

Title  Use Newton's method on the equation Z^n  1 to draw fractals in Visual Basic 6 
Description  This example shows how to use Newton's method on the equation Z^n  1 to draw fractals in Visual Basic 6. 
Keywords  Newton's method, polynomial, root, function, fractal 
Categories  Algorithms, Graphics 


Newton's method calculates the roots of equations. In other words, it finds the values of X for which F(X) = 0. For more information on the part of the code that calculates Newton's method, see Use Newton's method to find the roots of an equation.
To draw a fractal, the program applies Newton's method to complex numbers in a region of the complex plane. It then colors each point in the region to indicate the root of the equation to which Newton's method leads for that point.
For example, the equation Z^4  1 has four roots (values of Z that make the equation zero): 1, 1, i, and i.
To make the calculations easier to understand, the program uses the following Complex class.


Option Explicit
Public Re As Double
Public Im As Double
Public Function Clone() As Complex
Set Clone = NewComplex(Re, Im)
End Function
Public Sub Initialize(ByVal new_re As Double, ByVal new_im _
As Double)
Re = new_re
Im = new_im
End Sub
Public Function MagnitudeSquared() As Double
MagnitudeSquared = Re * Re + Im * Im
End Function
Public Function Magnitude() As Double
Magnitude = Sqr(MagnitudeSquared())
End Function
Public Function Times(ByVal x As Complex) As Complex
Set Times = NewComplex(Re * x.Re  Im * x.Im, Re * x.Im _
+ Im * x.Re)
End Function
Public Function TimesComponents(ByVal new_re As Double, _
ByVal new_im As
Double) As Complex
Set TimesComponents = NewComplex(Re * new_re  Im * _
new_im, Re * new_im
+ Im * new_re)
End Function
Public Function Plus(ByVal x As Complex) As Complex
Set Plus = NewComplex(Re + x.Re, Im + x.Im)
End Function
Public Function PlusComponents(ByVal new_re As Double, _
ByVal new_im As
Double) As Complex
Set PlusComponents = NewComplex(Re + new_re, Im + _
new_im)
End Function
Public Function DividedBy(ByVal denominator As Complex) As _
Complex
Dim conjugate As Complex
Dim numerator As Complex
Set conjugate = denominator.Clone
conjugate.Im = conjugate.Im
Set denominator = denominator.Times(conjugate)
Set numerator = Me.Times(conjugate)
numerator.Re = numerator.Re / denominator.Re
numerator.Im = numerator.Im / denominator.Re
Set DividedBy = numerator
End Function
Public Function IsCloseTo(ByVal x As Complex) As Boolean
IsCloseTo = (Abs(Re  x.Re) + Abs(Im  x.Im) < 0.001)
End Function
Public Function ToString() As String
If Im < 0 Then
ToString = Format$(Re) & "  " & Format$(Abs(Im)) & _
"i"
Else
ToString = Format$(Re) & " + " & Format$(Im) & "i"
End If
End Function


To make initializing Complex objects easier, it also uses the following factory function.


Public Function NewComplex(ByVal new_re As Double, ByVal _
new_im As Double)
As Complex
Dim result As Complex
Set result = New Complex
result.Initialize new_re, new_im
Set NewComplex = result
End Function


The following code shows how the program calculates the function Z^N  1 and its derivative N * Z^(N  1). Here the power N is given by the constant NUM_ROOTS (there are N roots to the equation Z^N  1).


' The function.
' F(x) = x^NUM_ROOTS  1.
Private Function F(ByVal x As Complex) As Complex
Dim result As Complex
Dim i As Integer
Set result = x.Clone() ' x.
' x^NUM_ROOTS.
For i = 1 To NUM_ROOTS  1
Set result = result.Times(x)
Next i
' x^NUM_ROOTS  1.
Set F = result.PlusComponents(1, 0)
End Function
' The function's derivative.
' dFdx(x) = NUM_ROOTS * x^(NUM_ROOTS  1).
Private Function dFdx(ByVal x As Complex) As Complex
Dim result As Complex
Dim i As Integer
Set result = x.Clone() ' x.
' x^(NUM_ROOTS  1).
For i = 1 To NUM_ROOTS  2
Set result = result.Times(x)
Next i
' NUM_ROOTS * x^(NUM_ROOTS  1).
Set dFdx = result.TimesComponents(NUM_ROOTS, 0)
End Function


The following code draws the fractal. The code loops over the points in an area and uses Newton's method to find a root of the equation starting with that point as its initial guess. It then looks through a list of the equation's roots to see which one it found and plots the point with a corresponding color.


' Draw the fractal.
Private Sub DrawFractal()
Const CUTOFF As Single = 0.0000001 ' Work until the
' epsilon squared < this.
Const MAX_ITER As Integer = 200 ' At most this many
' iterations.
Dim clr As Integer
Dim x As Complex
Dim x0 As Complex
Dim epsilon As Complex
Dim dx As Double
Dim dy As Double
Dim wid As Double
Dim hgt As Double
Dim iter As Integer
Dim r As Integer
Dim i As Integer
Dim j As Integer
DrawMode = vbCopyPen
Line (0, 0)(ScaleWidth, ScaleHeight), BackColor, BF
MousePointer = vbHourglass
DoEvents
' Adjust the aspect ratio.
AdjustAspect
' dx is the change in the real part
' (X value) for C. dy is the change in the
' imaginary part (Y value).
wid = ScaleWidth
hgt = ScaleHeight
dx = (VisibleXmax  VisibleXmin) / (wid  1)
dy = (VisibleYmax  VisibleYmin) / (hgt  1)
' Calculate the values.
Set x0 = NewComplex(Xmin, 0)
For i = 1 To wid
x0.Im = Ymin
For j = 1 To hgt
Set x = x0
iter = 0
Do
iter = iter + 1
If iter > MAX_ITER Then Exit Do
Set epsilon = _
F(x).DividedBy(dFdx(x)).TimesComponents(1, _
0)
Set x = x.Plus(epsilon)
Loop While epsilon.MagnitudeSquared() > CUTOFF
' Set which root this is.
If iter > MAX_ITER Then
clr = 0 ' Black.
Else
' Find the root.
clr = 1 ' White.
For r = 0 To UBound(m_Roots)
If x.IsCloseTo(m_Roots(r)) Then
clr = r + 2 ' +2 to skip black and
' white.
Exit For
End If
Next r
End If
' Set the pixel's color.
PSet (i, j), m_Colors(clr)
' Move to the next point.
x0.Im = x0.Im + dy
Next j
x0.Re = x0.Re + dx
DoEvents
Next i
Refresh
Picture = Image
MousePointer = vbCrosshair
DrawMode = vbInvert
End Sub


See the code for additional details.
Unfortunately this code is quite slow because of all of the heavylifting it does manipulating Complex objects. The VB .NET version is much faster.
For more information on Newton's method, see Eric W. Weisstein's article Newton's Method from MathWorldA Wolfram Web Resource.





