▁used ▁for ▁interpolation ▁of ▁a ▁monot onic ▁data ▁set , ▁the ▁interpol ated ▁function ▁will ▁not ▁necessarily ▁be ▁monot onic , ▁but ▁monot onic ity ▁can ▁be ▁preserved ▁by ▁adjust ing ▁the ▁tang ents . ▁ ▁Inter pol ation ▁on ▁the ▁unit ▁interval ▁with ▁matched ▁derivatives ▁at ▁end points ▁Consider ing ▁a ▁single ▁coordinate ▁of ▁the ▁points ▁ ▁and ▁ ▁as ▁the ▁values ▁that ▁a ▁function , ▁f ( x ), ▁takes ▁at ▁integer ▁ordin ates ▁x = n − 1 , ▁n , ▁n + 1 ▁and ▁n + 2 , ▁ ▁If , ▁in ▁addition , ▁the ▁tang ents ▁at ▁the ▁end points ▁are ▁defined ▁as ▁the ▁centered ▁differences ▁of ▁the ▁adjacent ▁points , ▁ ▁To ▁evaluate ▁the ▁interpol ated ▁f ( x ) ▁for ▁a ▁real ▁x , ▁first ▁separate ▁x ▁into ▁the ▁integer ▁portion , ▁n , ▁and ▁fraction al ▁portion , ▁u ▁ ▁Then ▁the ▁Cat m ull – R om ▁spl ine ▁is ▁▁▁▁▁ ▁denotes ▁the ▁floor ▁function ▁which ▁returns ▁the ▁largest ▁integer ▁no ▁larger ▁than ▁x ▁and ▁ ▁denotes ▁the ▁matrix ▁trans pose . ▁ ▁The ▁bottom ▁equality ▁is ▁dep ict ing ▁the ▁application ▁of ▁Hor ner ' s ▁method . ▁ ▁This ▁writing ▁is ▁relevant ▁for ▁tr ic ub ic ▁interpolation , ▁where ▁one ▁optimization ▁requires ▁you ▁to ▁compute ▁C INT u ▁six teen ▁times ▁with ▁the ▁same ▁u ▁and ▁different ▁p . ▁ ▁See ▁also ▁ ▁B ic ub ic ▁interpolation , ▁a ▁general ization ▁to ▁two ▁dimensions ▁ ▁Tr ic ub ic ▁interpolation