Lagrange

static void LagrangePolinomial()
        {
            double[] xn = { 1, 2, 3, 4, 5 };
            double[] fn = { 1, 16, 81, 256, 625 };
            double x = 0;
            double Sum = 0;
            double multiplier = 1;
            for (int i = 0; i < 4; i++)
            {
                x = (xn[i + 1] + xn[i]) / 2;
                Sum = 0;
                for(int j = 0; j < 5; j++)
                {
                    multiplier = 1;
                    for (int k = 0; k < 5; k++)
                    {
                        if(k == j ) continue;
                        multiplier *= (x - xn[k]) / (xn[j] - xn[k]);
                    }
                    Sum += fn[j] * multiplier;
                }
                Console.WriteLine("{0}  -  {1}  -  {2}", x, Sum, x * x * x * x);
            }
        }

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