LagrangeBasis

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LagrangeBasis - Lagrange polynomials on a set of nodes

	Calling Sequence
	LagrangeBasis(k, nodes, x)

Parameters

k	-	algebraic expression; the index
nodes	-	list of algebraic expressions; the nodes where the polynomial is known
x	-	algebraic expression; the argument

Description

•	LagrangeBasis(k,nodes,x) = w[k]*prod(x-nodes[j], j<>k) defines the th Lagrange polynomial of degree which is either or on the given nodes. By convention, the nodes are indexed from , so , and the barycentric weights are defined as .

•	At present, this can only be evaluated in Maple by prior use of the object-oriented representation obtained by and subsequent call to , which uses the numerically stable barycentric form to evaluate the polynomial .

Examples

(1)

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

(2)

That polynomial has the value 3 at , the value 0 at , the value 5 at , and the value 7 at .

P := Record(Value = Default[value], Variable = x, Degree = 3, Coefficient = coe, Dimension = [1, 1], Basis = LagrangeBasis, 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, LinearAlgebra[CompanionMatrix], NewtonBasis, OrthogonalSeries, PochhammerBasis, type,MatrixPolynomialObject