Finding the x intercept of a 4th degree polynomial using small increments - vba

Question

I am trying to find the x intercept of a 4th degree function by incrementing the x value. I feel like this way doesnt work always and isnt the most efficient way to do this, is there another way I am missing?
My code is:
Sub Findintercept()
Dim equation As Double, x As Double, A As Double, B As Double, C As Double, D As Double, E As Double
A = 0.000200878
B = -0.002203704
C = 0.0086
D = -0.02333
E = 0.02033
x = 0
equation = A * x ^ 4 + B * x ^ 3 + C * x ^ 2 + D * x + E
While (equation > 0.00001 Or equation < -0.00001)
If (x > 5) Then
MsgBox "Could not find intercept"
equation = 0
Else
x = x + 0.0001
equation = A * x ^ 4 + B * x ^ 3 + C * x ^ 2 + D * x + E
End If
Wend
MsgBox x
End Sub
Sometimes it fails to find the intercept hence the IF condition in the while loop. (Im always expecting the intercept to be less than 5!

Answer 1

Your method suffers from two problems:
You assume a step size to change x. The step could be too large, causing you to "walk past" the value your are looking for. To deal with this, you make a small step size, which can mean an excessively large number of iterations are needed to find the solution.
You always assume the same direction to change x. Even with seemingly small values for your step size, you could "walk past" the solution, and have no means to change direction. Or, your initial guess may be on the wrong side of the solution, and you never find an answer.
The Newton-Raphson method handles both of these issues neatly. You do still need to choose your initial guess somewhat close to the root you are looking for.
This method does have potential problems, but for polynomials such as the one you are dealing with, it is quite good.
Below is a simple VBA sub that implements this method. It solves your problem in 4 iterations. I recommend adjusting the initial guess (xii) a lot to see how it impacts the solution you get.
Sub SimpleNewtonRaphson()
Const Tol As Double = 1E-06
Const MaxIter As Long = 50
Dim xi As Double, xii As Double, deriv As Double
Dim IterCount As Long
' Initialize
xi = 0#
xii = 1#
IterCount = 0
' Method
Do While IterCount < MaxIter And Abs(xii - xi) > Tol
xi = xii
deriv = myDeriv(xi)
If deriv = 0# Then Exit Do
xii = xi - myFunc(xi) / deriv
IterCount = IterCount + 1
Loop
' Results
If deriv = 0 Then MsgBox "Ran into a 0 derivative, modify initial guess"
If IterCount >= MaxIter Then MsgBox "MaxIterations reached"
If Abs(xii - xi) <= Tol Then MsgBox "Solution found #" & vbCrLf & "F(" & xii & ") = " & myFunc(xii)
End Sub
... and two VBA functions for your equation and it's derivative ...
Function myFunc(x As Double) As Double
Const A As Double = 0.000200878
Const B As Double = -0.002203704
Const C As Double = 0.0086
Const D As Double = -0.02333
Const E = 0.02033
myFunc = A * x ^ 4 + B * x ^ 3 + C * x ^ 2 + D * x + E
End Function
Function myDeriv(x As Double) As Double
Const A As Double = 0.000200878
Const B As Double = -0.002203704
Const C As Double = 0.0086
Const D As Double = -0.02333
myDeriv = 4 * A * x ^ 3 + 3 * B * x ^ 2 + 2 * C * x + D
End Function