计算方法（二）用C#实现数值积分

在工程中，经常会遇到积分问题，这时原函数往往都是找不到的，因此就需要用计算方法的数值积分来求。

    public class Integral
    {
        /// <summary>
        /// 梯形公式
        /// </summary>
        /// <param name="fun">被积函数</param>
        /// <param name="up">积分上限</param>
        /// <param name="down">积分下限</param>
        /// <returns>积分值</returns>

        public static double TiXing(Func<double, double> fun, double up, double down)
        {
            return (up - down) /  * (fun(up) + fun(down));
        }
        /// <summary>
        /// 辛普森公式
        /// </summary>
        /// <param name="fun">被积函数</param>
        /// <param name="up">积分上限</param>
        /// <param name="down">积分下限</param>
        /// <returns>积分值</returns>
        public static double Simpson(Func<double, double> fun, double up, double down)
        {
            return (up - down) /  * (fun(up) + fun(down) +  * fun((up + down) / ));
        }
        /// <summary>
        /// 科特克斯公式
        /// </summary>
        /// <param name="fun">被积函数</param>
        /// <param name="up">积分上限</param>
        /// <param name="down">积分下限</param>
        /// <returns>积分值</returns>
        public static double Cotes(Func<double, double> fun, double up, double down)
        {
            double C = (up - down) /  * ( * fun(up) +  * fun(down) +  * fun((up +  * down) / )
                     +  * fun((up + down) / ) +  * fun(( * up + down) / ));
            return C;
        }

        /// <summary>
        /// 复化梯形公式
        /// </summary>
        /// <param name="fun">被积函数</param>
        /// <param name="N">区间划分快数</param>
        /// <param name="up">积分上限</param>
        /// <param name="down">积分下限</param>
        /// <returns>积分值</returns>
        public static double FuHuaTiXing(Func<double, double> fun, int N, double up, double down)
        {
            double h = (up - down) / N;
            double result = ;
            double x = down;
            for (int i = ; i < N - ; i++)
            {
                x += h;
                result += fun(x);
            }
            result = (fun(up) + result *  + fun(down)) * h / ;
            return result;
        }

        /// <summary>
        /// 复化辛浦生公式
        /// </summary>
        /// <param name="fun">被积函数</param>
        /// <param name="N">区间划分快数</param>
        /// <param name="up">积分上限</param>
        /// <param name="down">积分下限</param>
        /// <returns>积分值</returns>
        public static double FSimpson(Func<double, double> fun, int N, double up, double down)
        {
            double h = (up - down) / N;
            double result = ;
            for (int n = ; n < N; n++)
            {
                result += h /  * (fun(down) +  * fun(down + h / ) + fun(down + h));
                down += h;
            }
            return result;
        }
        /// <summary>
        /// 复化科特斯公式
        /// </summary>
        /// <param name="fun">被积函数</param>
        /// <param name="N">区间划分快数</param>
        /// <param name="up">积分上限</param>
        /// <param name="down">积分下限</param>
        /// <returns>积分值</returns>
        public static double FCotes(Func<double, double> fun, int N, double up, double down)
        {
            double h = (up - down) / N;
            double result = ;
            for (int n = ; n < N; n++)
            {
                result += h /  * ( * fun(down) +  * fun(down + h / ) +  * fun(down + h / ) +
                         * fun(down +  * h / ) +  * fun(down + h));
                down += h;
            }
            return result;
        }
        /// <summary>
        /// 龙贝格算法
        /// </summary>
        /// <param name="fun">被积函数</param>
        /// <param name="e">结果精度</param>
        /// <param name="up">积分上限</param>
        /// <param name="down">积分下限</param>
        /// <returns>积分值</returns>
        public static double Romberg(Func<double, double> fun, double e, double up, double down)
        {
            double R1 = , R2 = ;
            int k = ; //2的k次方即为N（划分的子区间数）
            R1 = ( * C(fun,  * (int)Math.Pow(, k), up, down) - C(fun, (int)Math.Pow(, k++), up, down)) / ;
            R2 = ( * C(fun,  * (int)Math.Pow(, k), up, down) - C(fun, (int)Math.Pow(, k++), up, down)) / ;
            while (Math.Abs(R2 - R1) > e)
            {
                R1 = R2;
                R2 = ( * C(fun,  * (int)Math.Pow(, k), up, down) - C(fun, (int)Math.Pow(, k++), up, down)) / ;
            }
            return R2;
        }
        private static double S(Func<double, double> fun, int N, double up, double down)
        {
            return ( * FuHuaTiXing(fun,  * N, up, down) - FuHuaTiXing(fun, N, up, down)) / ;
        }
        private static double C(Func<double, double> fun, int N, double up, double down)
        {
            return ( * S(fun,  * N, up, down) - S(fun, N, up, down)) / ;
        }
    }