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Cubic Splines 
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#1
Sep2613, 09:04 AM

P: 3

Hi,
I was wondering if anyone could help me with this problem on a closed natural cubic spline. given the equations below i need to compute the 0,1 and 2nd order derivatives lets say for X(0) in t = 0 and t =1 xi (t) = ai,x + bi,xt + ci,xt2 + di,xt3 yi (t) = ai,y + bi,yt + ci,yt2 + di,yt3 from that, i get the linear system of (below) for all i segments x1 (1) = a1 + b1 + c1 + d1 = aN = xN (0) x'1(1) = b1 + 2c1 + 3d1 = bN = x'N (0) x''1(1) = 2c1 + 6d1 = 2cN = x''N (0) . . . xi (1) = ai + bi + ci + di = ai−1 = xi−1 (0) x'i(1) = bi + 2ci + 3di = bi−1 = x'i−1(0) x''i(1) = 2ci + 6di = 2ci−1 = x''i−1(0) . . . xN (1) = aN + bN + cN + dN = aN−1 = xN−1 (0) x'N (1) = bN + 2cN + 3dN = bN−1 = x'N−1(0) x''N (1) = 2cN + 6dN = 2cN−1 = x''N−1(0) It is the next step that i do not understand where i am supposed to exploit these three groups of equations to get rid of b_i and d_i. let's check if it is possible: we have 3xN equations and 3xN variables (a_i aren't actually variables!) so it's ok.  look at the first three eq. we can easily get rid of of d_N and b_N by substitution...but we have only one b_0....  how can we do? we can exploit the equations between x_0 and x_1 ! express b_0 in function of a_0,a_1,c_0,c_1 and substitute it back in the first three equations  we have now N equations that involves only a_i and c_i variables... just take all a_i terms to the right, the c_i to the left and express everything in matricial form.  we are done. this leaves me with the matrix of 4/3 1/3 1/3 1/3 4/3 1/3 1/3 1/3 4/3 I do not understand how they arrive at this matrix? ANy help would be greatly appreciated 


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