Possible error in Special.BSpline function

[remyzerems]

It seems that there is an error in the BSpline function in the Special.cs file.
There may be a mismatch between the definition stated here : http://crsouza.com/2010/03/kernel-functions-for-machine-learning-applications/#bspline and the implementation on the Special class.

Original code :

        /// <summary>
        ///   Computes the Basic Spline of order <c>n</c>
        /// </summary>
        public static double BSpline(int n, double x)
        {
            // ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/hamers/PhD_bhamers.pdf
            // http://sepwww.stanford.edu/public/docs/sep105/sergey2/paper_html/node5.html

            if (n == Int32.MaxValue)
                throw new ArgumentOutOfRangeException("n");

            double a = 1.0 / Special.Factorial(n);    // <= preloading the 1/n!... ok
            double c;

            bool positive = true;
            for (int k = 0; k <= n + 1; k++)
            {
                c = Binomial(n + 1, k) * Tools.TruncatedPower(x + (n + 1.0) / 2.0 - k, n);
                a += positive ? c : -c;               // <= Then here you sum the n+1 parts of the formula which shouldn't be added to the 1/n! preload
                positive = !positive;
            }

            return a;           // <= returned "a" which does not correspond to the stated formula
        }

Code should look like this :

        /// <summary>
        ///   Computes the Basic Spline of order <c>n</c>
        /// </summary>
        public static double BSpline(int n, double x)
        {
            // ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/hamers/PhD_bhamers.pdf
            // http://sepwww.stanford.edu/public/docs/sep105/sergey2/paper_html/node5.html

            if (n == Int32.MaxValue)
                throw new ArgumentOutOfRangeException("n");

            double a = 0.0;  // Preload a with 0
            double c;

            bool positive = true;
            for (int k = 0; k <= n + 1; k++)
            {
                c = Binomial(n + 1, k) * Tools.TruncatedPower(x + (n + 1.0) / 2.0 - k, n);
                a += positive ? c : -c;  // Sum terms over k
                positive = !positive;
            }

            return 1.0 / Special.Factorial(n) * a;  // Finally apply the 1/n! factor
        }

What do you think ? I hope I'm not misunderstanding anything...

[cesarsouza]

Thank you - your observation and your solution were perfectly correct. Apologies for taking this long to be able to integrate it in the framework!

[remyzerems]

No worries ! The most important is that it has been fixed ! ;-)

[cesarsouza]

Integrated in release 3.4.0.