// RK4th.scala  basic step code
//  RUNGE-KUTTA 4th order step wise integration of
//  ordinary differential equation, given  f1(x,y) = dy/dx f(x,y)
//  and yStart at xStart,  find value  of f(x,y) at  x=xEnd

object RK4th
{
  def main(args: Array[String])
  {
    var k1:Double = 0.0
    var k2:Double = 0.0
    var k3:Double = 0.0
    var k4:Double = 0.0
    var x:Double = 0.0
    var y:Double = 0.0

    println("RK4th.scala running")
    println("f(x,y)=(x-1)(x-2)(x-3)(x-4)")
    println("RK4th, enter xStart, yStart, xEnd, h")
    var xStart:Double = 0.0
    var yStart:Double = 24.0
    var xEnd:Double = 5.0
    var yEnd:Double = 0.0
    var h:Double = 0.25 // power of 2 needed for near exact solution
    println("xStart= "+xStart)
    println("yStart= "+yStart)
    println("xEnd= "+xEnd)
    println("step size h= "+h)

    // set initial conditions
    x=xStart
    y=yStart
    while(x<xEnd)  // basic iteration, x incremented by step size h
    {
      k1=h*F1(x,y)
      k2=h*F1(x+h/2,y+k1/2)
      k3=h*F1(x+h/2,y+k2/2)
      k4=h*F1(x+h,y+k3)
      y=y+(k1+2*k2+2*k3+k4)/6 // 4th order estimate of y
      x=x+h
      println("x="+x+"    y="+y)
    }
    xEnd=x
    yEnd=y
    println("yEnd= "+yEnd+"  at xEnd="+x)
    println("y=0 expected at x=1,2,3,4")
    println("yEnd expected to be 24 at xEnd=5")
    
  }

  def F1(x:Double, y:Double):Double=
  {
    // derivitive of (x-1)(x-2)(x-3)(x-4)
    return 4.0*x*x*x - 30.0*x*x + 70.0*x - 50.0
  }

} // end RK4th.scala