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use std::f64::{consts::E, NAN};
use crate::{Elementary::*, Integrate, EULER_MASCHERONI};
/// An infinit number in the complex plane with an unknown or undefined complex argument.
///
/// For instance will approach complex infinity as x approaches 0. The limit could be either plus
/// or minus infinity wich makes its magnitude infinit and its complex argument undefined.
///
/// See [this article](https://mathworld.wolfram.com/ComplexInfinity.html) for further information.
pub const COMPLEX_INFINITY: f64 = NAN;
/// returns n! for numbers n ∈ ℕ
fn factorial_integer(numb: u128) -> u128 {
if numb == 0 {
1
} else {
let mut res: u128 = 1;
for i in 1..=numb {
res *= i;
}
res
}
}
// TODO: make the gamma function work for values < 1
/// Returns the value of 𝜞(z) as defined by ∫t^(z-1)e^(-t)dt evaluated from 0 to ∞.
pub fn gamma_function(z: f64) -> f64 {
let inner_funciton = Mul(
Pow(X.into(), Sub(Con(z).into(), Con(1.).into()).into()).into(),
Pow(Con(E).into(), Mul(X.into(), Con(-1.).into()).into()).into(),
);
// for whole numbers
if z.fract() == 0.0 {
// fraction part of the number is zero, meaning that the number is an integer
inner_funciton
.integrate()
.set_lower_bound(0.)
.set_upper_bound(100.)
.set_precision(1000)
.evaluate()
.unwrap()
.round_to(0)
} else {
inner_funciton
.integrate()
.set_lower_bound(0.)
.set_upper_bound(100.)
.set_precision(100000)
.evaluate()
.unwrap()
}
}
/// the polygamma function 𝛙m(z) describes the relationship between 𝜞(z) and its derivatives. For instance 𝛙0(z) =
/// 𝜞'(z)/𝜞(z). [See article](https://en.wikipedia.org/wiki/Polygamma_function)
pub fn polygamma_function(z: f64, m: usize) -> f64 {
if m == 0 {
digamma_function(z)
} else {
let inner_funciton = Pow(Log(Con(E).into(), X.into()).into(), Con(m as f64).into())
* Pow(X.into(), Con(z - 1.).into())
/ (Con(1.) - X);
-inner_funciton
.integrate()
.set_lower_bound(1e-10)
.set_upper_bound(1. - 1e-10)
.set_precision(10)
.evaluate()
.unwrap()
}
}
/// Special case of the polygamma function 𝛙m(z) where m=0, The integral definition of the function
/// then changes. [See article](https://en.wikipedia.org/wiki/Digamma_function#Integral_representations)
pub fn digamma_function(z: f64) -> f64 {
let inner_funciton = (Con(1.) - Pow(X.into(), Con(z - 1.).into())) / (Con(1.) - X);
let integral_value = inner_funciton
.integrate()
.set_lower_bound(0.)
.set_upper_bound(1. - 1e-10)
.set_precision(1000)
.evaluate()
.unwrap();
integral_value - EULER_MASCHERONI
}
/// Allows the usage of factorials i.e. `x!`
/// usually defined as:
/// * x! = x*(x-1)!
/// * 0! = 1
///
/// for x ∈ ℕ.
///
/// The definition of the factorial function can be expanded to the domain of all real numbers
/// using the [gamma function](https://en.wikipedia.org/wiki/Gamma_function):
///
/// x! = 𝜞(x+1)
///
pub trait Factorial {
type Output;
fn factorial(&self) -> Self::Output;
}
/// implement factorial method for natural number types
macro_rules! impl_factorial_natural {
(for $($t:ty), +) => {
$(impl Factorial for $t {
type Output = u128;
fn factorial(&self) -> Self::Output {
factorial_integer(self.clone() as u128)
}
})*
};
}
impl_factorial_natural!(for u8, u16, u32, u64, u128, usize);
/// implement factorial for integer number types
macro_rules! impl_factorial_integer {
(for $($t:ty), +) => {
$(impl Factorial for $t {
type Output = f64;
fn factorial(&self) -> Self::Output {
if self.is_negative() {
// the continuos factorial funciton approaches ±∞ (complex infinity) for any
// negative integer
NAN
} else {
factorial_integer(self.clone() as u128) as f64
}
}
})*
};
}
impl_factorial_integer!(for i8, i16, i32, i64, i128, isize);
/// implement factorial method for float types
macro_rules! impl_factorial_float {
(for $($t:ty), +) => {
$(impl Factorial for $t {
type Output = Self;
fn factorial(&self) -> Self::Output{
gamma_function(*self as f64 + 1.) as Self
}
})*
}
}
impl_factorial_float!(for f32, f64);
/// Allows the usage of rounding methods that are more specific than rust std's round() method.
pub trait Round {
/// Rounds self (a number) to the given number of decimal places. This method is
/// mainly made for the f32 and f64 types since integer types already have no decimal places.
///
/// Example:
/// ```rust
/// assert_eq!(23.3274.round_to(2), 23.33);
///
/// assert_eq!((1. / 3.).round_to(5), 0.33333);
///
/// // For integer types, rounding to a decimal point is the same as casting it to f64
/// assert_eq!(100_u8.round_to(10), 100.);
/// ```
fn round_to(&mut self, decimal_places: u32) -> f64;
/// Rounds self (a number) to the given number of significant figures.
/// Example:
/// ```rust
/// assert_eq!(14912387964_u128.with_significant_figures(5), 14912000000);
///
/// assert_eq!(-4095_i32.with_significant_figures(1), -4000);
///
/// assert_eq!(1234.5678_f64.with_significant_figures(6), 1234.57)
/// ```
fn with_significant_figures(&mut self, digits: u64) -> Self;
}
macro_rules! impl_round_float {
(for $($t:ty), +) => {
$(impl Round for $t {
fn round_to(&mut self, decimal_places: u32) -> f64 {
let mut value = *self as f64;
// move the decimal point
value *= 10_usize.pow(decimal_places) as f64;
// round to an integer
value = value.round();
// move the decimal point back
value /= 10_usize.pow(decimal_places) as f64;
// give self the value of the rounded number
*self = value as Self;
value
}
fn with_significant_figures(&mut self, digits: u64) -> Self {
let value = if *self >= 0. {
let order = (*self).log10().trunc() as i32;
if digits as i32 <= order {
((*self) as isize).with_significant_figures(digits) as Self
} else {
if *self >= 1. {
(*self * (10 as Self).powi((digits as i32 - order -1) as i32)).round() / (10 as Self).powi((digits as i32 - order -1) as i32)
} else {
(*self * (10 as Self).powi((digits as i32 - order) as i32)).round() / (10 as Self).powi((digits as i32 - order) as i32)
}
}
} else {
-1. *(*self *-1.).with_significant_figures(digits)
};
*self = value as Self;
value
}
})*
};
}
impl_round_float!(for f32, f64);
macro_rules! impl_round_int {
(for $($t:ty), +) => {
$(impl Round for $t {
#[allow(unused_variables)]
fn round_to(&mut self, decimal_places: u32) -> f64 {
*self as f64
}
fn with_significant_figures(&mut self, digits: u64) -> Self {
// move the decimal point to the appropriate spot so that we can round and then
// move it back
let order = (*self).ilog10() as u64;
let new_value = ((*self) as f64 / 10_f64.powi((order - digits +1) as i32)).round() * 10_f64.powi((order - digits +1) as i32);
// set new value
*self = new_value as Self;
*self
}
})*
};
}
impl_round_int!(for u8, u16, u32, u64, u128, usize, i8, i16, i32, i64, i128, isize);