1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295
/****************************************************************************** * Copyright 2019 Manuel Simon * This file is part of the norman library. * * Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or * https://www.apache.org/licenses/LICENSE-2.0> or the MIT license * <LICENSE-MIT or https://opensource.org/licenses/MIT>, at your * option. This file may not be copied, modified, or distributed * except according to those terms. *****************************************************************************/ #![doc(html_root_url = "https://docs.rs/norman/0.0.4")] //! The norman library provides everything you need for calculationg norms of //! or general distances between elements of vector spaces. //! //! Based on two traits—[`Norm`](crate::Norm) and [`Distance`](crate::Distance) //!—this crate implements different kinds of norm and distance //! functions for a wide variety of types—including complex numbers and arrays. //! //! # Usage //! //! If you only want to compute the _standard norm_ of an element, //! like the absolute value of a floating point number or the euclidean //! norm of a vector, then you can use the traits [`NormEucl`](special::NormEucl) //! or [`DistanceEucl`](special::DistanceEucl) without specifying a certain //! type of norm: //! //! ``` //! use ndarray::Array1; //! //! use norman::special::NormEucl; //! //! let a = Array1::from(vec![2.0f32, -4.0, -2.0]); //! //! assert_eq!(a.norm_eucl(), (2.0f32*2.0 + 4.0*4.0 + 2.0*2.0).sqrt()); //! ``` //! //! For a detailed description on how these traits are implemented, //! see the module documentation of [`special::implementation`]. //! //! However, there are many ways to define a norm on a vector. //! If you want more control over the exact kind of norm or distance function used, //! the full [`Norm`] and [`Distance`] traits are the right ones for you. //! //! These traits have a generic parameter `D`, the //! _descriptor_ of the norm or distance function. This makes it possible //! to implement not only a single type of norm function for a type, but //! multiple ones. //! //! E.g. [`ndarray::ArrayBase`] implements [`Norm<Sup>`](crate::desc::Sup), //! which yields the spuremum norm, and [`Norm<PNorm>`](crate::desc::PNorm) //! which yields the _p_-norm. //! //! ``` //! use ndarray::Array1; //! //! use norman::Norm; //! use norman::desc::{Sup, PNorm}; //! //! let a = Array1::from(vec![2.0f32, -4.0, -2.0]); //! //! assert_eq!(a.norm(Sup::new()), 4.0); //! assert_eq!(a.norm(PNorm::new(2)), (2.0f32*2.0 + 4.0*4.0 + 2.0*2.0).sqrt()); //! assert_eq!(a.norm(PNorm::new(1)), 2.0f32 + 4.0 + 2.0); //! ``` //! //! You see: The norm-function recieves one additional parameter which further //! describes the norm. There is only one supremum norm, so this one needs no //! further description—we will always call it with //! [`Sup::new()`](crate::desc::Sup::new). //! //! But the _p_-norms do need additional specification: We need to specify //! whether we want a 1-norm or a 2-norm. So we pass `PNorm::new(2)` //! or `PNorm::new(1)` as the additional parameter. //! //! These norms are implemented on [`ndarray::ArrayBase`](ndarray::ArrayBase) //! as long as the elments of the array implement [`Norm<Abs>`](crate::desc::Abs): //! //! ``` //! use num_complex::Complex; //! use ndarray::Array1; //! //! use norman::Norm; //! use norman::desc::{Sup, PNorm}; //! //! let a = Array1::from(vec![ //! Complex::new(- 2.0, 0.0), //! Complex::new( 3.0, 4.0), //! Complex::new(-15.0, 8.0), //! ]); //! //! assert_eq!(a.norm(Sup::new()), 17.0); //! assert_eq!( //! a.norm(PNorm::new(2)), //! (2.0f32*2.0 + 0.0*0.0 + 3.0*3.0 + 4.0*4.0 + 15.0*15.0 + 8.0*8.0).sqrt() //! ); //! assert_eq!(a.norm(PNorm::new(1)), (2.0f32 + 5.0 + 17.0)); //! ``` //! //! # The [`Distance`](crate::Distance) trait //! //! In many cases, you do not want to calculate the norm of a single value, //! but you have to retrieve the distance between two values. This //! would often be possible by calling `(a-b).norm(norm_descriptor)`, //! but e.g. for ndarrays this would imply calculating the difference //! and once storing it in a new ndarray, which will cause one unnecessary //! memory allocation. Instead you can use the `Distance` //! trait, which calculates the same result without storing the intermediate //! difference of the arrays: //! //! ``` //! use ndarray::Array1; //! //! use norman::{Norm, Distance}; //! use norman::desc::{Sup, PNorm}; //! //! let a = Array1::from(vec![ 2.0f32, -4.0, -2.0]); //! let b = Array1::from(vec![ 6.0f32, -1.0, 4.0]); //! //! assert_eq!(a.distance(&b, Sup::new()), (a.clone()-&b).norm(Sup::new())); //! assert_eq!(a.distance(&b, PNorm::new(2)), (a.clone()-b).norm(PNorm::new(2))); //! ``` //! //! For a detailed description of how norms and distances are implemented //! on various types, see the module documentation of [`implementation`]. //! //! # Accessing the implementations //! //! The implementations on types of external crates are behind feature gates, //! which have the same name as the crate. E.g. in order to use the `Norm` //! trait on an [`ndarray::Array1`], `norman` must be included with the feature //! "ndarray", i.e. the corresponding part of your Cargo.toml would look like: //! //! ```toml //! norman = { version = "0.0.4", features = ["ndarray"] } //! ``` //! //! However, [`ndarray`] and [`num_complex`] are declared as default features, //! so they do not need to be named explicitly. //! //! ## All crate features //! //! * `num-complex`: Unlocks the implementations on [`num_complex::Complex`]. //! * `ndarray`: Unlocks the implementations on [`ndarray::ArrayBase`]. //! * `array`: Unlocks the implementations on the array types [T; N] for //! N=0 to N=32. //! If [const generics](https://github.com/rust-lang/rust/issues/44580) //! land some day, this feature gate will probably be removed. pub mod desc; pub mod implementation; pub mod special; mod utility; use std::ops::{Div, DivAssign}; use num_traits::Num; /// The `Norm` trait is the core of the `norman` crate. /// /// It provides a [`norm`](Norm::norm) function which calculates a specific /// norm of the vector. /// /// The type `D` is the norm descriptor, which specifies the exact kind of norm; e.g. /// a supremum norm or a euclidean norm. See the [`desc`](crate::desc) module /// for several different norm descriptors. pub trait Norm<D> { /// The resulting type of the norm function. /// /// Mathematically, a norm is a mapping from a vector space _V_ into the non-negative /// real numbers, so `Output` will usually be a floating point type /// or in some cases an unsigned integer type. type Output: Num; /// Calculates the norm of `self`, specified by the descriptor `desc`. /// /// # Panics /// /// An implementation of `norm` should never panic. /// /// An exception may be made for types like the [`noisy_float`] /// floating point types that already have a special panicking behaviour /// to ensure that no invalid values occur. /// /// # Example /// /// ``` /// use num_complex::Complex; /// /// use norman::Norm; /// use norman::desc::Abs; /// /// assert_eq!(Norm::norm(&Complex::new(3.0, 4.0), Abs::new()), 5.0); /// ``` fn norm(&self, desc: D) -> Self::Output; } /// The abstract notion of the distance between two values. /// /// This can be used to calculate the distance between two arbitrary /// values without storing their difference as an intermediate result. pub trait Distance<D> { /// The resulting type of the distance function. /// /// Mathematically, a distance metric is a mapping /// from 2-tuples of vectors of a vector space _V_ /// into the non-negative real numbers, so `Output` will usually be a floating point type /// or in some cases an unsigned integer type. type Output: Num; /// Calculates the distance between `self` and `other`. /// /// # Panics /// /// An implementation of `distance` may panic if the operands /// do not fit together, e.g. have different sizes etc. /// /// # Example /// /// ``` /// use num_complex::Complex; /// /// use norman::Distance; /// use norman::desc::Abs; /// /// assert_eq!(Complex::new(2.0, 5.0).distance(&Complex::new(-1.0, 1.0), Abs::new()), 5.0); /// ``` fn distance(&self, other: &Self, desc: D) -> Self::Output; } /// Normalizes the vector `v` according to the norm `desc`, /// i.e. divides it by its norm. /// /// As long as the implementations of `Div` and `DivAssign` on `T` match, /// `v` will be equal to `normalized(v)` after calling this function. /// /// # Attention /// /// Due to numerical errors, `v` is **not** guaranteed to have exactly norm `1` /// after calling this function. /// /// On integer types this function will do complete nonsense since /// `DivAssign` is implemented as an integer division for integers. /// /// # Example /// /// ``` /// use norman::normalize; /// use norman::desc::Abs; /// /// let mut a = 0.25f32; /// normalize(&mut a, Abs::new()); /// assert_eq!(a, 1.0); /// let mut a = -3.0f32; /// normalize(&mut a, Abs::new()); /// assert_eq!(a, -1.0); /// ``` pub fn normalize<T: Norm<D, Output=R> + DivAssign<R>, R: Num, D>(v: &mut T, desc: D) { *v /= v.norm(desc); } /// Returns the normalization of `v` according to the norm `desc`, /// i.e. `v` divided by its norm. /// /// # Attention /// /// Due to numerical errors, the result is **not** guaranteed to have exactly norm `1` /// after calling this function. /// /// On integer types this function will do complete nonsense since /// `Div` is implemented as an integer division for integers. /// /// # Example /// /// ``` /// use norman::normalized; /// use norman::desc::Abs; /// /// assert_eq!(normalized( 0.25f32, Abs::new()), 1.0); /// assert_eq!(normalized(-3.00f32, Abs::new()), -1.0); /// ``` pub fn normalized<T: Norm<D, Output=R> + Div<R, Output=T>, D, R: Num>(v: T, desc: D) -> T { let norm = v.norm(desc); v / norm } /*impl<T: Copy + Norm<D> + Sub<Self, Output=Self>, D> Distance<D> for T { type Output = <Self as Norm<D>>::Output; fn distance(&self, other: &Self, desc: D) -> <Self as Distance<D>>::Output { (*self - *other).norm(desc) } }*/