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Cubic Splines MACM 316 4/15 Alternate formulation This linear system can be simplified considerably by defining mi = S ′′ i (xi) = 2ci or ci = m i 2 and thinking of the mi as unknowns instead. Then: • S ′′ i (xi+1) = S ′′ i+1 (xi+1) for i = 0, 1, . . . , n − 2: =⇒ 2ci + 6hidi − 2ci+1 = 0 =⇒ mi + 6hidi − mi+1 = 0 =⇒ di = m i+1−m i 6h i • Si(xi) = yi and Si(xi+1) = yi+1 for i = 0, 1, . . . , n − 1: =⇒ yi + hibi + h 2 i ci + h 3 i di = yi+1 Substitute ci and di from above: =⇒ bi = y i+1−y i h i − h i 2 mi − h i 6 (mi+1 − mi) • S ′ i (xi+1) = S ′ i+1 (xi+1) for i = 0, 1, . . . , n − 2: =⇒ bi + 2hici + 3h 2 i di = bi+1 Substitute bi, ci and di from above and simplify: himi + 2(hi + hi+1)mi+1 + hi+1mi+2 = 6 yi+2−y i+1 h i+1 − y i+1−yi hi November 1, 2012 c○ Steven Rauch and John Stockie
Cubic Splines MACM 316 5/15 Notice: These are n − 1 linear equations for n + 1 unknowns [m0, m1, m2, . . . , mn] T where mi = g ′′ i (xi). November 1, 2012 c○ Steven Rauch and John Stockie
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