ABSTRACT
Introducing Bk — s'k(xk), k = 1, • • •, n - 1, and Bn as unknowns, the C1 conditions, s'k(xk+i) = 4 + 1 ( ^ + 1 ) = #A:+i> Y i e l d
(4.5)
Equations (4.4) and (4.5) may be solved for Ck and Dk as functions of Bk and Bk+i'-
(4.6)
for k = 1, • • •, n — 1. If the Bk were prescribed ahead of time, then we would have a Hermite interpolation problem, which could be easily solved from these equations. We will return to this problem later. In the present case, however, we have the C2 conditions,
to consider. These are easily expressed as
and we thus obtain a linear system of equations,
(4.7)
As this is a system of n — 2 equations in n unknowns, B\, • • •, Bn, we need to add two extra conditions. This will be done in such a way so as to reduce the number of unknowns to n — 2 and to also guarantee the existence of a solution to (4.7). In principle, one could assign values to B\ and B<i and then solve (4.7) by a simple recursion, but from experience such asymmetric end conditions are strongly reflected in the resulting spline interpolant. It is bet ter to assign suitable values to Bi and Bn, as then the coefficient matr ix of (4.7) is symmetric, tridiagonal, and strictly diagonally dominant. For example, we could set:
1. B\ and Bn arbitrarily.
