ABSTRACT
In general, by a spline interpolant s G Cm[xi, xn] with knots xjt, we mean a set of n — 1 functions, s^, defined on [xk, #fc+i], respectively, k = 1, • • •, n — 1, tha t are stitched together so as to be m-times (m > 0) continuously different iate at the knots and tha t satisfy the interpolation conditions,
sk{xk)=Vk, sk(xk+i) = Vk+u A; = l , - - - , n - 1 . (2.2) For a polygonal pa th through the points {xk,Vk), k — l , . . . , n , we have m = 0 and the s^ are all line segments with endpoints (#¿, ?/¿), ¿ = k, k + 1. For m = 1, we will be connecting parabolic segments, and for m = 2, cubic polynomials as well as other functions. As we shall see with polynomials of degree five and m = 4, m > 2 is in general unsuitable, since, as we saw in Chapter 1, the unacceptable properties of polynomials of higher degrees again take effect. One could, in principle, choose a different function type on each interval for s^, but we avoid this for practical considerations.
