66 X Q Chen,Z M Gong.H Huang.SZ Ge,and L B Zhou 4.Si1(k1)=Sc)i=0.l,…,n-2. 5.Si(xt1)=S(x1）i=0,1,…,n-2. 6.S(xo)=f'(xo),and S')=f)(clamped boundary). To construct the cubic spline interpolation for a given function f,the conditions above are applied to the cubic polynomials: S;(x)=a;+b(x-x)+cx-x)2+di(x-x) For donation convenience,leth;=x-x.Then, S;(xi)=a;=f(xi) Si)=S;i)=aim=ai+bh+ch2+dhi hold for i=0,1,...,n-2.Applying the condition 4,we have bitt =bi+2cih +3d h2 i=0,l…,n-1 Defineand apply the condition 5,then ci1=C+3d,hi=0l,…,n-1 Solving for d in the above equations gives a=a,+hh+bc,+c） 3 bitl =bi+hi(ci+cit!) Finally,re-arranging the equation gives m=(ar In matrix form,the above equation becomes66 X Q Chen, Z M Gong, H Huang, S Z Ge, and L B Zhou 4. ( ) ( ) 1 ' 1 ' i+1 i+ = ii + S x S x i = L,,1,0 n − 2 . 5. ( ) ( ) 1 " 1 " i+1 i+ = ii + S x S x i = L,,1,0 n − 2 . 6. ( ) ( ) 0 ' 0 ' S x = xf , and ( ) ( ) n n S x xf ' ' = (clamped boundary). To construct the cubic spline interpolation for a given function f, the conditions above are applied to the cubic polynomials: ( ) ( ) ( ) ( ) 2 3 i i i i i i i i S x = a + b x − x + c x − x + d x − x For donation convenience, let . i i 1 i h = x − x + Then, ( ) ( ) ii i i S x = a = xf ( ) ( ) 2 3 i 1 i 1 ii 1 i 1 i ii ii ii S x = S x = a = a + hb + hc + hd + + + + hold for i=0,1,…,n-2. Applying the condition 4, we have 2 1 2 3 i i ii ii b = b + hc + hd + i = L,,1,0 n −1 Define ( ) 2 " n n S x c = and apply the condition 5, then i i ii c c 3 hd +1 = + i = L,,1,0 n −1 Solving for i d in the above equations gives ( ) ( ) 1i i ii 1i i i i i i ii b b ch c c c h a a hb + + + + = + + = + + + 1 2 1 2 3 Finally, re-arranging the equation gives ( ) ( ) ( ) 1 1 1 1 1 1 1 3 3 2 − − − − + − + + + = + − − i − i i i i i i i i ii ii a a h a a h h c h ch ch In matrix form, the above equation becomes