# numlib::contfracPeriodic

Periodic continued fraction expansions

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```numlib::contfracPeriodic(`p`, `q`, `n`)
```

## Description

`numlib::contfracPeriodic(p, q, n)` returns the continued fraction expansion of `p + q*sqrt(n)` as a sequence of two lists: the first one contains the non-periodic part, the second one contains the periodic part of the expansion.

The non-periodic part may be an empty list. No periodic part is returned for rational input, i.e., q = 0 or `n` square.

## Examples

### Example 1

The non-periodic part may start with zero. All other coefficients of a continued fraction expansion are positive:

`numlib::contfracPeriodic(2/7, 1/7, 2)`

The result agrees with that one of `contfrac`:

`op(contfrac(2/7 + 1/7 *sqrt(2)), 1)`

### Example 2

The golden mean is famous for its simple continued fraction expansion:

`numlib::contfracPeriodic(1/2, 1/2, 5)`

### Example 3

Since 81 is a perfect square, there is no periodic part in the continued fraction expansion of its square root:

`numlib::contfracPeriodic(0, 1, 81)`

## Parameters

 `p` A rational number `q` A rational number `n` A positive integer

## Return Values

If is a rational number, then `numlib::contfracPeriodic` returns one list, otherwise two lists of integers.

## Algorithms

A real number has a periodic continued fraction expansion if and only if it is of the form .