计算双重积分的程序

Write a program to calculate double integral numerically.

例子：

输入：给定以下积分。 $\int _{3.7}^{4.3}\int _{2.3}^{2.5}\sqrt{x^4+y^5}\:dxdy$在哪里$f(x, y)=\sqrt{x^4+y^5}\$输出： 3.915905

解释和方法：

1. 我们需要决定我们将使用什么方法来求解积分。
   在本例中，我们将使用 Simpson 1/3 方法进行 x 和 y 积分。
   为此，首先，我们需要确定步长。设 h 是关于 x 的积分步长，k 是关于 y 的积分步长。
   在这个例子中，我们取 h=0.1 和 k=0.15。
   请参阅辛普森 1/3 规则
2. 我们需要创建一个表格，其中包含所有 x 和 y 点的所有可能组合的函数f(x, y) 的值。
   

   x\y	y0	y1	y2	….	ym
   x0	f(x0, y0)	f(x0, y1)	f(x0, y2)	….	f(x0, ym)
   x1	f(x1, y0)	f(x1, y1)	f(x1, y2)	….	f(x1, ym)
   x2	f(x2, y0)	f(x2, y1)	f(x2, y2)	….	f(x2, ym)
   x3	f(x3, y0)	f(x3, y1)	f(x3, y2)	….	f(x3, ym)
   ….	….	….	….	….	….
   ….	….	….	….	….	….
   xn	f(xn, y0)	f(xn, y1)	f(xn, y2)	….	f(xn, ym)

   在给定的问题中，

   x0=2.3
   x2=2.4
   x3=3.5
   
   y0=3.7
   y1=3.85
   y2=4
   y3=4.15
   y4=4.3
   

3. 生成表格后，我们在表格的每一行上应用 Simpson 1/3 规则(或问题中要求的任何规则)以找到每个 x 处的积分 wrt y 并将值存储在数组 ax[] 中。
4. 我们再次对数组 ax[] 的值应用辛普森 1/3 规则(或任何要求的规则)来计算积分 wrt x。

下面是上面代码的实现：

C++

// C++ program to calculate
// double integral value
  
#include 
using namespace std;
  
// Change the function according to your need
float givenFunction(float x, float y)
{
    return pow(pow(x, 4) + pow(y, 5), 0.5);
}
  
// Function to find the double integral value
float doubleIntegral(float h, float k,
                     float lx, float ux,
                     float ly, float uy)
{
    int nx, ny;
  
    // z stores the table
    // ax[] stores the integral wrt y
    // for all x points considered
    float z[50][50], ax[50], answer;
  
    // Calculating the numner of points
    // in x and y integral
    nx = (ux - lx) / h + 1;
    ny = (uy - ly) / k + 1;
  
    // Calculating the values of the table
    for (int i = 0; i < nx; ++i) {
        for (int j = 0; j < ny; ++j) {
            z[i][j] = givenFunction(lx + i * h,
                                    ly + j * k);
        }
    }
  
    // Calculating the integral value
    // wrt y at each point for x
    for (int i = 0; i < nx; ++i) {
        ax[i] = 0;
        for (int j = 0; j < ny; ++j) {
            if (j == 0 || j == ny - 1)
                ax[i] += z[i][j];
            else if (j % 2 == 0)
                ax[i] += 2 * z[i][j];
            else
                ax[i] += 4 * z[i][j];
        }
        ax[i] *= (k / 3);
    }
  
    answer = 0;
  
    // Calculating the final integral value
    // using the integral obtained in the above step
    for (int i = 0; i < nx; ++i) {
        if (i == 0 || i == nx - 1)
            answer += ax[i];
        else if (i % 2 == 0)
            answer += 2 * ax[i];
        else
            answer += 4 * ax[i];
    }
    answer *= (h / 3);
  
    return answer;
}
  
// Driver Code
int main()
{
    // lx and ux are upper and lower limit of x integral
    // ly and uy are upper and lower limit of y integral
    // h is the step size for integration wrt x
    // k is the step size for integration wrt y
    float h, k, lx, ux, ly, uy;
  
    lx = 2.3, ux = 2.5, ly = 3.7,
    uy = 4.3, h = 0.1, k = 0.15;
  
    printf("%f", doubleIntegral(h, k, lx, ux, ly, uy));
    return 0;
}

Java

// Java program to calculate
// double integral value
class GFG{
  
  
// Change the function according to your need
static float givenFunction(float x, float y)
{
    return (float) Math.pow(Math.pow(x, 4) + 
                        Math.pow(y, 5), 0.5);
}
  
// Function to find the double integral value
static float doubleIntegral(float h, float k,
                    float lx, float ux,
                    float ly, float uy)
{
    int nx, ny;
  
    // z stores the table
    // ax[] stores the integral wrt y
    // for all x points considered
    float z[][] = new float[50][50], ax[] = new float[50], answer;
  
    // Calculating the numner of points
    // in x and y integral
    nx = (int) ((ux - lx) / h + 1);
    ny = (int) ((uy - ly) / k + 1);
  
    // Calculating the values of the table
    for (int i = 0; i < nx; ++i)
    {
        for (int j = 0; j < ny; ++j) 
        {
            z[i][j] = givenFunction(lx + i * h,
                                    ly + j * k);
        }
    }
  
    // Calculating the integral value
    // wrt y at each point for x
    for (int i = 0; i < nx; ++i) 
    {
        ax[i] = 0;
        for (int j = 0; j < ny; ++j)
        {
            if (j == 0 || j == ny - 1)
                ax[i] += z[i][j];
            else if (j % 2 == 0)
                ax[i] += 2 * z[i][j];
            else
                ax[i] += 4 * z[i][j];
        }
        ax[i] *= (k / 3);
    }
  
    answer = 0;
  
    // Calculating the final integral value
    // using the integral obtained in the above step
    for (int i = 0; i < nx; ++i)
    {
        if (i == 0 || i == nx - 1)
            answer += ax[i];
        else if (i % 2 == 0)
            answer += 2 * ax[i];
        else
            answer += 4 * ax[i];
    }
    answer *= (h / 3);
  
    return answer;
}
  
// Driver Code
public static void main(String[] args)
{
    // lx and ux are upper and lower limit of x integral
    // ly and uy are upper and lower limit of y integral
    // h is the step size for integration wrt x
    // k is the step size for integration wrt y
    float h, k, lx, ux, ly, uy;
  
    lx = (float) 2.3; ux = (float) 2.5; ly = (float) 3.7;
    uy = (float) 4.3; h = (float) 0.1; k = (float) 0.15;
  
    System.out.printf("%f", doubleIntegral(h, k, lx, ux, ly, uy));
}
}
  
/* This code contributed by PrinciRaj1992 */

Python3

# Python3 program to calculate 
# double integral value 
  
# Change the function according
# to your need 
def givenFunction(x, y): 
  
    return pow(pow(x, 4) + pow(y, 5), 0.5) 
  
# Function to find the double integral value 
def doubleIntegral(h, k, lx, ux, ly, uy): 
  
    # z stores the table 
    # ax[] stores the integral wrt y 
    # for all x points considered 
    z = [[None for i in range(50)]
               for j in range(50)]
    ax = [None] * 50
  
    # Calculating the numner of points 
    # in x and y integral 
    nx = round((ux - lx) / h + 1) 
    ny = round((uy - ly) / k + 1)
  
    # Calculating the values of the table 
    for i in range(0, nx): 
        for j in range(0, ny): 
            z[i][j] = givenFunction(lx + i * h, 
                                    ly + j * k) 
          
    # Calculating the integral value 
    # wrt y at each point for x 
    for i in range(0, nx): 
        ax[i] = 0
        for j in range(0, ny): 
              
            if j == 0 or j == ny - 1: 
                ax[i] += z[i][j] 
            elif j % 2 == 0:
                ax[i] += 2 * z[i][j] 
            else:
                ax[i] += 4 * z[i][j] 
          
        ax[i] *= (k / 3) 
      
    answer = 0
  
    # Calculating the final integral 
    # value using the integral 
    # obtained in the above step 
    for i in range(0, nx): 
        if i == 0 or i == nx - 1: 
            answer += ax[i] 
        elif i % 2 == 0:
            answer += 2 * ax[i] 
        else:
            answer += 4 * ax[i] 
      
    answer *= (h / 3) 
  
    return answer 
  
# Driver Code 
if __name__ == "__main__":
  
    # lx and ux are upper and lower limit of x integral 
    # ly and uy are upper and lower limit of y integral 
    # h is the step size for integration wrt x 
    # k is the step size for integration wrt y 
    lx, ux, ly = 2.3, 2.5, 3.7
    uy, h, k = 4.3, 0.1, 0.15
  
    print(round(doubleIntegral(h, k, lx, ux, ly, uy), 6)) 
      
# This code is contributed 
# by Rituraj Jain

C#

// C# program to calculate
// double integral value
using System;
  
class GFG
{
  
// Change the function according to your need
static float givenFunction(float x, float y)
{
        return (float) Math.Pow(Math.Pow(x, 4) + 
                            Math.Pow(y, 5), 0.5);
    }
      
    // Function to find the double integral value
    static float doubleIntegral(float h, float k,
                        float lx, float ux,
                        float ly, float uy)
    {
        int nx, ny;
      
        // z stores the table
        // ax[] stores the integral wrt y
        // for all x points considered
        float [, ] z = new float[50, 50];
        float [] ax = new float[50];
        float answer;
      
        // Calculating the numner of points
        // in x and y integral
        nx = (int) ((ux - lx) / h + 1);
        ny = (int) ((uy - ly) / k + 1);
      
        // Calculating the values of the table
        for (int i = 0; i < nx; ++i)
        {
            for (int j = 0; j < ny; ++j) 
            {
                z[i, j] = givenFunction(lx + i * h,
                                        ly + j * k);
            }
        }
      
        // Calculating the integral value
        // wrt y at each point for x
        for (int i = 0; i < nx; ++i) 
        {
            ax[i] = 0;
            for (int j = 0; j < ny; ++j)
            {
                if (j == 0 || j == ny - 1)
                    ax[i] += z[i, j];
                else if (j % 2 == 0)
                    ax[i] += 2 * z[i, j];
                else
                    ax[i] += 4 * z[i, j];
            }
            ax[i] *= (k / 3);
        }
      
        answer = 0;
      
        // Calculating the final integral value
        // using the integral obtained in the above step
        for (int i = 0; i < nx; ++i)
        {
            if (i == 0 || i == nx - 1)
                answer += ax[i];
            else if (i % 2 == 0)
                answer += 2 * ax[i];
            else
                answer += 4 * ax[i];
        }
        answer *= (h / 3);
      
        return answer;
    }
      
    // Driver Code
    public static void Main()
    {
        // lx and ux are upper and lower limit of x integral
        // ly and uy are upper and lower limit of y integral
        // h is the step size for integration wrt x
        // k is the step size for integration wrt y
        float h, k, lx, ux, ly, uy;
      
        lx = (float) 2.3; ux = (float) 2.5; ly = (float) 3.7;
        uy = (float) 4.3; h = (float) 0.1; k = (float) 0.15;
      
        Console.WriteLine(doubleIntegral(h, k, lx, ux, ly, uy));
    }
}
  
// This code contributed by ihritik

输出：

3.915905