merge: Created midpoint integration numerical method (TheAlgorithms#822)

* Created midpoint integration numerical method

* Auto-update DIRECTORY.md

* Added resources link

* Fixed doxumentation

* Fixed spelling error

Co-authored-by: ggkogkou <ggkogkou@ggkogkou.gr>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

Author: 3 people

DIRECTORY.md

@@ -174,6 +174,7 @@
   * [MatrixExponentiationRecursive](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/MatrixExponentiationRecursive.js)
   * [MatrixMultiplication](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/MatrixMultiplication.js)
   * [MeanSquareError](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/MeanSquareError.js)
+  * [MidpointIntegration](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/MidpointIntegration.js)
   * [ModularBinaryExponentiationRecursive](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/ModularBinaryExponentiationRecursive.js)
   * [NumberOfDigits](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/NumberOfDigits.js)
   * [Palindrome](https://github.com/TheAlgorithms/Javascript/blob/master/Maths/Palindrome.js)

Maths/MidpointIntegration.js

@@ -0,0 +1,54 @@
+/**
+*
+* @title Midpoint rule for definite integral evaluation
+* @author [ggkogkou](https://github.com/ggkogkou)
+* @brief Calculate definite integrals with midpoint method
+*
+* @details The idea is to split the interval in a number N of intervals and use as interpolation points the xi
+* for which it applies that xi = x0 + i*h, where h is a step defined as h = (b-a)/N where a and b are the
+* first and last points of the interval of the integration [a, b].
+*
+* We create a table of the xi and their corresponding f(xi) values and we evaluate the integral by the formula:
+* I = h * {f(x0+h/2) + f(x1+h/2) + ... + f(xN-1+h/2)}
+*
+* N must be > 0 and a<b. By increasing N, we also increase precision
+*
+* [More info link](https://tutorial.math.lamar.edu/classes/calcii/approximatingdefintegrals.aspx)
+*
+*/
+
+function integralEvaluation (N, a, b, func) {
+  // Check if all restrictions are satisfied for the given N, a, b
+  if (!Number.isInteger(N) || Number.isNaN(a) || Number.isNaN(b)) { throw new TypeError('Expected integer N and finite a, b') }
+  if (N <= 0) { throw Error('N has to be >= 2') } // check if N > 0
+  if (a > b) { throw Error('a must be less or equal than b') } // Check if a < b
+  if (a === b) return 0 // If a === b integral is zero
+
+  // Calculate the step h
+  const h = (b - a) / N
+
+  // Find interpolation points
+  let xi = a // initialize xi = x0
+  const pointsArray = []
+
+  // Find the sum {f(x0+h/2) + f(x1+h/2) + ... + f(xN-1+h/2)}
+  let temp
+  for (let i = 0; i < N; i++) {
+    temp = func(xi + h / 2)
+    pointsArray.push(temp)
+    xi += h
+  }
+
+  // Calculate the integral
+  let result = h
+  temp = 0
+  for (let i = 0; i < pointsArray.length; i++) temp += pointsArray[i]
+
+  result *= temp
+
+  if (Number.isNaN(result)) { throw Error('Result is NaN. The input interval does not belong to the functions domain') }
+
+  return result
+}
+
+export { integralEvaluation }

Maths/test/MidpointIntegration.test.js

@@ -0,0 +1,16 @@
+import { integralEvaluation } from '../MidpointIntegration'
+
+test('Should return the integral of f(x) = sqrt(x) in [1, 3] to be equal 2.797434', () => {
+  const result = integralEvaluation(10000, 1, 3, (x) => { return Math.sqrt(x) })
+  expect(Number(result.toPrecision(6))).toBe(2.79743)
+})
+
+test('Should return the integral of f(x) = sqrt(x) + x^2 in [1, 3] to be equal 11.46410161', () => {
+  const result = integralEvaluation(10000, 1, 3, (x) => { return Math.sqrt(x) + Math.pow(x, 2) })
+  expect(Number(result.toPrecision(10))).toBe(11.46410161)
+})
+
+test('Should return the integral of f(x) = log(x) + Pi*x^3 in [5, 12] to be equal 15809.9141543', () => {
+  const result = integralEvaluation(20000, 5, 12, (x) => { return Math.log(x) + Math.PI * Math.pow(x, 3) })
+  expect(Number(result.toPrecision(10))).toBe(15809.91415)
+})