If it is known or if it is desired that the n points (x1, y1), (x2, y2), ... , (xn, yn)to be interpolated he on a single cycle of a periodic curve with period xn - x1, then an interpolating cubic spline S(x) must satisfy
S(x1) = S(xn)
Sʹ(x1) = Sʹ(xn)
Sʹʹ(x1) = S"(xn)
(a) Show that these three periodicity conditions require that
y1 = yn
M1 = Mn
4M1 + M2 + Mn-1 = 6(yn-1 - 2y1 + y2) / h2
(b) Using the three equations in part (a) and Equations 15, construct an (n - 1) × (n - 1) linear system for M1, M2, ..., Mn-1 in matrix form.