Relationship of Continued Fraction and Difference Equation

By Kardi Teknomo, PhD.

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You can use the bottom up approach in MS Excel by calculating , then , then copy from to 1 and the final result is .

For example of bottom up approach, see section how do we compute continued fraction . To use MS Excel, download the MS Excel sheet companion of this tutorial

Another better approach is using top down approach where the continued fraction is expressed as ratio of two difference equations

Where

, with two alternative initial conditions

1. , , , and the difference equation work for , or
2. alternatively, we can use dummy variables , , , , and the difference equation work for

The two-difference equations actually produce the same sequence for

Example:

To compute using top down approach we can use the first initial conditions and produce

Using the second initial condition, we obtain the same results

See the MS Excel sheet companion of this tutorial and you can try any sequence of continued fraction numbers.

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi. Continued Fraction. http://people.revoledu.com/kardi/tutorial/ContinuedFraction/index.html