and has been shown to perform well empirically (Rivlin, 1990). Chebyshev nodes are
also known to possess a further convenient property, i.e. equi-oscillation 3(Judd, 1998).
As important as the choice of the nodes interpolants is that of a family of functions from
which the approximant P will be drawn. We suggest using a Chebyshev polynomial. This
is defined as4:
Γi(s) = cos(ia cos(s)) i = 0,1,..., n
with Γ0(s)=1,Γ1(s)=s,and Γn+1(s) =2sΓn - Γn-1 (s)
Therefore:
n
V(s)=∑ciΓi(s)
i=0
(6)
where c0 =
1
n+1
n
∑V(sk) and
i=0
ci
2n
—∑Vv(sk)cos(iacos(sk)), i = 1,.∙∙,n
n + 1 i=0
A Chebyshev basis polynomial, in conjunction with Chebyshev interpolation nodes,
produces an efficient interpolation equation which is very accurate and stable over n .
However, in our case, to solve the problem in (3), the polynomial we choose should be
able to replicate, not just the function V at s1,s2,...,sn , but also its derivatives
s1',s2' ,...,sn' . Therefore the approximant that solves our problem can be defined as
follows5:
n
∑ciΓi(s) =V (si),∀i = 1,..., n1
i=1
n
∑ciΓ'i(s')=V '(si),∀i=1,...,n2
i =1
3 This property states that the maximum error of a cubic function, for example, shall be reached at least five
times, and the sign of this error should alternate between the interpolation points.
4 Note that in this application we use the general formula for the Chebyshev basis, however there exists also
a recursive formula.
5 Note that, although one can also use Hermite polynomials to approximate the functional and the slopes,
the latter are inefficient (Judd, 1998).