Newton-Raphson Square Roots

gistfile1.scala

/**
 *
 * Newton-Raphson Square Roots
 * as is shown in "why functional programming matters"
 * http://www.cs.utexas.edu/~shmat/courses/cs345/whyfp.pdf
 */

object NewtonRoots {

  //algorithm to approach the square root of n
  def next(n:Double)(x:Double) = (x + n/x)/2

  //create a stream composed of repeatedly applying f
  def repeat(f:Double=>Double, a:Double):Stream[Double] = a #:: repeat(f,f(a))

  //look for a good enough match in the stream
  def within(eps:Double, s:Stream[Double]):Double = s match {
    case a #:: b #:: rest => if( (a-b).abs < eps) b else within(eps, b#::rest)
  }

   //look for a good enough match in the stream.
   // A better way is to see how close a/b is to 1,
   // to avoid the rounding errors
  def relative(eps:Double, s:Stream[Double]):Double = s match {
    case a #:: b #:: rest => if( (a/b -1).abs < eps) b else within(eps, b#::rest)
  }

  //find the square root of 3 starting from the approximation
  // value 1, until approximations are within 0.01
  def main(args: Array[String]) {
    println( within(0.01, repeat( next(3), 1 ) ) )
  }

}