New Appoach for Random Number Generation???

[Namir]

Hi All!

I thought of a new class of random numbers! Pick two different approximations of the same function, call them f1(x) and f2(x). Perform the following:

Code:

Using a seed of X in [0,1].
Calculate f1(x) and f2(x).
Calculate D = Abs(f1(x) – f2(x))
Calculate L = fix(log10(Diff)) to remove factional part
Calculate x = Diff / 10 ^ L
Calculate x = Frac(10000 * x)

Functions f1(x) and f2(x) should NOT return extremely small or ig values for argumens in [0, 1]. The last calculation step above obtains some numerical noise suitable as random values.

Here is sample code in Excel VBA using two approximations for the Normal CDF:

Code:

Option Explicit

Function Frac(ByVal x As Double) As Double
  Frac = x - Fix(x)
End Function

Function phi3(ByVal x As Double) As Double
' Page 1977
  Dim y As Double, Pi As Double
  Dim K As Double
  
  Pi = 4 * Atn(1)
  K = Sqr(2 / Pi)
  y = K * x * (1 + 0.44715 * x * x)
  
  phi3 = 0.5 * (1 + WorksheetFunction.Tanh(y))
End Function

Function phi4(ByVal x As Double) As Double
' Hammaker 1978
  Dim y As Double, Pi As Double
  Dim K As Double
  
  Pi = 4 * Atn(1)
  K = Sqr(2 / Pi)
  y = 0.806 * x * (1 - 0.018 * x)
  
  phi4 = 0.5 * (1 + -Sqr(1 - Exp(-y * y)))
End Function

Function Rand34(ByVal x As Double) As Double
  Dim Diff As Double
  Dim L As Integer
  
  Diff = Abs(phi3(x) - phi4(x))
  L = Fix(Log(Diff) / Log(10))
  x = Diff / 10 ^ L
  x = Frac(10000 * x)
  Rand34 = x
End Function

Sub go()
  Const max = 10000
  Dim x As Double
  Dim I As Integer
  
  Randomize Timer
  x = Rnd(1)
  For I = 1 To max
    DoEvents
    If x = 0 Then x = Frac(Log(4 * Atn(1) + I) / Log(10))
    x = Rand34(x)
    Cells(I, 1) = x
  Next I
End Sub

The resulting numbers have a mean close to 0.5 and standard deviation close to 0.28.

You can use simpler approximations for the Normal or other functions. Ultimately, your choices for f1(x) and f2(x) are HUGE!!!

Namir