Continued Fractions and the Modified Lentz's Method

This articles explains how to evaluate continued fractions and offers VBA code for Excel implementation

Continued fractions are of the form below

Continued fractions can be used to express certain numbers, such as π. They are also used in calculating bessel functions of fractional orders. A general method to evaluate continued functions is known as the modified Lentz's method.

The algorithm is consists of the following steps adapted from Press et. al.

1. f₀=b₀. If f₀=0 then let f₀=10^-30 or an extremely small value ≈ 0.
2. C₀=f₀
3. D₀=0
4. For i=1 to a pre-determined number of iterations, e.g. 10000.
   D_i=b_i+a_iD_i-1. If D_i=0, let D_i=10^-30 or an extremely small value ≈ 0.
   C_i=b_i+^a_i/_Ci-1. If C_i=0, let C_i=10^-30 or an extremely small value ≈ 0.
   D_i=¹/_Di
   Chg_i=C_iD_i
   f_i=f_i-1Chg_i
5. Exit when |Chg_i-1| is smaller than a specified threshold.

These steps assume that you should stop when the change in f(x) becomes sufficiently small. By iterating the above twice, it becomes obvious how the algorithm works

The VBA implementation of the algorithm is as follows

Sub lentz()

'Declare your variables

Dim anow As Double, bnow As Double

Dim achange As Double, bchange As Double

Dim Cnow As Double, Dnow As Double

Dim fnow As Double, n As Long, Chgnow As Double

Dim small1 As Double, stop1 As Double

Dim answer As Range

'Set your variables

anow = 1

bnow = 2

achange = 1

bchange = 1

n = 9999

small1 = 10 ^ -30

stop1 = 10 ^ -15

Set answer = Range("B3")

'Start iterating

If bnow = 0 Then

fnow = small1

Else

fnow = bnow

End If

Cnow = fnow

Dnow = 0

For i = 1 To n

anow = anow + achange

bnow = bnow + bchange

Dnow = anow * Dnow + bnow

If Dnow = 0 Then Dnow = small1

Cnow = anow / Cnow + bnow

If Cnow = 0 Then Cnow = small1

Dnow = 1 / Dnow

Chgnow = Cnow * Dnow

fnow = fnow * Chgnow

If Abs(Chgnow - 1) <>

answer.Value = fnow

End

End If

Next i

answer.Value = fnow

End Sub

The code above is also adpated from Press et. al. However, it assumes that a_i and b_i will increase by 1 for each iteration.

References

Numerical Recipes The Art of Scientific Computing 3ed

William H. Press, Saul, A. Teukolsky, William T. Vetterling and Brian P. Flannery 2007

Cambridge University Press

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