ABSTRACT
The most simply constructed rational function, defined on [xk,xk+\], involving (just as for cubic splines) four parameters is probably ([128,161])
(6.1)
If the Sk and the subsequent rational spline interpolant that they form are to be twice continuously differentiable on [xi,£n] and in particular, continous there, then the pole of s^,
(6.2)
must lie outside of [a;fc,xfc+i], i.e., either
(6.3)
In order to see how to handle the difficulties caused by this restriction, we will first of all consider the simplest possible rational function having one
pole and two parameters (just as for polygonal paths (2.4)), i.e.,
(6-4)
Here, the Dk, because of the stipulated continuity, must also satisfy (6.3). The interpolation conditions (2.2), in this case, become
(6.5) (6.6)
If dk = 0, i.e., yk+i = yk, and yk+i ^ 0, then Dk = 0 and sk = yk. If Vk — Vk+i — 0, then Dk is arbitrary and sk = 0. If dk ^ 0 and yk = 0, then Ck = 0, i. e., sk = 0, and hence the interpolation problem has no solution. Finally, if dk ^ 0 and yk+\ = 0, then the constant Dk is undefined and there is also no solution. There remains the case of yk ^ 2/fc+i, Vk i1 0, ?/fc+i 7^ 0, for which there arises the restriction that sgn(yk) = sgn(yk+i), as sk cannot have any zeros. Altogether, we see that the interpolation problem with nonconstant denominator in (6.4) is only solvable in the cases:
(6.7)
By substituting (6.6) into (6.3), we see that no pole arises when (6.7) holds. If the segments (6.4) are, as for polygonal paths, to be joined continuously, then we must necessarily have yk > 0, k = l , - - - ,n or yk < 0, fc = l , " - , n a s well as the corresponding conditions of (6.7). The resulting spline interpolants would then be composed of hyperbolic segments and would certainly not be as useful as are the corresponding polygonal paths. An exception would be if one of the alternatives of (6.7) holds at the same time for all k = 1, • • •, n — 1, i. e., either
(6.8)
or
In that case, since
the global spline interpolant, s, is strictly decreasing or increasing and either convex or concave. This follows from the facts that the denominators of (6.9) and (6.10) have no zeros in [xk,xk+\], and consequently are of constant sign there, and that the numerators are constant. This property goes over naturally to rationals of the form (6.1) as the linear part, Ak -f Bk(x — xjç), becomes constant after taking one derivative and zero after taking the second derivative.
