Magnets.swift

func positionsOfZeroNetForce(_ magnetPositions: [Double]) -> [Double] {
    
    let threshold = 0.0000000000001
    
    // Utility function to give 1/fn(m0) + 1/fn(m1) + ... + 1/fn(mn)
    // Used for working out the force at x, and its derivitive
    func sumReciprocals(_ fn: (Double)->(Double)) -> Double {
        return magnetPositions.map(fn).map {
            guard $0 != 0 else { return Double.nan }
            return 1.0/$0
            }.reduce(0, +)
    }
    
    let pairsOfMagnetPositions = (0..<(magnetPositions.count-1)).map {
        (m0: magnetPositions[$0], m1: magnetPositions[$0+1])
    }
    
    return pairsOfMagnetPositions.map {
        // Newton's method for successively approximating roots of a function
        // https://en.wikipedia.org/wiki/Newton%27s_method
        var x = $0.0+($0.1-$0.0)/2 // Initial approximation = halfway between positions
        var accuracy = threshold*2
        repeat {
            let forceAtX = sumReciprocals { $0-x }
            let derivitiveOfForceAtX = sumReciprocals { ($0-x)*($0-x) }
            let xn = x - forceAtX/derivitiveOfForceAtX
            accuracy = abs(xn - x)
            x = xn
        } while threshold < accuracy
        return x
    }
}