ABSTRACT
Now replace kby k-l'm (5.6) and substitute in (5.9), (5.10), and (5.11). This gives
(5.12)
Similarly, replace k by k - 1 in (5.7) and substitute in (5.10) and (5.11). This results in
(5.13)
Together, the systems (5.12) and (5.13) yield 2(n — 2) equations in the 2n unknowns £?&, C&, k = 1, • • •, n. If we prescribe values for Z?i, Bn, Ci, and Cn, i. e., the first and second (up to a factor of 1/2) derivatives at the endpoints, we are left with 2(n — 2) unknowns B2, • • •, # n - i and C2, • • •, Cn_i . The coefficient matrix then has the form,
(5.14)
of the simple structure of the given system, this may be computationally more expensive than direct methods due to the large number of iterations that it requires. A corresponding Fortran subroutine, QUINTP, for natural quintic spline interpolants {C\ = D\ = Cn — Dn = 0) is given in [154]. The Algol procedure "quintic" of [159] handles alternatively the end conditions that we described (Bi,Ci,Bn,Cn prescribed), natural end conditions, or prescribed values for C\,E\,Cn and En.
