# Boring Math Library - Recursive functions
The purpose of this project is to explore different ways to efficiently
implement recursive functions in Python.
* Recursive functions
* [Ackermann Function](#ackermann-s-function)
* [Fibonacci Sequences](#fibonacci-sequences)
* CLI applications
* [Ackermann CGI programs](#ackermann-cgi-scripts)
Part of the "Boring Math" PyPI **bm** namespace for mathematical hobby
projects.
* **Repositories**
* [bm.recursive-functions][1] project on *PyPI*
* [Source code][2] on *GitHub*
* **Detailed documentation**
* [Detailed API documentation][3] on *GH-Pages*
### Recursive functions
#### Ackermann's Function
* Function **ackermann_list**(m: int, n: int) -> int
* an example of a total computable function that is not primitive recursive
* becomes numerically intractable after `m=4`
* see CLI section below for mathematical definition
#### Fibonacci Sequences
* Function **fibonacci**(f0: int=0, f1: int=1) -> Iterator[int]
* returns a *Fibonacci* sequence iterator where
* `f(0) = f0` and `f(1) = f1`
* `f(n) = f(n-1) + f(n-2)`
* yield defaults to `0, 1, 1, 2, 3, 5, 8, 13, 21, ...`
* rev_fibonacci**(f0: int=0, f1: int=1) -> Iterator[int]
* returns a *Reverse Fibonacci* sequence iterator where
* `rf(0) = f0` and `rf(1) = f1`
* `rf(n) = rf(n-1) - rf(n-2)`
* `rf(0) = fib(-1) = 1`
* `rf(1) = fib(-2) = -1`
* `rf(2) = fib(-3) = 2`
* `rf(3) = fib(-4) = -3`
* `rf(4) = fib(-5) = 5`
* yield defaults to `1, -1, 2, -3, 5, -8, 13, -21, ...`
---
## CLI Applications
Implemented in an OS and package build tool independent way via the
project.scripts section of pyproject.toml.
### Ackermann CLI scripts
Ackermann, a student of Hilbert, discovered early examples of totally
computable functions that are not primitively recursive.
A [fairly standard][4] definition of the Ackermann function is
recursively defined for `m,n >= 0` by
```
ackermann(0,n) = n+1
ackermann(m,0) = ackermann(m-1,1)
ackermann(m,n) = ackermann(m-1, ackermann(m, n-1))
```
#### CLI program **ackermann_list**
* Given two non-negative integers, evaluates Ackermann's function
* Implements the recursion via a Python array
* **Usage:** `ackerman_list m n`
---
[1]: https://pypi.org/project/bm.recursive-functions/
[2]: https://github.com/grscheller/bm-recursive-functions/
[3]: https://grscheller.github.io/boring-math-docs/recursive-functions/
[4]: https://mathworld.wolfram.com/AckermannFunction.html
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