28 Numerics library [numerics]

28.9 Basic linear algebra algorithms [linalg]

28.9.10 Transpose in-place transformation [linalg.transp]

28.9.10.1 Introduction [linalg.transp.intro]

layout_transpose is an mdspan layout mapping policy that swaps the two indices, extents, and strides of any unique mdspan layout mapping policy.
The transposed function takes an mdspan representing a matrix, and returns a new mdspan representing the transpose of the input matrix.

28.9.10.2 Exposition-only helpers for layout_transpose and transposed [linalg.transp.helpers]

The exposition-only transpose-extents function takes an extents object representing the extents of a matrix, and returns a new extents object representing the extents of the transpose of the matrix.
The exposition-only alias template transpose-extents-t<InputExtents> gives the type of transpose-extents(e) for a given extents object e of type InputExtents.
template<class IndexType, size_t InputExtent0, size_t InputExtent1> constexpr extents<IndexType, InputExtent1, InputExtent0> transpose-extents(const extents<IndexType, InputExtent0, InputExtent1>& in); // exposition only
Returns: extents<IndexType, InputExtent1, InputExtent0>(in.extent(1), in.extent(0))
template<class InputExtents> using transpose-extents-t = decltype(transpose-extents(declval<InputExtents>())); // exposition only

28.9.10.3 Class template layout_transpose [linalg.transp.layout.transpose]

layout_transpose is an mdspan layout mapping policy that swaps the two indices, extents, and strides of any mdspan layout mapping policy.
namespace std::linalg { template<class Layout> class layout_transpose { public: using nested_layout_type = Layout; template<class Extents> struct mapping { private: using nested-mapping-type = typename Layout::template mapping<transpose-extents-t<Extents>>; // exposition only public: using extents_type = Extents; using index_type = typename extents_type::index_type; using size_type = typename extents_type::size_type; using rank_type = typename extents_type::rank_type; using layout_type = layout_transpose; constexpr explicit mapping(const nested-mapping-type&); constexpr const extents_type& extents() const noexcept { return extents_; } constexpr index_type required_span_size() const { return nested-mapping_.required_span_size(); template<class Index0, class Index1> constexpr index_type operator()(Index0 ind0, Index1 ind1) const { return nested-mapping_(ind1, ind0); } constexpr const nested-mapping-type& nested_mapping() const noexcept { return nested-mapping_; } static constexpr bool is_always_unique() noexcept { return nested-mapping-type::is_always_unique(); } static constexpr bool is_always_exhaustive() noexcept { return nested-mapping-type::is_always_exhaustive(); } static constexpr bool is_always_strided() noexcept { return nested-mapping-type::is_always_strided(); } constexpr bool is_unique() const { return nested-mapping_.is_unique(); } constexpr bool is_exhaustive() const { return nested-mapping_.is_exhaustive(); } constexpr bool is_strided() const { return nested-mapping_.is_strided(); } constexpr index_type stride(size_t r) const; template<class OtherExtents> friend constexpr bool operator==(const mapping& x, const mapping<OtherExtents>& y); }; private: nested-mapping-type nested-mapping_; // exposition only extents_type extents_; // exposition only }; }
Layout shall meet the layout mapping policy requirements ([mdspan.layout.policy.reqmts]).
Mandates:
  • Extents is a specialization of std​::​extents, and
  • Extents​::​rank() equals 2.
constexpr explicit mapping(const nested-mapping-type& map);
Effects:
  • Initializes nested-mapping_ with map, and
  • initializes extents_ with transpose-extents(map.extents()).
constexpr index_type stride(size_t r) const;
Preconditions:
Returns: nested-mapping_.stride(r == 0 ? 1 : 0)
template<class OtherExtents> friend constexpr bool operator==(const mapping& x, const mapping<OtherExtents>& y);
Constraints: The expression x.nested-mapping_ == y.nested-mapping_ is well-formed and its result is convertible to bool.
Returns: x.nested-mapping_ == y.nested-mapping_.

28.9.10.4 Function template transposed [linalg.transp.transposed]

The transposed function takes a rank-2 mdspan representing a matrix, and returns a new mdspan representing the input matrix's transpose.
The input matrix's data are not modified, and the returned mdspan accesses the input matrix's data in place.
template<class ElementType, class Extents, class Layout, class Accessor> constexpr auto transposed(mdspan<ElementType, Extents, Layout, Accessor> a);
Mandates: Extents​::​rank() == 2 is true.
Let ReturnExtents be transpose-extents-t<Extents>.
Let R be mdspan<ElementType, ReturnExtents, ReturnLayout, Accessor>, where ReturnLayout is:
  • layout_right if Layout is layout_left;
  • otherwise, layout_left if Layout is layout_right;
  • otherwise, layout_stride if Layout is layout_stride;
  • otherwise, layout_blas_packed<OppositeTriangle, OppositeStorageOrder>, if Layout is
    layout_blas_packed<Triangle, StorageOrder> for some Triangle and StorageOrder, where
    • OppositeTriangle is conditional_t<is_same_v<Triangle, upper_triangle_t>, lower_triangle_t, upper_triangle_t> and
    • OppositeStorageOrder is conditional_t<is_same_v<StorageOrder, column_major_t>, row_major_t, column_major_t>
  • otherwise, NestedLayout if Layout is layout_transpose<NestedLayout> for some NestedLayout;
  • otherwise, layout_transpose<Layout>.
Returns: With ReturnMapping being the type typename ReturnLayout​::​template mapping<ReturnExtents>:
  • if Layout is layout_left, layout_right, or a specialization of layout_blas_packed, R(a.data_handle(), ReturnMapping(transpose-extents(a.mapping().extents())), a.accessor())
  • otherwise, if Layout is layout_stride, R(a.data_handle(), ReturnMapping(transpose-extents(a.mapping().extents()), array{a.mapping().stride(1), a.mapping().stride(0)}), a.accessor())
  • otherwise, if Layout is a specialization of layout_transpose, R(a.data_handle(), a.mapping().nested_mapping(), a.accessor())
  • otherwise, R(a.data_handle(), ReturnMapping(a.mapping()), a.accessor())
[Example 1: void test_transposed(mdspan<double, extents<size_t, 3, 4>> a) { const auto num_rows = a.extent(0); const auto num_cols = a.extent(1); auto a_t = transposed(a); assert(num_rows == a_t.extent(1)); assert(num_cols == a_t.extent(0)); assert(a.stride(0) == a_t.stride(1)); assert(a.stride(1) == a_t.stride(0)); for (size_t row = 0; row < num_rows; ++row) { for (size_t col = 0; col < num_rows; ++col) { assert(a[row, col] == a_t[col, row]); } } auto a_t_t = transposed(a_t); assert(num_rows == a_t_t.extent(0)); assert(num_cols == a_t_t.extent(1)); assert(a.stride(0) == a_t_t.stride(0)); assert(a.stride(1) == a_t_t.stride(1)); for (size_t row = 0; row < num_rows; ++row) { for (size_t col = 0; col < num_rows; ++col) { assert(a[row, col] == a_t_t[row, col]); } } } — end example]