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Cubic Splines MACM 316 6/15 Endpoint Conditions: Natural Spline ➤ Now, we’ll deal with the two missing equations . . . ➤ Note: Derivative matching conditions are applied only at interior points, which suggests applying some constraints on the derivative(s) at x0 and xn. ➤ There is no unique choice, but several common ones are 1. Natural or Free Spline (zero curvature): m0 = 0 and mn = 0 for taken together gives an (n + 1) × (n + 1) system: ⎡ 1 0 0 ⎢ h0 2(h0 ⎢ + h1) h1 ⎢ 0 h1 2(h1 + h2) ⎢ 0 0 h2 ⎢ . ⎣ 0 0 · · · 0 h2 2(h2 + h3) ... hn−2 0 ⎤ · · · 0 ⎡ ⎤ ⎥ m0 ⎥ ⎢ ⎥ 0 ⎥ ⎢ m1 ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ m2 ⎥ h3 . ⎥ ⎢ ⎥ . .. . ⎥ ⎢ m3 ⎥ .. ⎥ ⎢ ⎥ ⎥ ⎣ 2(hn−2 + hn−1) hn−1 ⎦ . ⎦ mn 0 1 ⎡ ⎤ 0 ⎢ y2−y1 y1−y0 ⎢ − ⎥ h1 h0 ⎥ ⎢ y3−y2 y2−y1 ⎥ ⎢ − ⎥ h2 h1 ⎢ y4−y3 y3−y2 ⎥ = 6 ⎢ − ⎥ ⎢ h3 h2 ⎥ ⎢ . ⎥ ⎢ yn−yn−1 yn−1−yn−2 ⎥ ⎣ − ⎦ hn−1 hn−2 0 . . . or you could just drop the equations for m0 and mn and write it as an (n − 1) × (n − 1) system. November 1, 2012 c○ Steven Rauch and John Stockie
Cubic Splines MACM 316 7/15 Interpretation of a Natural Spline • The word “spline” comes from the draftsman’s spline, which is shown here constrained by 3 pegs. • There is no force on either end to bend the spline beyond the last pegs, which results in a flat shape (zero curvature). • This is why the S ′′ = 0 end conditions are called natural. November 1, 2012 c○ Steven Rauch and John Stockie
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