Iterating over expressions

xiterable and inner types

xtensor provides two base classes for making expressions iterable: xconst_iterable and xiterable. They define the API for iterating as described in Concepts. For an expression to be iterable, it must inherit directly or indirectly from one of these classes. For instance, the xbroadcast class is defined as following:

template <class CT, class X>
class xbroadcast : public xexpression<xbroadcast<CT, X>>,
                   public xconst_iterable<xbroadcast<CT, X>>
    // ...

Some of the methods provided by xconst_iterable or xiterable may need to be refined in the inheriting class. In that case, a common pattern is to make the inheritance private, import the methods we need with using declaration and redefine the methods whose behavior differ from the one provided in the base class. This is what is done in xfunction_base:

template <class F, class R, class... CT>
class xfunction_base : private xconst_iterable<xfunction_base<F, R, CT...>>

    using self_type = xfunction_base<F, R, CT...>;
    using iterable_base = xconst_iterable<self_type>;

    using iterable_base::begin;
    using iterable_base::end;
    using iterable_base::cbegin;
    using iterable_base::cend;
    using iterable_base::rbegin;
    using iterable_base::rend;
    using iterable_base::crbegin;
    using iterable_base::crend;

    template <layout_type L = DL>
    const_storage_iterator storage_begin() const noexcept;
    template <layout_type L = DL>
    const_storage_iterator storage_end() const noexcept;
    template <layout_type L = DL>
    const_storage_iterator storage_cbegin() const noexcept;
    template <layout_type L = DL>
    const_storage_iterator storage_cend() const noexcept;

    template <layout_type L = DL>
    const_reverse_storage_iterator storage_rbegin() const noexcept;
    template <layout_type L = DL>
    const_reverse_storage_iterator storage_rend() const noexcept;
    template <layout_type L = DL>
    const_reverse_storage_iterator storage_crbegin() const noexcept;
    template <layout_type L = DL>
    const_reverse_storage_iterator storage_crend() const noexcept;

The implementation of the iterator methods defined in xconst_iterable and xiterable rely on a few types and methods that must be defined in the inheriting class.

First, as stated in the xiterable section, the xiterable_inner_types structure must be specialized as illustrated below:

template <class F, class R, class... CT>
struct xiterable_inner_types<xfunction_base<F, R, CT...>>
    using inner_shape_type = promote_shape_t<typename std::decay_t<CT>::shape_type...>;
    using const_stepper = xfunction_stepper<F, R, CT...>;
    using stepper = const_stepper;

Then the inheriting class must define the following methods:

template <class S>
const_stepper stepper_begin(const S& shape) const noexcept;
template <class S>
const_stepper stepper_end(const S& shape, layout_type l) const noexcept;

If the expression class inherits from xiterable instead of xconst_iterable, the non-const counterparts of the previous methods must also be defined. Every method implemented in one of the base class eventually calls one of these stepper methods, whose mechanics is explained hereafter.


Steppers are the low-level tools for iterating over expressions. They provide a raw API for “stepping” of a given amount in a given dimension, dereferencing the stepper, and moving it to the beginning or the end of the expression:

reference operator*() const;

void step(size_type dim, size_type n = 1);
void step_back(size_type dim, size_type n = 1);
void reset(size_type dim);
void reset_back(size_type dim);

void to_begin();
void to_end(layout_type l);

The reset and reset_back methods are shortcut to step_back and step called with dim and shape[dim] - 1. The steppers are initialized with a “position” (that may be an index, a pointer to the underlying buffer of an container-based expression, etc…) in the expression, and can then be used to browse the expression in any direction:


In this diagram, the data is stored in row-major order, and we step in the first dimension (dimension index starts at 0). The positions of the stepper are represented by the red dots.

The to_end method takes a layout parameter, because the ending positions of a stepper depend on the layout used to iterate. Indeed, if we call step_back after a call to to_end, we want the stepper to point to the last element. To ensure this for both row-major order and column-major order iterations, the ending positions must be set as shown below:


The red dots are the position of a stepper iterating in column-major while the green ones are the positions of a stepper iterating in row-major order. Thus, if we assume that p is a pointer to the last element (the square containing 11), the ending positions of the stepper are p + 1 in row-major, and p + 3 in column-major order.

A stepper is specific to an expression type, therefore implementing a new kind of expression usually requires to implement a new kind of stepper. However xtensor provides a generic xindexed_stepper class, that can be used with any kind of expressions. Even though it is generally not optimal, authors of new expression types can make use of the generic index stepper in a first implementation.


The steppers of container-based expressions rely on strides and backstrides for stepping. A naive implementation of the step method would be:

template <class C>
inline void xstepper<C>::step(size_type dim, size_type n)
    m_it += n * p_c->strides()[dim];

where m_it is an iterator on the underlying buffer, and p_c a pointer to the container-based expression.

However, this implementation fails when broadcasting is involved. Consider the following expression:

xarray<int> a = {{0, 1,  2,  3},
                 {4, 5,  6,  7},
                 {8, 9, 10, 11}};
xarray<int> b = {0, 1, 2, 3};
auto r = a + b;

r is an xfunction representing the sum of a and b. The stepper specific to this expression holds the steppers of the arguments of the function; calling step or step_back results in calling step or step_back of the steppers of a and b.

According to the broadcasting rules, the shape of r is { 3, 4}. Thus, calling r.stepper_begin().step(1, 1) will eventually call b.stepper_begin().step(1, 1), leading to undefined behavior since the shape of b is {4}. To avoid that, a broadcasting offset is added to the stepper:

template <class C>
inline void xstepper<C>::step(size_type dim, size_type n)
    if (dim >= m_offset)
        m_it += difference_type(n * p_c->strides()[dim - m_offset]);

This implementation takes into account that the broadcasting is done on the last dimension and dimensions are stored in ascending order; here dimension 1 of a corresponds to dimension 0 of b.

This implementation ensures that a step in dimension 0 of the function updates the stepper of a while the stepper of b remains unchanged; on the other hand, stepping in dimension 1 will update both steppers, as illustrated below:


The red dots are initial stepper positions, the green dots and blue dots are the positions of the steppers after calling step with different dimension arguments.


xtensor iterator is implemented in the xiterator class. This latter provides a STL compliant iterator interface, and is built upon the steppers. Whereas the steppers are tied to the expression they refer to, xiterator is generic enough to work with any kind of stepper.

An iterator holds a stepper and a multi-dimensional index. A call to operator++ increases the index and calls the step method of the stepper accordingly. The way the index is increased depends on the layout used for iterating. For a row-major order iteration over a container with shape {3, 4}, the index iterating sequence is:

{0, 0}
{0, 1}
{0, 2}
{0, 3}
{1, 0}
{1, 1}
{1, 2}
{1, 3}
{2, 0}
{2, 1}
{2, 2}
{2, 3}

When a member of an index reaches its maximum value, it is reset to 0 and the member in the next dimension is increased. This translates into the calls of two methods of the stepper, first reset and then step. This is illustrated by the following picture:


The green arrows represent the iteration from {0, 0} to {0, 3}. The blue arrows illustrate what happens when the index is increased from {0, 3} to {1, 0}: first the stepper is reset to {0, 0}, then step(0, 1) is called, setting the stepper to the position {1, 0}.

xiterator implements a random access iterator, providing operator-- and operator[] methods. The implementation of these methods is similar to the one of operator++.