#### Simpson Integration

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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
*
*
*/

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 doesn't belong to the functions domain") }

return result
}

export { integralEvaluation }
```  