

Title  Use Newton's method to find the roots of an equation in Visual Basic .NET 
Description  This example shows how to use Newton's method to find the roots of an equation in Visual Basic .NET. 
Keywords  Newton's method, equation, root, zero 
Categories  Algorithms, Graphics, VB.NET 


Newton's method calculates the roots of equations. In other words, it finds
the values of X for which F(X) = 0.
This program graphs the equation X^3/3  2*X + 5. When you click on the
graph, it uses Newton's method to find a root of the equation, starting from
the X value that you clicked.


' The function.
Private Function F(ByVal x As Double) As Single
Return x * x * x / 3  2 * x * x + 5
End Function
' Draw the background graph.
Private Sub DrawGraph(ByVal gr As Graphics)
gr.Clear(Me.BackColor)
' Scale for convenience.
gr.ScaleTransform(SCALE_FACTOR, SCALE_FACTOR)
gr.TranslateTransform( _
Me.ClientRectangle.Width / 2, _
Me.ClientRectangle.Height / 2, _
Drawing2D.MatrixOrder.Append)
' Draw the axes.
Dim blue_pen As New Pen(Color.Blue, 0)
For x As Integer = 10 To 10
gr.DrawLine(blue_pen, x, 0.5!, x, 0.5!)
Next x
For y As Integer = 10 To 10
gr.DrawLine(blue_pen, 0.5!, y, 0.5!, y)
Next y
gr.DrawLine(blue_pen, 10, 0, 10, 0)
gr.DrawLine(blue_pen, 0, 10, 0, 10)
blue_pen.Dispose()
' Draw the curve.
Dim red_pen As New Pen(Color.Red, 0)
Dim y1 As Single
Dim y2 As Single = F(10)
Const STEP_SIZE As Single = 0.1
For x As Single = 10 + STEP_SIZE To 10 Step STEP_SIZE
y1 = y2
y2 = F(x)
gr.DrawLine(red_pen, x  STEP_SIZE, y1, x, y2)
Next x
red_pen.Dispose()
End Sub


The way Newton's method works is it starts from an initial guess X0 (given
by the point you clicked). It then estimates the difference between that
value and root:
epsilon = F(x) / dFdx(x)
The method then sets its next guess for x to be the current value plus
epsilon (X(n+1) = X(n) + epsilon). It repeats this process until epsilon is
smaller than some cutoff value.


' The function's derivative.
Private Function dFdx(ByVal x As Double) As Single
Return x * x  4 * x
End Function
' Find the roots by using Newton's method.
Private Sub Form1_MouseDown(ByVal sender As Object, ByVal e _
As
System.Windows.Forms.MouseEventArgs) Handles _
MyBase.MouseDown
' Clear previous results.
Dim gr As Graphics = Me.CreateGraphics()
DrawGraph(gr)
' Convert the X coordinate into world coordinates.
Dim m As Matrix = gr.Transform()
m.Invert()
Dim pts() As PointF = {New PointF(e.X, e.Y)}
m.TransformPoints(pts)
Dim x0 As Single = pts(0).X
Const CUTOFF As Single = 0.0000001
Dim epsilon As Single
Dim the_pen As New Pen(Color.Black, 0)
Dim iterations As Integer = 0
Do
iterations += 1
gr.DrawLine(the_pen, x0, 1.5!, x0, 1.5!)
epsilon = F(x0) / dFdx(x0)
x0 = x0 + epsilon
Loop While Abs(epsilon) > CUTOFF
gr.DrawLine(the_pen, x0, 3.0!, x0, 3.0!)
the_pen.Dispose()
Me.Text = x0 & " +/" & epsilon & " in " & iterations & _
" iterations"
End Sub


Newton's method uses the function's derivative to make its next X value
guess. If the curve is relatively flat for the current value of X, then the
derivative (which is the slope of the function) gives a value far away from
the current one and the method is unstable. In this example, try clicking on
a point where X is at or near 0 and see what happens.
For more information on Newton's method, see Eric W. Weisstein's article
Newton's Method from MathWorldA Wolfram Web Resource.





