Step by Step solution about How to resolve the algorithm Numerical integration step by step in the Kotlin programming language

This Kotlin code implements numerical integration using different methods. Numerical integration is a technique used to estimate the area under the curve of a function.

Here's a detailed breakdown of the code:

1. Function Type Alias:

   typealias Func = (Double) -> Double

   This type alias defines a type Func that represents a function that takes a Double as an input and returns a Double as an output. This type alias is used to represent the function to be integrated.

2. integrate Function:

   fun integrate(a: Double, b: Double, n: Int, f: Func) {

   The integrate function is the main integration function. It takes four parameters:

   • a: Lower bound of the integration interval.
   • b: Upper bound of the integration interval.
   • n: Number of subintervals to use for integration.
   • f: The function to be integrated, represented as a Func instance.

3. Initialization of h and sum:

   val h = (b - a) / n
   val sum = DoubleArray(5)

   • h is the step size or width of each subinterval.
   • sum is an array of doubles used to store the results of different integration methods.

4. Loop for Integration:

   for (i in 0 until n) {

   This loop iterates over each subinterval and calculates the values of the function at different points within the subinterval.

5. Calculation of Function Values:

   val x = a + i * h
   sum[0] += f(x)
   sum[1] += f(x + h / 2.0)
   sum[2] += f(x + h)
   sum[3] += (f(x) + f(x + h)) / 2.0
   sum[4] += (f(x) + 4.0 * f(x + h / 2.0) + f(x + h)) / 6.0

   For each subinterval, it calculates the function values at different points and adds them to the corresponding elements of the sum array. These values are used to calculate the integrals using different methods.

6. Printing the Results:

   val methods = listOf("LeftRect ", "MidRect  ", "RightRect", "Trapezium", "Simpson  ")
   for (i in 0..4) println("${methods[i]} = ${"%f".format(sum[i] * h)}")

   After the loop completes, it prints the results of the integration using different methods. LeftRect is the left-hand Riemann sum, MidRect is the midpoint rule, RightRect is the right-hand Riemann sum, Trapezium is the trapezoidal rule, and Simpson is Simpson's rule.

7. main Function:

   fun main(args: Array<String>) {

   The main function is the entry point of the program.

8. Integration Examples:

   integrate(0.0, 1.0, 100) { it * it * it }
   integrate(1.0, 100.0, 1_000) { 1.0 / it }
   integrate(0.0, 5000.0, 5_000_000) { it }
   integrate(0.0, 6000.0, 6_000_000) { it }

   These are examples of integration for different functions and different numbers of subintervals.

Overall, this code demonstrates how to numerically integrate a function using different methods and provides the results for specified intervals and functions.

// version 1.1.2

typealias Func = (Double) -> Double

fun integrate(a: Double, b: Double, n: Int, f: Func) {
    val h = (b - a) / n
    val sum = DoubleArray(5)
    for (i in 0 until n) {
        val x = a + i * h
        sum[0] += f(x)
        sum[1] += f(x + h / 2.0)
        sum[2] += f(x + h)
        sum[3] += (f(x) + f(x + h)) / 2.0
        sum[4] += (f(x) + 4.0 * f(x + h / 2.0) + f(x + h)) / 6.0
    }
    val methods = listOf("LeftRect ", "MidRect  ", "RightRect", "Trapezium", "Simpson  ")
    for (i in 0..4) println("${methods[i]} = ${"%f".format(sum[i] * h)}")
    println()
}

fun main(args: Array<String>) {
    integrate(0.0, 1.0, 100) { it * it * it }
    integrate(1.0, 100.0, 1_000) { 1.0 / it }
    integrate(0.0, 5000.0, 5_000_000) { it }
    integrate(0.0, 6000.0, 6_000_000) { it }
}