One Dimensional Cubic Splines

On an interval [Xy i, of a variable x, there exists one cubic function 5, (x) and only one such that  

Now we consider an interval [a. h] of the variable x, which is subdivided by a mesh of points  

An alternative analytical and stable interpolation technology has recently been extensively used the continued fraction 229—232) 

We call this function a cubic spline on A. The definition of such a function requires a knowledge of the intermediate slopes  

Standard algorithms exist to solve such a band system in an efficient and stable way. The solution spline function 5 (x) exists and is unique. Moreover, it is an excellent interpolation function which converges to /(x) when N if lim y .(/ y) = 0. 

From the continuity requirement imposed onat jiy (/ = 1, 2. .. N — 1) the set of linear equations results  

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The first term corresponds to the potentid energy of a cyclopropane molecule in the FF configuration with the ring angle CCC = 2 a (Fig. 4-a). The calculated energy curve is pictured in Fig. 5 there appears no barrier to the reclosure motion of the diradical FF (a). This curve will be analytically approximated by means of one-dimensional cubic spline functions. 

MATLAB has several functions for interpolation. The function = interpl(x, y, x) takes the values of the independent variable x and the dependent variable y (base points) and does the one-dimensional interpolation based on x, to find yj. The default method of interpolation is linear. However, the user can choose the method of interpolation in the fourth input argument from nearest (nearest neighbor interpolation), linear (linear interpolation), spline (cubic spline interpolation), and cubic (cubic inteipolation). If the vector of independent variable is not equally spaced, the function interplq may be used instead. It is faster than interpl because it does not check the input arguments. MATLAB also has the function sp/ine to perform one-dimensional interpolation by cubic. splines, using nat-a- not method, ft can also return coefficients of piecewise poiynomiais, if required. The functions interp2, inte.rp3, and interpn perform two-, three-, and n-dimensional interpolation, respectively. 

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