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) / 2 * (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) / 6 * (fun(up) + fun(down) + 4 * fun((up + down) / 2)); } /// <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) / 90 * (7 * fun(up) + 7 * fun(down) + 32 * fun((up + 3 * down) / 4) + 12 * fun((up + down) / 2) + 32 * fun((3 * up + down) / 4)); 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 = 0; double x = down; for (int i = 0; i < N - 1; i++) { x += h; result += fun(x); } result = (fun(up) + result * 2 + fun(down)) * h / 2; 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 = 0; for (int n = 0; n < N; n++) { result += h / 6 * (fun(down) + 4 * fun(down + h / 2) + 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 = 0; for (int n = 0; n < N; n++) { result += h / 90 * (7 * fun(down) + 32 * fun(down + h / 4) + 12 * fun(down + h / 2) + 32 * fun(down + 3 * h / 4) + 7 * 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 = 0, R2 = 0; int k = 0; //2的k次方即为N（划分的子区间数） R1 = (64 * C(fun, 2 * (int)Math.Pow(2, k), up, down) - C(fun, (int)Math.Pow(2, k++), up, down)) / 63; R2 = (64 * C(fun, 2 * (int)Math.Pow(2, k), up, down) - C(fun, (int)Math.Pow(2, k++), up, down)) / 63; while (Math.Abs(R2 - R1) > e) { R1 = R2; R2 = (64 * C(fun, 2 * (int)Math.Pow(2, k), up, down) - C(fun, (int)Math.Pow(2, k++), up, down)) / 63; } return R2; } private static double S(Func<double, double> fun, int N, double up, double down) { return (4 * FuHuaTiXing(fun, 2 * N, up, down) - FuHuaTiXing(fun, N, up, down)) / 3; } private static double C(Func<double, double> fun, int N, double up, double down) { return (16 * S(fun, 2 * N, up, down) - S(fun, N, up, down)) / 15; } }