LagrangeBasis
Lagrange polynomials on a set of nodes
Calling Sequence
Parameters
Description
Examples
LagrangeBasis(k, nodes, x)
k
-
algebraic expression; the index
nodes
list of algebraic expressions; the nodes where the polynomial is known
x
algebraic expression; the argument
LagrangeBasisk,nodes,x=wk∏j≠kx−nodesj defines the kth Lagrange polynomial of degree n which is either 1 or 0 on the given nodes. By convention, the nodes are indexed from 0, so nodes=x0,x1,...,xn , and the barycentric weights wk are defined as wk=∏j≠k1xk−xj.
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 the numerically stable barycentric form to evaluate the polynomial p.
nodes≔−1,−13,13,1
p≔3LagrangeBasis0,nodes,x+5LagrangeBasis2,nodes,x+7LagrangeBasis3,nodes,x
p≔3LagrangeBasis0,−1,−13,13,1,x+5LagrangeBasis2,−1,−13,13,1,x+7LagrangeBasis3,−1,−13,13,1,x
That polynomial has the value 3 at x=−1, the value 0 at x=−13, the value 5 at x=13, and the value 7 at x=1.
P≔convertp,MatrixPolynomialObject,x
P≔RecordValue=Defaultvalue,Variable=x,Degree=3,Coefficient=coe,Dimension=1,1,Basis=LagrangeBasis,BasisParameters=−1,−13,13,1,IsMonic=mon,OutputOptions=shape=,storage=rectangular,order=Fortran_order,fill=0,attributes=
P:-Degree
3
Note that the result returned by convert...,MatrixPolynomialObject represents a matrix polynomial; hence these results are 1 by 1 matrices.
seqP:-Valuenodesk1,1,k=1..nopsnodes
3,0,5,7
P:-Value0.3
4.729812500
factorP:-Valuet1,1
−33t2−26t−353t+116
See Also
BernsteinBasis
convert/MatrixPolynomialObject
LinearAlgebra[CompanionMatrix]
NewtonBasis
OrthogonalSeries
PochhammerBasis
type/MatrixPolynomialObject
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