fixedPoint

package example
import math._

object week2Application {

  val tolerance = 0.0001                          //> tolerance  : Double = 1.0E-4

  def isCloseEnough(x: Double, y: Double) =
    abs((x - y) / x) < tolerance                  //> isCloseEnough: (x: Double, y: Double)Boolean

  def fixedPoint(f: Double => Double)(firstGuess: Double) = {

    def iterate(guess: Double): Double = {
      val next = f(guess)
      if (isCloseEnough(guess, next)) next
      else iterate(next)
    }
    iterate(firstGuess)
  }                                               //> fixedPoint: (f: Double => Double)(firstGuess: Double)Double

  fixedPoint(x=>1+x/2)(1.0)                       //> res0: Double = 1.9998779296875

  def avgDamp(f:Double=>Double)(x:Double)=(f(x)+x)/2
                                                  //> avgDamp: (f: Double => Double)(x: Double)Double

  def sqrt(x:Double):Double=
  fixedPoint(avgDamp(y=>x/y))(1.0)                //> sqrt: (x: Double)Double

 sqrt(2)                                          //> res1: Double = 1.4142135623746899
}