A parabolic runout spline is the interpolating function you

A parabolic runout spline is the interpolating function you get by changing the condition f\"(x_1) - f\"(x_n) = 0 to the condition that p_1 (x) and p_n-1(x) should be quadratic polynomials, that is, p\'\"_1 = p\"\'_n - 1 = 0 (Since the p_1\'s are cubic polynomials the third derivatives are constant, so it doesn\'t matter where we evaluate them.). Modify the file splineaat2.m so that it computes the matrix relevant to this modified problem. Call the modified file 8plinemat2pr.m. (Hand in a a print-out of the modified file and an explanation of your changes.) Use your new file to graph the parabolic runout spline for the points (1,1), (2,1), (3,2), (4,4) and (5,3). (The easiest way to do this is to change splinemat2 to splineaat2pr inside the file plotsplino2.m and call the modified file plot8pline2pr.nl. Use this new file to plot the modified spline.) Hand in a plot of both the parabolic runout spline and the cubic spline on the same graph.

Solution

The fundamental idea behind cubic spline interpolation is based on the engineer’s tool used to draw smooth curves through a number of points. This spline consists of weights attached to a flat surface at the points to be connected. A flexible strip is then bent across each of these weights, resulting in a pleasingly smooth curve. The mathematical spline is similar in principle. The points, in this case, are numerical data. The weights are the coefficients on the cubic polynomials used to interpolate the data. These coefficients ’bend’ the line so that it passes through each of the data points without any erratic behavior or breaks in continuity.

The Four Properties of Cubic Splines Our spline will need to conform to the following stipulations. 1. The piecewise function S›xfi will interpolate all data points. 2. S›xfi will be continuous on the interval flx1, xn‡ 3. S v ›xfi will be continuous on the interval flx1, xn‡ 4. S vv ›xfi will be continuous on the interval flx1, xn‡ Since the piecewiece function S›xfi will interpolate all of the data points, we can conclude that S›xifi = yi ) for i = 1, 2, ..., n ? 1. Since xi 5 flxi, xi+1‡, S›xifi = si›xifi and we can use equation (2) to produce yi = si›xifi yi = ai›xi ? xifi 3 + bi›xi ? xifi 2 + ci›xi ? xifi + di yi = di (6 ) for each i = 1, 2, ..., n ? 1. Since the curve S›xfi must be continuous across its entire interval, it can be concluded that each sub-function must join at the data points, so si›xifi = si?1›xifi (7 ) for i = 2, 3, ..., n. From equation (2), si›xifi = di and si?1›xifi = ai?1›xi ? xi?1fi 3 + bi?1›xi ? xi?1fi 2 + ci?1›xi ? xi?1fi + di?1 (8 ) so di = ai?1›xi ? xi?1fi 3 + bi?1›xi ? xi?1fi 2 + ci?1›xi ? xi?1fi + di?