Cardinal B-Splines

We have also written code that computes Cardinal B splines of order two through six, see Equation 8. Figure 2 demonstrates that the Cardinal B splines our code generates are symmetric about their center, as stated in [9]. We also checked, using Matlab, that they obey the sum property they are supposed to, namely, the sum of all integer values of the spline function is one. To confirm that our function calculated the spline properly, we made use of a number of mathematical properties of the spline function, as given in [9]. The most important ones are symmetry, limited range where the function is nonzero, and the fact that the sum of the function evaluated for all integers is one. The first two properties are readily evident from 2, made in Matlab.

Figure 2: Cardinal B-Splines for order 2 through 6

We originally created this graph as a debugging tool. The spline function was not giving correct results, so this graph allowed us to see exactly where the function was incorrect. The varying asymmetries in the graphs of the different order functions pointed us directly to where the errors were in the code. Higher order splines have nonzero values for a larger domain, allowing them to use more values in interpolation, though obviously at the cost of more computational time. We confirmed the sum property using output from our code and numerical tools in matlab. This property is important because we multiply the Cardinal B-Spline value by the charge of a particle and then make use of its values at integer valued mesh points. This sum property ensures that the charge field will not have its magnitude distorted by interpolation.