ABSTRACT

Some plotted examples of such L¿ are given, for example, in [121], p. 83. Then, since Li(xk) = ¿ifc, we have

(1.2)

This representation requires 0(n2) arithmetic operations for each evaluation of Pn-i-A more economical and also numerically more stable form can

be obtained as follows. Set

(1.3)

Notice tha t these factors are independent of the point of evaluation, v, and thus need only be computed once. Also, the special case of (1.2) with Di = 1, i = 1, • • •, n, yields the relation,

Together, these may be used to rewrite (1.2) in the so-called barycentric representation,

(1.4)

of the Lagrange interpolating polynomial. This formula is well-defined for x T¿ XÍ and may be extended continuously by setting pn-\{xi) :— Vi, i — 1, • • •, n. Using (1.4) requires only a further 0(n) operations per evaluation. For numerical reasons ([171]) it is good policy to renumber the interpolation nodes X{ so tha t

(1.5) holds, where x~ = ^ J27=i Xi' Since the values (1.3) are independent of the yi, the barycentric representation is especially recommended when several polynomials with different y i but the same nodes x{ are to be evaluated.