practice and research. However, the segments have low resolution in the through-plane ...

Fabio Nobile, Francesca Bonizzoni

We study the Darcy boundary value problem with log-normal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solu ...