taylor.swift

taylor.swift

//: Calculate the value of pi by integrating arctan's Taylor Series


/*:
 arctan(1) - arctan(0) == integral of 1/(1+x^2) from 0 to 1
 arctan(1) - 0 = pi/4
 the sum 1/(1+x^2) is geometric, we can rewrite in the form of Sn = 1/1-r in order to easily write it as a taylor series
 Sn = 1/(1 - (-x^2))
 taylor series for arctan is therefore (-x^2)^n
 = (-1)^n * (x^2n)
 1 - x^2 + x^4 - x^6 + ...
 integrate that, and you get...
 1 - x^3/3 + x^5/5 - x^6/6 + ...
 starting with an index of zero at, the nth term is:
 (-1)^n * x/(2n+1)
 we're integrating from 0 to 1, and the term will end up being zero for x=0, so we can simplify to:
 pi/4 = (-1)^n * 1/(2n+1)
 pi = 4 * ((-1)^n * 1/(2n+1))
*/

import Foundation

var sum = 0.0
var errorAtIndex = [Double]()

for i in 0..<1000 {
    let doubleTypeN = Double(i)
    let value = Double(pow(-1, doubleTypeN)) * Double(1/(2*doubleTypeN + 1)) * 4
    let percentError = abs(sum - .pi) / .pi * 100
    errorAtIndex.append(percentError)
    sum += value
}

print("Calculated value of pi at n=1000: \(sum)")
// "Calculated value of pi at n=1000: 3.140592653839794"

// the swift playground can render this as a graph.
errorAtIndex.map() { $0 }


print("Percent error at n=1000: \(errorAtIndex.last!)")
// "Percent error at n=1000: 0.03186284348822856"

print("Error margin at n=1000: \(abs(.pi - sum))")
// "Error margin at n=1000: 0.000999999749998981"