Question

Natural Cubic spline If the and Derivatives at the endpoints are known, ine. Pi" (x)= Y, 4 and Puni (xu) z Yu" for some numbers y," and ya theni 25, + z = 3y[x, 1x23 - į sxy," (al Sunit zou z 3y [xn-1 , xu] + & Sx uni Yu" (6) 10 Derive equations (a) and (b) for the natural cubic Spline

Answer 1

Derivation Natural Cubic Apline <xn = b and you A Cubic Interpolating Buch that Suppobe cele have a = x0 <--- yo spline for these data és a a fimation s(a) that is twice continuo ubly differentiable on [a, b] batifles sexy) = y; for ico...in and is a cubic polynomial for & in [xi, niti and i zo.. The following denivation shows how to construct the Cubec speeces" Sila). izo....nas end points conditions. S(30) SiLC) n-1 using the natural = S"lon) ご。 S" (960) Yiti - Yi and Ayi = yiti xi neiti Notation Axi = i=0 for Bufpobe Silje) = ai + bel 22 *;) + ci( x - xi)?+di (x-)? éco -in-1 Si(x): S112) s( xito) +1)=Yiti we have two interpolation conditions on sexil and = yi si (xit) ai yil With this, the The first immediately gives 3 Yit Becond giweb, yi? Ibraxi + cia xe? + de dai

bi +2 CPD Hi tadi ani? to iti & subatooeting Yi from both sides and dividing by Axi. we obtain Ayi ini =0. bit cibrit di AXE2 siti (a) =S (M&+? CI The differentiability condikono asithat. S'em) and S"() erast and one continuous throughout [a,b]. These condition & require that Sallret) fero é with, Si'(o) = bi + 20 ; ( 36-4199 + Idilx-ap)? for each is the first requirement becomep, é = 0 ...,n - 2 n-2 20 i + with, si"?) edil dari for eache, becomeb, the second denuate or just = 2liti, qci +6 de ani M-2 ģ=0, 7 Citi Cat + 3d dri =

Since, ulo also have a Danal 6 dnes Š"(.*n) S"(xn) = 2C1- 1-1 was com define, con S"camo /2 and extend the above relation blir é znat, obtaining iso. Citi 10 Leancy ci to d² AXE that the natural end conditione moting S'' (20).- O semaining ne 3nal s' land imply that , Coacno Thus we have n bils, Çi's and ndi's for total of 3n-t cofficients to be determined, Equation 0. and linear equations that these cofficients. must got tisfy we will makipulate 0.0 md (111) linear equation & in the cis alone. From ® » we obtain to get a bystem of Citi ci P.O. Mall de 30 x thm. .yeeld this exp for di respion beeb &fituting C:) (zo- ARE syi X (ait bitli axit tal w

120 bi + aci one + Axi (ciri-ci) - biti 1. substracting @ and @ and reassonging we get + 2 birl Ayi Ani teit bri 2 į brilliti - C) ZO. n-2 ¿-1 as replacing i by + 2 sain (ci-a. Ayia + Ciu Anin 1 bi Anin i=12 en-1 we obtain > bi én for bi Substituting this this expression for oni) eit sai citi 2lonil Axistin Dyi ДУвс, és! + n-1 2 w Ani onial vo system fince, Co =(n=o. conftitute o the equation ull cofficients a Claaseny of 1-1 equations in the man

since, Anexo for each i this bystem is Strictly diagenally dominant and asa consequence of this. determines enni uniquely. byptem ve bili from di b from Ho Having solued the and the for cars ull com obtain