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// Copyright 2014-2020 bluss and ndarray developers.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.
#![crate_name = "ndarray"]
#![doc(html_root_url = "https://docs.rs/ndarray/0.15/")]
#![doc(html_logo_url = "https://rust-ndarray.github.io/images/rust-ndarray_logo.svg")]
#![allow(
unstable_name_collisions, // our `PointerExt` collides with upcoming inherent methods on `NonNull`
clippy::deref_addrof,
clippy::manual_map, // is not an error
clippy::while_let_on_iterator, // is not an error
clippy::incompatible_msrv, // false positive PointerExt::offset
)]
#![doc(test(attr(deny(warnings))))]
#![doc(test(attr(allow(unused_variables))))]
#![doc(test(attr(allow(deprecated))))]
#![cfg_attr(not(feature = "std"), no_std)]
// Enable the doc_cfg nightly feature for including feature gate flags in the documentation
#![cfg_attr(docsrs, feature(doc_cfg))]
#![cfg_attr(docsrs, doc(auto_cfg))]
#![warn(missing_docs)]
//! The `ndarray` crate provides an *n*-dimensional container for general elements
//! and for numerics.
//!
//! In *n*-dimensional we include, for example, 1-dimensional rows or columns,
//! 2-dimensional matrices, and higher dimensional arrays. If the array has *n*
//! dimensions, then an element in the array is accessed by using that many indices.
//! Each dimension is also called an *axis*.
//!
//! To get started, functionality is provided in the following core types:
//! - **[`ArrayBase`]**:
//! The *n*-dimensional array type itself.<br>
//! It is used to implement both the owned arrays and the views; see its docs
//! for an overview of all array features.<br>
//! - The main specific array type is **[`Array`]**, which owns
//! its elements.
//! - A reference type, **[`ArrayRef`]**, that contains most of the functionality
//! for reading and writing to arrays.
//! - A reference type, **[`LayoutRef`]**, that contains most of the functionality
//! for reading and writing to array layouts: their shape and strides.
//!
//! ## Highlights
//!
//! - Generic *n*-dimensional array
//! - [Slicing](ArrayBase#slicing), also with arbitrary step size, and negative
//! indices to mean elements from the end of the axis.
//! - Views and subviews of arrays; iterators that yield subviews.
//! - Higher order operations and arithmetic are performant
//! - Array views can be used to slice and mutate any `[T]` data using
//! `ArrayView::from` and `ArrayViewMut::from`.
//! - [`Zip`] for lock step function application across two or more arrays or other
//! item producers ([`NdProducer`] trait).
//!
//! ## Crate Status
//!
//! - Still iterating on and evolving the crate
//! + The crate is continuously developing, and breaking changes are expected
//! during evolution from version to version. We adopt the newest stable
//! rust features if we need them.
//! + Note that functions/methods/traits/etc. hidden from the docs are not
//! considered part of the public API, so changes to them are not
//! considered breaking changes.
//! - Performance:
//! + Prefer higher order methods and arithmetic operations on arrays first,
//! then iteration, and as a last priority using indexed algorithms.
//! + The higher order functions like [`.map()`](ArrayRef::map),
//! [`.map_inplace()`](ArrayRef::map_inplace), [`.zip_mut_with()`](ArrayRef::zip_mut_with),
//! [`Zip`] and [`azip!()`](azip) are the most efficient ways
//! to perform single traversal and lock step traversal respectively.
//! + Performance of an operation depends on the memory layout of the array
//! or array view. Especially if it's a binary operation, which
//! needs matching memory layout to be efficient (with some exceptions).
//! + Efficient floating point matrix multiplication even for very large
//! matrices; can optionally use BLAS to improve it further.
//!
//! - **MSRV: Requires Rust 1.64 or later**
//!
//! ## Crate Feature Flags
//!
//! The following crate feature flags are available. They are configured in your
//! `Cargo.toml`. See [`crate::doc::crate_feature_flags`] for more information.
//!
//! - `std`: Rust standard library-using functionality (enabled by default)
//! - `serde`: serialization support for serde 1.x
#![cfg_attr(
not(feature = "rayon"),
doc = "//! - `rayon`: Parallel iterators, parallelized methods, and the `par_azip!` macro."
)]
#![cfg_attr(
feature = "rayon",
doc = "//! - `rayon`: Parallel iterators, parallelized methods, the [`parallel`] module and [`par_azip!`]."
)]
//! - `approx` Implementations of traits from the [`approx`] crate.
//! - `blas`: transparent BLAS support for matrix multiplication, needs configuration.
//! - `matrixmultiply-threading`: Use threading from `matrixmultiply`.
//!
//! ## Documentation
//!
//! * The docs for [`ArrayBase`] provide an overview of
//! the *n*-dimensional array type. Other good pages to look at are the
//! documentation for the [`s![]`](s!) and
//! [`azip!()`](azip!) macros.
//!
//! * If you have experience with NumPy, you may also be interested in
//! [`ndarray_for_numpy_users`](doc::ndarray_for_numpy_users).
//!
//! ## The ndarray ecosystem
//!
//! `ndarray` provides a lot of functionality, but it's not a one-stop solution.
//!
//! `ndarray` includes matrix multiplication and other binary/unary operations out of the box.
//! More advanced linear algebra routines (e.g. SVD decomposition or eigenvalue computation)
//! can be found in [`ndarray-linalg`](https://crates.io/crates/ndarray-linalg).
//!
//! The same holds for statistics: `ndarray` provides some basic functionalities (e.g. `mean`)
//! but more advanced routines can be found in [`ndarray-stats`](https://crates.io/crates/ndarray-stats).
//!
//! If you are looking to generate random arrays instead, check out [`ndarray-rand`](https://crates.io/crates/ndarray-rand).
//!
//! For conversion between `ndarray`, [`nalgebra`](https://crates.io/crates/nalgebra) and
//! [`image`](https://crates.io/crates/image) check out [`nshare`](https://crates.io/crates/nshare).
extern crate alloc;
#[cfg(not(feature = "std"))]
extern crate core as std;
#[cfg(feature = "std")]
extern crate std;
#[cfg(feature = "blas")]
extern crate cblas_sys;
#[cfg(any(doc, docsrs))]
pub mod doc;
use alloc::fmt::Debug;
#[cfg(target_has_atomic = "ptr")]
use alloc::sync::Arc;
#[cfg(not(target_has_atomic = "ptr"))]
use portable_atomic_util::Arc;
use core::ptr::NonNull;
use std::marker::PhantomData;
pub use crate::dimension::dim::*;
pub use crate::dimension::{Axis, AxisDescription, Dimension, IntoDimension, RemoveAxis};
pub use crate::dimension::{DimAdd, DimMax};
pub use crate::dimension::IxDynImpl;
pub use crate::dimension::NdIndex;
pub use crate::error::{ErrorKind, ShapeError};
pub use crate::indexes::{indices, indices_of};
pub use crate::order::Order;
pub use crate::slice::{MultiSliceArg, NewAxis, Slice, SliceArg, SliceInfo, SliceInfoElem, SliceNextDim};
use crate::iterators::Baseiter;
use crate::iterators::{ElementsBase, ElementsBaseMut};
pub use crate::arraytraits::AsArray;
pub use crate::linalg_traits::LinalgScalar;
#[cfg(feature = "std")]
pub use crate::linalg_traits::NdFloat;
pub use crate::stacking::{concatenate, stack};
pub use crate::impl_views::IndexLonger;
pub use crate::math_cell::MathCell;
pub use crate::shape_builder::{Shape, ShapeArg, ShapeBuilder, StrideShape};
#[macro_use]
mod macro_utils;
#[macro_use]
mod private;
mod impl_ref_types;
mod aliases;
#[macro_use]
mod itertools;
mod argument_traits;
#[cfg(feature = "serde")]
mod array_serde;
mod arrayformat;
mod arraytraits;
pub use crate::argument_traits::AssignElem;
mod data_repr;
mod data_traits;
pub use crate::aliases::*;
pub use crate::data_traits::{Data, DataMut, DataOwned, DataShared, RawData, RawDataClone, RawDataMut, RawDataSubst};
mod free_functions;
pub use crate::free_functions::*;
pub use crate::iterators::iter;
mod error;
mod extension;
mod geomspace;
mod indexes;
mod iterators;
mod layout;
mod linalg_traits;
#[cfg(feature = "std")]
mod finite_bounds;
mod linspace;
#[cfg(feature = "std")]
pub use crate::linspace::{linspace, range, Linspace};
mod logspace;
#[cfg(feature = "std")]
pub use crate::logspace::{logspace, Logspace};
mod math_cell;
mod numeric_util;
mod order;
mod partial;
mod shape_builder;
#[macro_use]
mod slice;
mod split_at;
mod stacking;
mod low_level_util;
#[macro_use]
mod zip;
mod dimension;
pub use crate::zip::{FoldWhile, IntoNdProducer, NdProducer, Zip};
pub use crate::layout::Layout;
/// Implementation's prelude. Common types used everywhere.
mod imp_prelude
{
pub use crate::dimension::DimensionExt;
pub use crate::prelude::*;
pub use crate::ArcArray;
pub use crate::{
CowRepr,
Data,
DataMut,
DataOwned,
DataShared,
Ix,
Ixs,
RawData,
RawDataMut,
RawViewRepr,
RemoveAxis,
ViewRepr,
};
}
pub mod prelude;
/// Array index type
pub type Ix = usize;
/// Array index type (signed)
pub type Ixs = isize;
/// An *n*-dimensional array.
///
/// The array is a general container of elements.
/// The array supports arithmetic operations by applying them elementwise, if the
/// elements are numeric, but it supports non-numeric elements too.
///
/// The arrays rarely grow or shrink, since those operations can be costly. On
/// the other hand there is a rich set of methods and operations for taking views,
/// slices, and making traversals over one or more arrays.
///
/// In *n*-dimensional we include for example 1-dimensional rows or columns,
/// 2-dimensional matrices, and higher dimensional arrays. If the array has *n*
/// dimensions, then an element is accessed by using that many indices.
///
/// The `ArrayBase<S, D>` is parameterized by `S` for the data container and
/// `D` for the dimensionality.
///
/// Type aliases [`Array`], [`ArcArray`], [`CowArray`], [`ArrayView`], and
/// [`ArrayViewMut`] refer to `ArrayBase` with different types for the data
/// container: arrays with different kinds of ownership or different kinds of array views.
///
/// ## Contents
///
/// + [Array](#array)
/// + [ArcArray](#arcarray)
/// + [CowArray](#cowarray)
/// + [Array Views](#array-views)
/// + [Indexing and Dimension](#indexing-and-dimension)
/// + [Loops, Producers and Iterators](#loops-producers-and-iterators)
/// + [Slicing](#slicing)
/// + [Subviews](#subviews)
/// + [Arithmetic Operations](#arithmetic-operations)
/// + [Broadcasting](#broadcasting)
/// + [Conversions](#conversions)
/// + [Constructor Methods for Owned Arrays](#constructor-methods-for-owned-arrays)
/// + [Methods For All Array Types](#methods-for-all-array-types)
/// + [Methods For 1-D Arrays](#methods-for-1-d-arrays)
/// + [Methods For 2-D Arrays](#methods-for-2-d-arrays)
/// + [Methods for Dynamic-Dimensional Arrays](#methods-for-dynamic-dimensional-arrays)
/// + [Numerical Methods for Arrays](#numerical-methods-for-arrays)
///
/// ## `Array`
///
/// [`Array`] is an owned array that owns the underlying array
/// elements directly (just like a `Vec`) and it is the default way to create and
/// store n-dimensional data. `Array<A, D>` has two type parameters: `A` for
/// the element type, and `D` for the dimensionality. A particular
/// dimensionality's type alias like `Array3<A>` just has the type parameter
/// `A` for element type.
///
/// An example:
///
/// ```
/// // Create a three-dimensional f64 array, initialized with zeros
/// use ndarray::Array3;
/// let mut temperature = Array3::<f64>::zeros((3, 4, 5));
/// // Increase the temperature in this location
/// temperature[[2, 2, 2]] += 0.5;
/// ```
///
/// ## `ArcArray`
///
/// [`ArcArray`] is an owned array with reference counted
/// data (shared ownership).
/// Sharing requires that it uses copy-on-write for mutable operations.
/// Calling a method for mutating elements on `ArcArray`, for example
/// [`view_mut()`](ArrayRef::view_mut) or [`get_mut()`](ArrayRef::get_mut),
/// will break sharing and require a clone of the data (if it is not uniquely held).
///
/// ## `CowArray`
///
/// [`CowArray`] is analogous to [`std::borrow::Cow`].
/// It can represent either an immutable view or a uniquely owned array. If a
/// `CowArray` instance is the immutable view variant, then calling a method
/// for mutating elements in the array will cause it to be converted into the
/// owned variant (by cloning all the elements) before the modification is
/// performed.
///
/// ## Array Views
///
/// [`ArrayView`] and [`ArrayViewMut`] are read-only and read-write array views
/// respectively. They use dimensionality, indexing, and almost all other
/// methods the same way as the other array types.
///
/// Methods for `ArrayBase` apply to array views too, when the trait bounds
/// allow.
///
/// Please see the documentation for the respective array view for an overview
/// of methods specific to array views: [`ArrayView`], [`ArrayViewMut`].
///
/// A view is created from an array using [`.view()`](ArrayRef::view),
/// [`.view_mut()`](ArrayRef::view_mut), using
/// slicing ([`.slice()`](ArrayRef::slice), [`.slice_mut()`](ArrayRef::slice_mut)) or from one of
/// the many iterators that yield array views.
///
/// You can also create an array view from a regular slice of data not
/// allocated with `Array` — see array view methods or their `From` impls.
///
/// Note that all `ArrayBase` variants can change their view (slicing) of the
/// data freely, even when their data can’t be mutated.
///
/// ## Indexing and Dimension
///
/// The dimensionality of the array determines the number of *axes*, for example
/// a 2D array has two axes. These are listed in “big endian” order, so that
/// the greatest dimension is listed first, the lowest dimension with the most
/// rapidly varying index is the last.
///
/// In a 2D array the index of each element is `[row, column]` as seen in this
/// 4 × 3 example:
///
/// ```ignore
/// [[ [0, 0], [0, 1], [0, 2] ], // row 0
/// [ [1, 0], [1, 1], [1, 2] ], // row 1
/// [ [2, 0], [2, 1], [2, 2] ], // row 2
/// [ [3, 0], [3, 1], [3, 2] ]] // row 3
/// // \ \ \
/// // column 0 \ column 2
/// // column 1
/// ```
///
/// The number of axes for an array is fixed by its `D` type parameter: `Ix1`
/// for a 1D array, `Ix2` for a 2D array etc. The dimension type `IxDyn` allows
/// a dynamic number of axes.
///
/// A fixed size array (`[usize; N]`) of the corresponding dimensionality is
/// used to index the `Array`, making the syntax `array[[` i, j, ...`]]`
///
/// ```
/// use ndarray::Array2;
/// let mut array = Array2::zeros((4, 3));
/// array[[1, 1]] = 7;
/// ```
///
/// Important traits and types for dimension and indexing:
///
/// - A [`struct@Dim`] value represents a dimensionality or index.
/// - Trait [`Dimension`] is implemented by all
/// dimensionalities. It defines many operations for dimensions and indices.
/// - Trait [`IntoDimension`] is used to convert into a
/// `Dim` value.
/// - Trait [`ShapeBuilder`] is an extension of
/// `IntoDimension` and is used when constructing an array. A shape describes
/// not just the extent of each axis but also their strides.
/// - Trait [`NdIndex`] is an extension of `Dimension` and is
/// for values that can be used with indexing syntax.
///
///
/// The default memory order of an array is *row major* order (a.k.a “c” order),
/// where each row is contiguous in memory.
/// A *column major* (a.k.a. “f” or fortran) memory order array has
/// columns (or, in general, the outermost axis) with contiguous elements.
///
/// The logical order of any array’s elements is the row major order
/// (the rightmost index is varying the fastest).
/// The iterators `.iter(), .iter_mut()` always adhere to this order, for example.
///
/// ## Loops, Producers and Iterators
///
/// Using [`Zip`] is the most general way to apply a procedure
/// across one or several arrays or *producers*.
///
/// [`NdProducer`] is like an iterable but for
/// multidimensional data. All producers have dimensions and axes, like an
/// array view, and they can be split and used with parallelization using `Zip`.
///
/// For example, `ArrayView<A, D>` is a producer, it has the same dimensions
/// as the array view and for each iteration it produces a reference to
/// the array element (`&A` in this case).
///
/// Another example, if we have a 10 × 10 array and use `.exact_chunks((2, 2))`
/// we get a producer of chunks which has the dimensions 5 × 5 (because
/// there are *10 / 2 = 5* chunks in either direction). The 5 × 5 chunks producer
/// can be paired with any other producers of the same dimension with `Zip`, for
/// example 5 × 5 arrays.
///
/// ### `.iter()` and `.iter_mut()`
///
/// These are the element iterators of arrays and they produce an element
/// sequence in the logical order of the array, that means that the elements
/// will be visited in the sequence that corresponds to increasing the
/// last index first: *0, ..., 0, 0*; *0, ..., 0, 1*; *0, ...0, 2* and so on.
///
/// ### `.outer_iter()` and `.axis_iter()`
///
/// These iterators produce array views of one smaller dimension.
///
/// For example, for a 2D array, `.outer_iter()` will produce the 1D rows.
/// For a 3D array, `.outer_iter()` produces 2D subviews.
///
/// `.axis_iter()` is like `outer_iter()` but allows you to pick which
/// axis to traverse.
///
/// The `outer_iter` and `axis_iter` are one dimensional producers.
///
/// ## `.rows()`, `.columns()` and `.lanes()`
///
/// [`.rows()`][gr] is a producer (and iterable) of all rows in an array.
///
/// ```
/// use ndarray::Array;
///
/// // 1. Loop over the rows of a 2D array
/// let mut a = Array::zeros((10, 10));
/// for mut row in a.rows_mut() {
/// row.fill(1.);
/// }
///
/// // 2. Use Zip to pair each row in 2D `a` with elements in 1D `b`
/// use ndarray::Zip;
/// let mut b = Array::zeros(a.nrows());
///
/// Zip::from(a.rows())
/// .and(&mut b)
/// .for_each(|a_row, b_elt| {
/// *b_elt = a_row[a.ncols() - 1] - a_row[0];
/// });
/// ```
///
/// The *lanes* of an array are 1D segments along an axis and when pointed
/// along the last axis they are *rows*, when pointed along the first axis
/// they are *columns*.
///
/// A *m* × *n* array has *m* rows each of length *n* and conversely
/// *n* columns each of length *m*.
///
/// To generalize this, we say that an array of dimension *a* × *m* × *n*
/// has *a m* rows. It's composed of *a* times the previous array, so it
/// has *a* times as many rows.
///
/// All methods: [`.rows()`][gr], [`.rows_mut()`][grm],
/// [`.columns()`][gc], [`.columns_mut()`][gcm],
/// [`.lanes(axis)`][l], [`.lanes_mut(axis)`][lm].
///
/// [gr]: ArrayRef::rows
/// [grm]: ArrayRef::rows_mut
/// [gc]: ArrayRef::columns
/// [gcm]: ArrayRef::columns_mut
/// [l]: ArrayRef::lanes
/// [lm]: ArrayRef::lanes_mut
///
/// Yes, for 2D arrays `.rows()` and `.outer_iter()` have about the same
/// effect:
///
/// + `rows()` is a producer with *n* - 1 dimensions of 1 dimensional items
/// + `outer_iter()` is a producer with 1 dimension of *n* - 1 dimensional items
///
/// ## Slicing
///
/// You can use slicing to create a view of a subset of the data in
/// the array. Slicing methods include [`.slice()`], [`.slice_mut()`],
/// [`.slice_move()`], and [`.slice_collapse()`].
///
/// The slicing argument can be passed using the macro [`s![]`](s!),
/// which will be used in all examples. (The explicit form is an instance of
/// [`SliceInfo`] or another type which implements [`SliceArg`]; see their docs
/// for more information.)
///
/// If a range is used, the axis is preserved. If an index is used, that index
/// is selected and the axis is removed; this selects a subview. See
/// [*Subviews*](#subviews) for more information about subviews. If a
/// [`NewAxis`] instance is used, a new axis is inserted. Note that
/// [`.slice_collapse()`] panics on `NewAxis` elements and behaves like
/// [`.collapse_axis()`] by preserving the number of dimensions.
///
/// [`.slice()`]: ArrayRef::slice
/// [`.slice_mut()`]: ArrayRef::slice_mut
/// [`.slice_move()`]: Self::slice_move
/// [`.slice_collapse()`]: LayoutRef::slice_collapse
///
/// When slicing arrays with generic dimensionality, creating an instance of
/// [`SliceInfo`] to pass to the multi-axis slicing methods like [`.slice()`]
/// is awkward. In these cases, it's usually more convenient to use
/// [`.slice_each_axis()`]/[`.slice_each_axis_mut()`]/[`.slice_each_axis_inplace()`]
/// or to create a view and then slice individual axes of the view using
/// methods such as [`.slice_axis_inplace()`] and [`.collapse_axis()`].
///
/// [`.slice_each_axis()`]: ArrayRef::slice_each_axis
/// [`.slice_each_axis_mut()`]: ArrayRef::slice_each_axis_mut
/// [`.slice_each_axis_inplace()`]: Self::slice_each_axis_inplace
/// [`.slice_axis_inplace()`]: Self::slice_axis_inplace
/// [`.collapse_axis()`]: LayoutRef::collapse_axis
///
/// It's possible to take multiple simultaneous *mutable* slices with
/// [`.multi_slice_mut()`] or (for [`ArrayViewMut`] only)
/// [`.multi_slice_move()`].
///
/// [`.multi_slice_mut()`]: ArrayRef::multi_slice_mut
/// [`.multi_slice_move()`]: ArrayViewMut#method.multi_slice_move
///
/// ```
/// use ndarray::{arr2, arr3, s, ArrayBase, DataMut, Dimension, NewAxis, Slice};
///
/// // 2 submatrices of 2 rows with 3 elements per row, means a shape of `[2, 2, 3]`.
///
/// let a = arr3(&[[[ 1, 2, 3], // -- 2 rows \_
/// [ 4, 5, 6]], // -- /
/// [[ 7, 8, 9], // \_ 2 submatrices
/// [10, 11, 12]]]); // /
/// // 3 columns ..../.../.../
///
/// assert_eq!(a.shape(), &[2, 2, 3]);
///
/// // Let’s create a slice with
/// //
/// // - Both of the submatrices of the greatest dimension: `..`
/// // - Only the first row in each submatrix: `0..1`
/// // - Every element in each row: `..`
///
/// let b = a.slice(s![.., 0..1, ..]);
/// let c = arr3(&[[[ 1, 2, 3]],
/// [[ 7, 8, 9]]]);
/// assert_eq!(b, c);
/// assert_eq!(b.shape(), &[2, 1, 3]);
///
/// // Let’s create a slice with
/// //
/// // - Both submatrices of the greatest dimension: `..`
/// // - The last row in each submatrix: `-1..`
/// // - Row elements in reverse order: `..;-1`
/// let d = a.slice(s![.., -1.., ..;-1]);
/// let e = arr3(&[[[ 6, 5, 4]],
/// [[12, 11, 10]]]);
/// assert_eq!(d, e);
/// assert_eq!(d.shape(), &[2, 1, 3]);
///
/// // Let’s create a slice while selecting a subview and inserting a new axis with
/// //
/// // - Both submatrices of the greatest dimension: `..`
/// // - The last row in each submatrix, removing that axis: `-1`
/// // - Row elements in reverse order: `..;-1`
/// // - A new axis at the end.
/// let f = a.slice(s![.., -1, ..;-1, NewAxis]);
/// let g = arr3(&[[ [6], [5], [4]],
/// [[12], [11], [10]]]);
/// assert_eq!(f, g);
/// assert_eq!(f.shape(), &[2, 3, 1]);
///
/// // Let's take two disjoint, mutable slices of a matrix with
/// //
/// // - One containing all the even-index columns in the matrix
/// // - One containing all the odd-index columns in the matrix
/// let mut h = arr2(&[[0, 1, 2, 3],
/// [4, 5, 6, 7]]);
/// let (s0, s1) = h.multi_slice_mut((s![.., ..;2], s![.., 1..;2]));
/// let i = arr2(&[[0, 2],
/// [4, 6]]);
/// let j = arr2(&[[1, 3],
/// [5, 7]]);
/// assert_eq!(s0, i);
/// assert_eq!(s1, j);
///
/// // Generic function which assigns the specified value to the elements which
/// // have indices in the lower half along all axes.
/// fn fill_lower<S, D>(arr: &mut ArrayBase<S, D>, x: S::Elem)
/// where
/// S: DataMut,
/// S::Elem: Clone,
/// D: Dimension,
/// {
/// arr.slice_each_axis_mut(|ax| Slice::from(0..ax.len / 2)).fill(x);
/// }
/// fill_lower(&mut h, 9);
/// let k = arr2(&[[9, 9, 2, 3],
/// [4, 5, 6, 7]]);
/// assert_eq!(h, k);
/// ```
///
/// ## Subviews
///
/// Subview methods allow you to restrict the array view while removing one
/// axis from the array. Methods for selecting individual subviews include
/// [`.index_axis()`], [`.index_axis_mut()`], [`.index_axis_move()`], and
/// [`.index_axis_inplace()`]. You can also select a subview by using a single
/// index instead of a range when slicing. Some other methods, such as
/// [`.fold_axis()`], [`.axis_iter()`], [`.axis_iter_mut()`],
/// [`.outer_iter()`], and [`.outer_iter_mut()`] operate on all the subviews
/// along an axis.
///
/// A related method is [`.collapse_axis()`], which modifies the view in the
/// same way as [`.index_axis()`] except for removing the collapsed axis, since
/// it operates *in place*. The length of the axis becomes 1.
///
/// Methods for selecting an individual subview take two arguments: `axis` and
/// `index`.
///
/// [`.axis_iter()`]: ArrayRef::axis_iter
/// [`.axis_iter_mut()`]: ArrayRef::axis_iter_mut
/// [`.fold_axis()`]: ArrayRef::fold_axis
/// [`.index_axis()`]: ArrayRef::index_axis
/// [`.index_axis_inplace()`]: LayoutRef::index_axis_inplace
/// [`.index_axis_mut()`]: ArrayRef::index_axis_mut
/// [`.index_axis_move()`]: Self::index_axis_move
/// [`.collapse_axis()`]: LayoutRef::collapse_axis
/// [`.outer_iter()`]: ArrayRef::outer_iter
/// [`.outer_iter_mut()`]: ArrayRef::outer_iter_mut
///
/// ```
///
/// use ndarray::{arr3, aview1, aview2, s, Axis};
///
///
/// // 2 submatrices of 2 rows with 3 elements per row, means a shape of `[2, 2, 3]`.
///
/// let a = arr3(&[[[ 1, 2, 3], // \ axis 0, submatrix 0
/// [ 4, 5, 6]], // /
/// [[ 7, 8, 9], // \ axis 0, submatrix 1
/// [10, 11, 12]]]); // /
/// // \
/// // axis 2, column 0
///
/// assert_eq!(a.shape(), &[2, 2, 3]);
///
/// // Let’s take a subview along the greatest dimension (axis 0),
/// // taking submatrix 0, then submatrix 1
///
/// let sub_0 = a.index_axis(Axis(0), 0);
/// let sub_1 = a.index_axis(Axis(0), 1);
///
/// assert_eq!(sub_0, aview2(&[[ 1, 2, 3],
/// [ 4, 5, 6]]));
/// assert_eq!(sub_1, aview2(&[[ 7, 8, 9],
/// [10, 11, 12]]));
/// assert_eq!(sub_0.shape(), &[2, 3]);
///
/// // This is the subview picking only axis 2, column 0
/// let sub_col = a.index_axis(Axis(2), 0);
///
/// assert_eq!(sub_col, aview2(&[[ 1, 4],
/// [ 7, 10]]));
///
/// // You can take multiple subviews at once (and slice at the same time)
/// let double_sub = a.slice(s![1, .., 0]);
/// assert_eq!(double_sub, aview1(&[7, 10]));
/// ```
///
/// ## Arithmetic Operations
///
/// Arrays support all arithmetic operations the same way: they apply elementwise.
///
/// Since the trait implementations are hard to overview, here is a summary.
///
/// ### Binary Operators with Two Arrays
///
/// Let `A` be an array or view of any kind. Let `B` be an array
/// with owned storage (either `Array` or `ArcArray`).
/// Let `C` be an array with mutable data (either `Array`, `ArcArray`
/// or `ArrayViewMut`).
/// The following combinations of operands
/// are supported for an arbitrary binary operator denoted by `@` (it can be
/// `+`, `-`, `*`, `/` and so on).
///
/// - `&A @ &A` which produces a new `Array`
/// - `B @ A` which consumes `B`, updates it with the result, and returns it
/// - `B @ &A` which consumes `B`, updates it with the result, and returns it
/// - `C @= &A` which performs an arithmetic operation in place
///
/// Note that the element type needs to implement the operator trait and the
/// `Clone` trait.
///
/// ```
/// use ndarray::{array, ArrayView1};
///
/// let owned1 = array![1, 2];
/// let owned2 = array![3, 4];
/// let view1 = ArrayView1::from(&[5, 6]);
/// let view2 = ArrayView1::from(&[7, 8]);
/// let mut mutable = array![9, 10];
///
/// let sum1 = &view1 + &view2; // Allocates a new array. Note the explicit `&`.
/// // let sum2 = view1 + &view2; // This doesn't work because `view1` is not an owned array.
/// let sum3 = owned1 + view1; // Consumes `owned1`, updates it, and returns it.
/// let sum4 = owned2 + &view2; // Consumes `owned2`, updates it, and returns it.
/// mutable += &view2; // Updates `mutable` in-place.
/// ```
///
/// ### Binary Operators with Array and Scalar
///
/// The trait [`ScalarOperand`] marks types that can be used in arithmetic
/// with arrays directly. For a scalar `K` the following combinations of operands
/// are supported (scalar can be on either the left or right side, but
/// `ScalarOperand` docs has the detailed conditions).
///
/// - `&A @ K` or `K @ &A` which produces a new `Array`
/// - `B @ K` or `K @ B` which consumes `B`, updates it with the result and returns it
/// - `C @= K` which performs an arithmetic operation in place
///
/// ### Unary Operators
///
/// Let `A` be an array or view of any kind. Let `B` be an array with owned
/// storage (either `Array` or `ArcArray`). The following operands are supported
/// for an arbitrary unary operator denoted by `@` (it can be `-` or `!`).
///
/// - `@&A` which produces a new `Array`
/// - `@B` which consumes `B`, updates it with the result, and returns it
///
/// ## Broadcasting
///
/// Arrays support limited *broadcasting*, where arithmetic operations with
/// array operands of different sizes can be carried out by repeating the
/// elements of the smaller dimension array. See
/// [`.broadcast()`](ArrayRef::broadcast) for a more detailed
/// description.
///
/// ```
/// use ndarray::arr2;
///
/// let a = arr2(&[[1., 1.],
/// [1., 2.],
/// [0., 3.],
/// [0., 4.]]);
///
/// let b = arr2(&[[0., 1.]]);
///
/// let c = arr2(&[[1., 2.],
/// [1., 3.],
/// [0., 4.],
/// [0., 5.]]);
/// // We can add because the shapes are compatible even if not equal.
/// // The `b` array is shape 1 × 2 but acts like a 4 × 2 array.
/// assert!(
/// c == a + b
/// );
/// ```
///
/// ## Conversions
///
/// ### Conversions Between Array Types
///
/// This table is a summary of the conversions between arrays of different
/// ownership, dimensionality, and element type. All of the conversions in this
/// table preserve the shape of the array.
///
/// <table>
/// <tr>
/// <th rowspan="2">Output</th>
/// <th colspan="5">Input</th>
/// </tr>
///
/// <tr>
/// <td>
///
/// `Array<A, D>`
///
/// </td>
/// <td>
///
/// `ArcArray<A, D>`
///
/// </td>
/// <td>
///
/// `CowArray<'a, A, D>`
///
/// </td>
/// <td>
///
/// `ArrayView<'a, A, D>`
///
/// </td>
/// <td>
///
/// `ArrayViewMut<'a, A, D>`
///
/// </td>
/// </tr>
///
/// <!--Conversions to `Array<A, D>`-->
///
/// <tr>
/// <td>
///
/// `Array<A, D>`
///
/// </td>
/// <td>
///
/// no-op
///
/// </td>
/// <td>
///
/// [`a.into_owned()`][.into_owned()]
///
/// </td>
/// <td>
///
/// [`a.into_owned()`][.into_owned()]
///
/// </td>
/// <td>
///
/// [`a.to_owned()`][.to_owned()]
///
/// </td>
/// <td>
///
/// [`a.to_owned()`][.to_owned()]
///
/// </td>
/// </tr>
///
/// <!--Conversions to `ArcArray<A, D>`-->
///
/// <tr>
/// <td>
///
/// `ArcArray<A, D>`
///
/// </td>
/// <td>
///
/// [`a.into_shared()`][.into_shared()]
///
/// </td>
/// <td>
///
/// no-op
///
/// </td>
/// <td>
///
/// [`a.into_owned().into_shared()`][.into_shared()]
///
/// </td>
/// <td>
///
/// [`a.to_owned().into_shared()`][.into_shared()]
///
/// </td>
/// <td>
///
/// [`a.to_owned().into_shared()`][.into_shared()]
///
/// </td>
/// </tr>
///
/// <!--Conversions to `CowArray<'a, A, D>`-->
///
/// <tr>
/// <td>
///
/// `CowArray<'a, A, D>`
///
/// </td>
/// <td>
///
/// [`CowArray::from(a)`](CowArray#impl-From<ArrayBase<OwnedRepr<A>%2C%20D>>)
///
/// </td>
/// <td>
///
/// [`CowArray::from(a.into_owned())`](CowArray#impl-From<ArrayBase<OwnedRepr<A>%2C%20D>>)
///
/// </td>
/// <td>
///
/// no-op
///
/// </td>
/// <td>
///
/// [`CowArray::from(a)`](CowArray#impl-From<ArrayBase<ViewRepr<%26%27a%20A>%2C%20D>>)
///
/// </td>
/// <td>
///
/// [`CowArray::from(a.view())`](CowArray#impl-From<ArrayBase<ViewRepr<%26%27a%20A>%2C%20D>>)
///
/// </td>
/// </tr>
///
/// <!--Conversions to `ArrayView<'b, A, D>`-->
///
/// <tr>
/// <td>
///
/// `ArrayView<'b, A, D>`
///
/// </td>
/// <td>
///
/// [`a.view()`][.view()]
///
/// </td>
/// <td>
///
/// [`a.view()`][.view()]
///
/// </td>
/// <td>
///
/// [`a.view()`][.view()]
///
/// </td>
/// <td>
///
/// [`a.view()`][.view()] or [`a.reborrow()`][ArrayView::reborrow()]
///
/// </td>
/// <td>
///
/// [`a.view()`][.view()]
///
/// </td>
/// </tr>
///
/// <!--Conversions to `ArrayViewMut<'b, A, D>`-->
///
/// <tr>
/// <td>
///
/// `ArrayViewMut<'b, A, D>`
///
/// </td>
/// <td>
///
/// [`a.view_mut()`][.view_mut()]
///
/// </td>
/// <td>
///
/// [`a.view_mut()`][.view_mut()]
///
/// </td>
/// <td>
///
/// [`a.view_mut()`][.view_mut()]
///
/// </td>
/// <td>
///
/// illegal
///
/// </td>
/// <td>
///