merge: Created composite Simpson's integration method. Tests included. (#819)

* Created composite Simpson's integration method.Tests included

* Minor corrections

* Auto-update DIRECTORY.md

* Styled with standard.js

* chore: remove blank line

* chore: remove blank line

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

Maths/SimpsonIntegration.js

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+/*
+*
+* @file
+* @title Composite Simpson's rule for definite integral evaluation
+* @author: [ggkogkou](https://github.com/ggkogkou)
+* @brief Calculate definite integrals using composite Simpson's numerical method
+*
+* @details The idea is to split the interval in an EVEN 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/3 * {f(x0) + 4*f(x1) + 2*f(x2) + ... + 2*f(xN-2) + 4*f(xN-1) + f(xN)}
+*
+* That means that the first and last indexed i f(xi) are multiplied by 1,
+* the odd indexed f(xi) by 4 and the even by 2.
+*
+* N must be even number and a<b. By increasing N, we also increase precision
+*
+* More info: [Wikipedia link](https://en.wikipedia.org/wiki/Simpson%27s_rule#Composite_Simpson's_rule)
+*
+*/
+
+function integralEvaluation (N, a, b, func) {
+  // Check if N is an even integer
+  let isNEven = true
+  if (N % 2 !== 0) isNEven = false
+
+  if (!Number.isInteger(N) || Number.isNaN(a) || Number.isNaN(b)) { throw new TypeError('Expected integer N and finite a, b') }
+  if (!isNEven) { throw Error('N is not an even number') }
+  if (N <= 0) { throw Error('N has to be >= 2') }
+
+  // Check if a < b
+  if (a > b) { throw Error('a must be less or equal than b') }
+  if (a === b) return 0
+
+  // 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) + 4*f(x1) + 2*f(x2) + ... + 2*f(xN-2) + 4*f(xN-1) + f(xN)}
+  let temp
+  for (let i = 0; i < N + 1; i++) {
+    if (i === 0 || i === N) temp = func(xi)
+    else if (i % 2 === 0) temp = 2 * func(xi)
+    else temp = 4 * func(xi)
+
+    pointsArray.push(temp)
+    xi += h
+  }
+
+  // Calculate the integral
+  let result = h / 3
+  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 doesnt belong to the functions domain') }
+
+  return result
+}
+
+export { integralEvaluation }