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NewtonBasis

Newton polynomials on a set of nodes

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

NewtonBasis(k, nodes, x)

Parameters

k

-

algebraic expression; the index

nodes

-

list of algebraic expressions; nodes where the polynomial is known

x

-

algebraic expression; the argument

Description

• 

The kth Newton polynomial of degree k is defined by

NewtonBasisk,nodes,x=j=0k1xnodesj

  

By convention, the nodes are indexed from 0, so nodes=[x0,x1,...,xn].

• 

At present, this can only be evaluated in Maple by prior use of the object-oriented representation obtained by P:=convert(p,MatrixPolynomialObject,x) and subsequent call to P:-Value(<x-value>), which uses Horner's method to evaluate the polynomial p.

Examples

nodes1&comma;13&comma;13&comma;1

nodes−1&comma;13&comma;13&comma;1

(1)

p3NewtonBasis0&comma;nodes&comma;x&plus;5NewtonBasis2&comma;nodes&comma;x&plus;7NewtonBasis3&comma;nodes&comma;x

p3NewtonBasis0&comma;−1&comma;13&comma;13&comma;1&comma;x+5NewtonBasis2&comma;−1&comma;13&comma;13&comma;1&comma;x+7NewtonBasis3&comma;−1&comma;13&comma;13&comma;1&comma;x

(2)

The coefficients of that polynomial can be interpreted in terms of divided differences of the values of p at the nodes.

Pconvertp&comma;MatrixPolynomialObject&comma;x

PRecordValue=Defaultvalue&comma;Variable=x&comma;Degree=3&comma;Coefficient=coe&comma;Dimension=1&comma;1&comma;Basis=NewtonBasis&comma;BasisParameters=−1&comma;13&comma;13&comma;1&comma;IsMonic=mon&comma;OutputOptions=shape=&comma;storage=rectangular&comma;order=Fortran_order&comma;fill=0&comma;attributes=

(3)

P:-Degree

3

(4)

Note that the result returned by convert...,MatrixPolynomialObject represents a matrix polynomial; hence these results are 1 by 1 matrices.

seqP:-Valuenodesk1&comma;1&comma;k&equals;1..nopsnodes

3,3,679,2599

(5)

P:-Value0.3

6.924555556

(6)

factorP:-Valuet1&comma;1

7t3+12t2+539t+359

(7)

See Also

BernsteinBasis

convert/MatrixPolynomialObject

LagrangeBasis

LinearAlgebra[CompanionMatrix]

OrthogonalSeries

PochhammerBasis

type/MatrixPolynomialObject