GY-63_MS5611/libraries/statHelpers/README.md

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# statHelpers

Arduino library with a number of statistic helper functions.

## Description

This library contains functions that have the goal to help with 
some basic statistical calculations.


## Functions

### Permutation

returns how many different ways one can choose a set of k elements 
from a set of n. The order does matter. 
The limits mentioned is the n for which all k still work.

- **uint32_t permutations(n, k)** exact up to 12
- **uint64_t permutations64(n, k)** exact up to 20
- **double dpermutations(n, k)** not exact up to 34 (4 byte) or 170 (8 byte)

If you need a larger n but k is near 0 the functions will still work, but 
to which k differs per value for n. (no formula found, and an overflow
detection takes overhead).


- **nextPermutation<Type>(array, size)** given an array of type T it finds the next permutation
of that array in a lexicographical way.  ABCD --> ABDC. 
Based upon // http://www.nayuki.io/page/next-lexicographical-permutation-algorithm although 
other same code examples exist.


### Factorial

- **uint32_t factorial(n)** exact up to 12!
- **uint64_t factorial64(n)** exact up to 20!  (Print 64 bit ints with my printHelpers)
- **double dfactorial(n)** not exact up to 34! (4 byte) or 170! (8 byte)
- **double stirling(n)** approximation function for factorial (right magnitude)

**dfactorial()** is quite accurate over the whole range.
**stirling()** is up to 3x faster for large n (> 100), 
but accuracy is less than the **dfactorial()**, see example.


### SemiFactorial

- **uint32_t semiFactorial(n)** exact up to 20!!
- **uint64_t semiFactorial64(n)** exact up to 33!!  (Print 64 bit integers with my printHelpers)
- **double dSemiFactorial(n)** not exact up to 56!! (4 byte) or 300!! (8 byte)

SemiFactorial are defined for
- **odd** values:  n x (n-2) x (n-4) ... x 1
- **even** values: n x (n-2) x (n-4) ... x 2

Notes:  
```n! = n!! x (n-1)!!``` this formula allows to calculate the value of n! indirectly


### Combinations

returns how many different ways one can choose a set of k elements 
from a set of n. The order does not matter. 
The number of combinations grows fast so n is limited per function.
The limits mentioned is the n for which all k still work.

- **uint32_t combinations(n, k)**   n = 0 .. 30 (iterative version)
- **uint64_t combinations64(n, k)**  n = 0 .. 61 (iterative version)
- **uint32_t rcombinations(n, k)**    n = 0 .. 30 (recursive version, slightly slower)
- **uint64_t rcombinations64(n, k)**  n = 0 .. 61 (recursive version, slightly slower)
- **double dcombinations(n, k)**  n = 0 .. 125 (4bit)  n = 0 .. 1020 (8 bit) 

If you need a larger n but k is near 0 or near n the functions will still work, 
but for which k differs per value for n. (no formula found, and an overflow
detection takes overhead). 


- **combPascal(n, k)** n = 0 .. 30 but due to double recursion per iteration it takes
time and a lot of it for larger values. Added for recreational purposes, limited tested.


## Notes

- **perm1** is a sketch in the examples that shows a recursive permutation 
algorithm. It generates all permutations of a given char string. 
It allows you to process every instance.
It is added to this library as it fits in the context.


### Experimental - large numbers

- **void bigFactorial(uint32_t n, double &mantissa, uint32_t &exponent)** returns a double mantissa between 0 and 10, and an integer exponent. 
- **void bigPermutation(uint32_t n, uint32_t k, double &mantissa, uint32_t &exponent)** returns a double mantissa between 0 and 10, and an integer exponent. 
- **void bigCombination(uint32_t n, uint32_t k, double &mantissa, uint32_t &exponent)** returns a double mantissa between 0 and 10, and an integer exponent. 

An experimental **bigFactorial(n)** calculation to get an idea of the big numbers. it can calculate factorials up to an exponent of 4294967295 max.  100.000.000! can be done in 38 minutes on an ESP32 @240 Mhz.  Maximum value for n is **518678059! ==  2.1718890e4294967292** a number that took ~10 hrs to calculate. 

An experimental **bigPermutation(n, k)** calculation, to handle big numbers too. Maximum value for n is **518678059** to have full range support. For small(er) values of k, n can even be much larger, but not larger than 4294967295 max.

An experimental **bigCombination(n, k)** calculation for big numbers. Not investigated what its maximum value is, but it should be higher than **518678059** as the number of combinations is always smaller than number of permutations.


#### Experimental - not investigated yet

To have support for huge numbers one could upgrade the code to use uint64_t as parameter and internally but calculating these values could take a lot of time, although **bigPermutations64(n, k)** and **bigCombinations64(n, k)** would work fast for small values of k. 

- **void bigFactorial64(uint64_t n, double &mantissa, uint64_t &exponent)** returns a double mantissa between 0 and 10, and an integer exponent. 
- **void bigPermutation64(uint64_t n, uint64_t k, double &mantissa, uint64_t &exponent)** returns a double mantissa between 0 and 10, and an integer exponent. 
- **void bigCombination64(uint64_t n, uint64_t k, double &mantissa, uint64_t &exponent)** returns a double mantissa between 0 and 10, and an integer exponent. 



## Future

- code & example for get Nth Permutation
- investigate valid range detection for a given (n, k) for combinations and permutations.
- investigate a bigFloat class to do math for permutations and combinations to substantially larger values.


## Operation

See examples