Newton

  static List<List<double>> CountSepaetedDifference(double[] fn)
        {
            List<List<double>> separatedDifference = new List<List<double>>();

            for (int i = 0; i < fn.Length; i++)
            {
                List <double> tmp = new List<double>();
                tmp.Add(fn[i]);
                separatedDifference.Add(tmp);
            }

            int n = fn.Length - 1;

            for (int i = 0; i < fn.Length - 1; i++)
            {
                for (int j = 0; j < n; j++)
                {
                    separatedDifference[j].Add((separatedDifference[j+1][i] - separatedDifference[j][i]) / (i + 1) );
                }
                n--;
            }
            return separatedDifference;
        }
        static void Newton()
        {
            double[] xn = { 1, 2, 3, 4, 5 };
            double[] fn = { 1, 16, 81, 256, 625 };
            double x = 0;
            List<List<double>> separetedDifference = CountSepaetedDifference(fn);
            double Nn = 0;
            double multiplier = 1;

            for (int i = 0; i < xn.Length - 1; i++)
            {
                x = (xn[i] + xn[i + 1]) / 2;
                Nn = separetedDifference[0][0];
                multiplier = 1;

                for (int j = 0; j < xn.Length - 1; j++)
                {
                    multiplier *= (x - xn[j]);
                    Nn += separetedDifference[0][j + 1] * multiplier;
                }

                Console.WriteLine("{0}  -  {1}  -  {2}", x, Nn, x * x * x * x);
            }
        }

        static void Main(string[] args)
        {
            Newton();
        }