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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

nodes := [-1,-1/3,1/3,1];

nodes−1&comma;13&comma;13&comma;1

(1)

p := 3*NewtonBasis(0,nodes,x) + 5*NewtonBasis(2,nodes,x) +
7*NewtonBasis(3,nodes,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.

P := convert( p, MatrixPolynomialObject, 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.

seq( P:-Value( nodes[k] )[1,1], k=1..nops(nodes) );

3,3,679,2599

(5)

P:-Value(0.3);

6.924555556

(6)

factor( P:-Value(t)[1,1] );

7t3+12t2+539t+359

(7)

See Also

BernsteinBasis

convert/MatrixPolynomialObject

LagrangeBasis

LinearAlgebra[CompanionMatrix]

OrthogonalSeries

PochhammerBasis

type/MatrixPolynomialObject