xtensor’s core is built upon key concepts captured in interfaces that are put together in derived classes through CRTP (Curiously Recurring Template Pattern) and multiple inheritance. Interfaces and classes that model expressions implement value semantic. CRTP and value semantic achieve static polymorphism and avoid performance overhead of virtual methods and dynamic dispatching.


xt::xexpression is the base class for all expression classes. It is a CRTP base whose template parameter must be the most derived class in the hierarchy. For instance, if A inherits from B which in turn inherits from xt::xexpression, then B should be a template class whose template parameter is A and should forward this parameter to xt::xexpression:

#include <xtensor/xexpression.hpp>

template <class T>
class B : public xexpression<T>
    // ...

class A : public B<A>
    // ...

xt::xexpression only provides three overloads of a same function, that cast an xt::xexpression object to the most inheriting type, depending on the nature of the object (lvalue, const lvalue or rvalue):

derived_type& derived_cast() & noexcept;
const derived_type& derived_cast() const & noexcept;
derived_type derived_cast() && noexcept;


The iterable concept is modeled by two classes, xconst_iterable and xiterable, defined in xtensor/xiterable.hpp. xconst_iterable provides types and methods for iterating on constant expressions, similar to the ones provided by the STL containers. Unlike the STL, the methods of xconst_iterable and xiterable are templated by a layout parameter that allows you to iterate over a N-dimensional expression in row-major order or column-major order.


Row-major layout means that elements that only differ by their last index are contiguous in memory. Column-major layout means that elements that only differ by their first index are contiguous in memory.

template <class L>
const_iterator begin() const noexcept;
template <class L>
const_iterator end() const noexcept;
template <class L>
const_iterator cbegin() const noexcept;
template <class L>
const_iterator cend() const noexcept;

template <class L>
const_reverse_iterator rbegin() const noexcept;
template <class L>
const_reverse_iterator rend() const noexcept;
template <class L>
const_reverse_iterator crbegin() const noexcept;
template <class L>
const_reverse_iterator crend() const noexcept;

This template parameter is defaulted to XTENSOR_DEFAULT_TRAVERSAL (see Configuration), so that xtensor expressions can be used in generic code such as:

std::copy(a.cbegin(), a.cend(), b.begin());

where a and b can be arbitrary types (from xtensor, the STL or any external library) supporting standard iteration.

xiterable inherits from xconst_iterable and provides non-const counterpart of methods defined in xconst_iterable. Like xt::xexpression, both are CRTP classes whose template parameter must be the most derived type.

Besides traditional methods for iterating, xconst_iterable and xiterable provide overloads taking a shape parameter. This allows to iterate over an expression as if it was broadcast to the given shape:

#include <algorithm>
#include <iterator>
#include <iostream>
#include <xtensor/xarray.hpp>

int main(int argc, char* argv[])
    xt::xarray<int> a = { 1, 2, 3 };
    std::vector<std::size_t> shape = { 2, 3 };
    std::copy(a.cbegin(shape), a.cend(shape), std::output_iterator(std::cout, " "));
    // output: 1 2 3 1 2 3

Iterators returned by methods defined in xconst_iterable and xiterable are random access iterators.


The xsemantic_base interface provides methods for assigning an expression:

template <class E>
disable_xexpression<E, derived_type&> operator+=(const E&);

template <class E>
derived_type& operator+=(const xexpression<E>&);

and similar methods for operator-=, operator*=, operator/=, operator%=, operator&=, operator|= and operator^=.

The first overload is meant for computed assignment involving a scalar; it allows to write code like

#include <xtensor/xarray.hpp>
#include <xtensor/xio.hpp>

int main(int argc, char* argv)
    xarray<int> a = { 1, 2, 3 };
    a += 4;
    std::cout << a << std::endl;
    // outputs { 5, 6, 7 }

We rely on SFINAE to remove this overload from the overload resolution set when the parameter that we want to assign is not a scalar, avoiding ambiguity.

Operator-based methods taking a general xt::xexpression parameter don’t perform a direct assignment. Instead, the result is assigned to a temporary variable first, in order to prevent issues with aliasing. Thus, if a and b are expressions, the following

a += b

is equivalent to

temporary_type tmp = a + b;

Temporaries can be avoided with the assign-based methods:

template <class E>
derived_type& plus_assign(const xexpression<E>&);
template <class E>
derived_type&> minus_assign(const xexpression<E>&);
template <class E>
derived_type& multiplies_assign(const xexpression<E>&);
template <class E>
derived_type& divides_assign(const xexpression<E>&);
template <class E>
derived_type& modulus_assign(const xexpression<E>&);

xsemantic_base is a CRTP class whose parameter must be the most derived type in the hierarchy. It inherits from xt::xexpression and forwards its template parameter to this latter one.

xsemantic_base also provides a assignment operator that takes an xt::xexpression in its protected section:

template <class E>
derived_type& operator=(const xexpression<E>&);

Like computed assignment operators, it evaluates the expression inside a temporary before calling the assign method. Classes inheriting from xsemantic_base must redeclare this method either in their protected section (if they are not final classes) or in their public section. In both cases, they should forward the call to their base class.

Two refinements of this concept are provided, xcontainer_semantic and xview_semantic. Refer to the Assignment section for more details about semantic classes and how they’re involved in expression assignment.

xsemantic classes hierarchy:



The xcontainer class provides methods for container-based expressions. It does not hold any data, this is delegated to inheriting classes. It assumes the data are stored using a strided-index scheme. xcontainer defines the following methods:

Shape, strides and size

size_type size() const noexcept;
size_type dimension() const noexcept;

const inner_shape_type& shape() const noexcept;
const inner_strides_type& strides() const noexcept;
const inner_backstrides_type& backstrides() const noexcept;

Data access methods

template <class... Args>
const_reference operator()(Args... args) const;

template <class... Args>
const_reference at(Args... args) const;

template <class S>
disable_integral_t<S, const_reference> operator[](const S& index) const;

template <class I>
const_reference operator[](std::initializer_list<I> index) const;

template <class It>
const_reference element(It first, It last) const;

const storage_type& storage() const;

(and their non-const counterpart)

Broadcasting methods

template <class S>
bool broadcast_shape(const S& shape) const;

Lower-level methods are also provided, meant for optimized assignment and BLAS bindings. They are covered in the Assignment section.

If you read the entire code of xcontainer, you’ll notice that two types are defined for shape, strides and backstrides: shape_type and inner_shape_type, strides_type and inner_strides_type, and backstrides_type and inner_backstrides_type. The distinction between inner_shape_type and shape_type was motivated by the xtensor-python wrapper around NumPy data structures, where the inner shape type is a proxy on the shape section of the NumPy arrayobject. It cannot have a value semantics on its own as it is bound to the entire NumPy array.

xstrided_container inherits from xcontainer; it represents a container that holds its shape and strides. It provides methods for reshaping the container:

template <class S = shape_type>
void resize(D&& shape, bool force = false);

template <class S = shape_type>
void resize(S&& shape, layout_type l);

template <class S = shape_type>
void resize(S&& shape, const strides_type& strides);

template <class S = shape_type>
void reshape(S&& shape, layout_type l);

Both xstrided_container and xcontainer are CRTP classes whose template parameter must be the most derived type in the hierarchy. Besides, xcontainer inherits from xiterable, thus providing iteration methods.



The xfunction class is used to model mathematical operations and functions. It provides similar methods to the ones defined in xcontainer, and embeds the functor describing the operation and its operands. It inherits from xconst_iterable, thus providing iteration methods.