NewtonBasis

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NewtonBasis - Newton polynomials on a set of nodes

	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 th Newton polynomial of degree is defined by

NewtonBasis(k, nodes, x) = product(x-nodes[j], j = 0 .. k-1)

The degree of the th Newton polynomial is . By convention, the nodes are indexed from , so .

•	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 .

Examples

(1)

p := 3*NewtonBasis(0, [-1, -1/3, 1/3, 1], x)+5*NewtonBasis(2, [-1, -1/3, 1/3, 1], x)+7*NewtonBasis(3, [-1, -1/3, 1/3, 1], x)

(2)

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

P := Record(Value = Default[value], Variable = x, Degree = 3, Coefficient = coe, Dimension = [1, 1], Basis = NewtonBasis, BasisParameters = [[-1, -1/3, 1/3, 1]], IsMonic = mon, OutputOptions = [shape = [], storage = rectangular, order = Fortran_order, fill = 0, attributes = []])

(3)

(4)

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

(5)

(6)

(7)

	See Also
	BernsteinBasis, convert/MatrixPolynomialObject, LagrangeBasis, LinearAlgebra[CompanionMatrix], OrthogonalSeries, PochhammerBasis, type/MatrixPolynomialObject