Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Cubic Splines MACM 316 9/15<br />
Endpoint Conditions: Not-A-Knot<br />
3. Not-A-Knot Spline (third derivative matching):<br />
S ′′′<br />
0 (x1) = S ′′′<br />
1 (x1) and S ′′′<br />
n−2 (xn−1) = S ′′′<br />
n−1 (xn−1)<br />
Using S ′′′<br />
i (x) = 6di and di = mi+1−mi, these conditions<br />
6hi become<br />
and<br />
h1(m1 − m0) = h0(m2 − m1)<br />
hn−1(mn−1 − mn−2) = hn−2(mn − mn−1)<br />
The matrix in this case is<br />
⎡<br />
−h1 h0 + h1 −h0<br />
⎢ h0 2(h0 + h1) h1<br />
⎢ 0 h1 2(h1 + h2)<br />
⎢ . 0<br />
...<br />
⎢<br />
⎣ 0 · · · 0<br />
· · ·<br />
0<br />
h2<br />
. ..<br />
hn−2<br />
· · ·<br />
0<br />
. ..<br />
2(hn−2 + hn−1)<br />
0<br />
.<br />
.<br />
0<br />
hn−1<br />
⎤<br />
⎥<br />
⎦<br />
0 · · · · · · −hn−1 hn−2 + hn−1 −hn−2<br />
and the first/last RHS entries are zero.<br />
Note: Matlab’s spline function implements the not-a-knot<br />
condition (by default) as well as the clamped spline, but<br />
not the natural spline. Why not? (see Homework #4).<br />
November 1, 2012 c○ Steven Rauch and John Stockie
Change language