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Cubic Splines MACM 316 2/15 Derivation Using the definition, let’s determine equations relating the coefficients and keep a count of the number of equations: # eqns. Interpolation and continuity: Si(xi) = yi for i = 0, 1, . . . , n − 1 n Si(xi+1) = yi+1 for i = 0, 1, . . . , n − 1 n (based on both of these, S(x) is continuous) Derivative continuity: S ′ i (xi+1) = S ′ i+1 (xi+1) for i = 0, 1, . . . , n − 2 n − 1 S ′′ i (xi+1) = S ′′ i+1 (xi+1) for i = 0, 1, . . . , n − 2 n − 1 There are still 2 equations missing! (later) Total # of equations: 4n − 2 November 1, 2012 c○ Steven Rauch and John Stockie
Cubic Splines MACM 316 3/15 Derivation (cont’d) We need expressions for the derivatives of Si: Si(x) = ai + bi(x − xi) + ci(x − xi) 2 + di(x − xi) 3 S ′ i (x) = bi + 2ci(x − xi) + 3di(x − xi) 2 S ′′ i (x) = 2ci + 6di(x − xi) It is very helpful to introduce the hi = xi+1 − xi. Then the spline conditions can be written as follows: • Si(xi) = yi for i = 0, 1, . . . , n − 1: ai = yi • Si(xi+1) = yi+1 for i = 0, 1, . . . , n − 1: ai + hibi + h 2 i ci + h 3 i di = yi+1 • S ′ i (xi+1) = S ′ i+1 (xi+1) for i = 0, 1, . . . , n − 2: bi + 2hici + 3h 2 i di − bi+1 = 0 • S ′′ i (xi+1) = S ′′ i+1 (xi+1) for i = 0, 1, . . . , n − 2: 2ci + 6hidi − 2ci+1 = 0 The boxed equations above can be written as a large linear system for the 4n unknowns [a0, b0, c0, d0, a1, b1, c1, d1, . . . , an−1, bn−1, cn−1, dn−1] T November 1, 2012 c○ Steven Rauch and John Stockie
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