--- /dev/null
+==================
+Matrix Types
+==================
+
+.. contents::
+ :local:
+
+.. _matrixtypes:
+
+Clang provides a C/C++ language extension that allows users to directly express
+fixed-size 2-dimensional matrices as language values and perform arithmetic on
+them.
+
+This feature is currently experimental, and both its design and its
+implementation are in flux.
+
+Draft Specification
+===================
+
+Matrix Type
+-----------
+
+A matrix type is a scalar type with an underlying *element type*, a constant
+number of *rows*, and a constant number of *columns*. Matrix types with the same
+element type, rows, and columns are the same type. A value of a matrix type
+includes storage for ``rows * columns`` values of the *element type*. The
+internal layout, overall size and alignment are implementation-defined.
+
+The maximum of the product of the number of rows and columns is
+implementation-defined. If that implementation-defined limit is exceeded, the
+program is ill-formed.
+
+Currently, the element type of a matrix is only permitted to be one of the
+following types:
+
+* an integer type (as in C2x 6.2.5p19), but excluding enumerated types and ``_Bool``
+* the standard floating types ``float`` or ``double``
+* a half-precision floating point type, if one is supported on the target
+
+Other types may be supported in the future.
+
+Matrix Type Attribute
+---------------------
+
+Matrix types can be declared by adding the ``matrix_type`` attribute to the
+declaration of a *typedef* (or a C++ alias declaration). The underlying type
+of the *typedef* must be a valid matrix element type. The
+attribute takes two arguments, both of which must be integer constant
+expressions that evaluate to a value greater than zero. The first specifies the
+number of rows, and the second specifies the number of columns. The underlying
+type of the *typedef* becomes a matrix type with the given dimensions and an
+element type of the former underlying type.
+
+If a declaration of a *typedef-name* has a ``matrix_type`` attribute, then all
+declaration of that *typedef-name* shall have a matrix_type attribute with the
+same element type, number of rows, and number of columns.
+
+Standard Conversions
+--------------------
+
+The standard conversions are extended as follows. Note that these conversions
+are intentionally not listed as satisfying the constraints for assignment,
+which is to say, they are only permitted as explicit casts, not as implicit
+conversions.
+
+A value of matrix type can be converted to another matrix type if the number of
+rows and columns are the same and the value's elements can be converted to the
+element type of the result type. The result is a matrix where each element is
+the converted corresponding element.
+
+A value of any real type (as in C2x 6.2.5p17) can be converted to a matrix type
+if it can be converted to the element type of the matrix. The result is a
+matrix where all elements are the converted original value.
+
+If the number of rows or columns differ between the original and resulting
+type, the program is ill-formed.
+
+
+Arithmetic Conversions
+----------------------
+
+The usual arithmetic conversions are extended as follows.
+
+Insert at the start:
+
+* If both operands are of matrix type, no arithmetic conversion is performed.
+* If one operand is of matrix type and the other operand is of a real type,
+ convert the real type operand to the matrix type
+ according to the standard conversion rules.
+
+Matrix Type Element Access Operator
+-----------------------------------
+
+An expression of the form ``E1 [E2] [E3]``, where ``E1`` has matrix type ``cv
+M``, is a matrix element access expression. Let ``T`` be the element type
+of ``M``, and let ``R`` and ``C`` be the number of rows and columns in ``M``
+respectively. The index expressions shall have integral or unscoped
+enumeration type and shall not be uses of the comma operator unless
+parenthesized. The first index expression shall evaluate to a
+non-negative value less than ``R``, and the second index expression shall
+evaluate to a non-negative value less than ``C``, or else the expression has
+undefined behavior. If ``E1`` is a prvalue, the result is a prvalue with type
+``T`` and is the value of the element at the given row and column in the matrix.
+Otherwise, the result is a glvalue with type ``cv T`` and with the same value
+category as ``E1`` which refers to the element at the given row and column in
+the matrix.
+
+Programs containing a single subscript expression into a matrix are ill-formed.
+
+**Note**: We considered providing an expression of the form
+``postfix-expression [expression]`` to access columns of a matrix. We think
+that such an expression would be problematic once both column and row major
+matrixes are supported: depending on the memory layout, either accessing columns
+or rows can be done efficiently, but not both. Instead, we propose to provide
+builtins to extract rows and columns from a matrix. This makes the operations
+more explicit.
+
+Matrix Type Binary Operators
+----------------------------
+
+Each matrix type supports the following binary operators: ``+``, ``-`` and ``*``. The ``*``
+operator provides matrix multiplication, while ``+`` and ``-`` are performed
+element-wise. There are also scalar versions of the operators, which take a
+matrix type and the matrix element type. The operation is applied to all
+elements of the matrix using the scalar value.
+
+For ``BIN_OP`` in ``+``, ``-``, ``*`` given the expression ``M1 BIN_OP M2`` where
+at least one of ``M1`` or ``M2`` is of matrix type and, for `*`, the other is of
+a real type:
+
+* The usual arithmetic conversions are applied to ``M1`` and ``M2``. [ Note: if ``M1`` or
+ ``M2`` are of a real type, they are broadcast to matrices here. — end note ]
+* ``M1`` and ``M2`` shall be of the same matrix type.
+* The result is equivalent to Res in the following where col is the number of
+ columns and row is the number of rows in the matrix type:
+
+.. code-block:: c++
+
+ decltype(M1) Res;
+ for (int C = 0; C < col; ++C)
+ for (int R = 0; R < row; ++R)
+ Res[R][C] = M1[R][C] BIN_OP M2[R][C];
+
+Given the expression ``M1 * M2`` where ``M1`` and ``M2`` are of matrix type:
+
+* The usual arithmetic conversions are applied to ``M1`` and ``M2``.
+* The type of ``M1`` shall have the same number of columns as the type of ``M2`` has
+ rows. The element types of ``M1`` and ``M2`` shall be the same type.
+* The resulting type, ``MTy``, is a matrix type with the common element type,
+ the number of rows of ``M1`` and the number of columns of ``M2``.
+* The result is equivalent to ``Res`` in the following where ``EltTy`` is the
+ element type of ``MTy``, ``col`` is the number of columns, ``row`` is the
+ number of rows in ``MTy`` and ``inner`` is the number of columns of ``M1``:
+
+.. code-block:: c++
+
+ MTy Res;
+ for (int C = 0; C < col; ++C) {
+ for (int R = 0; R < row; ++R) {
+ EltTy Elt = 0;
+ for (int K = 0; K < inner; ++K) {
+ Elt += M1[R][K] * M2[K][C];
+ }
+ Res[R][C] = Elt;
+ }
+
+All operations on matrix types match the behavior of the element type with
+respect to signed overflows.
+
+With respect to floating-point contraction, rounding and environment rules,
+operations on matrix types match the behavior of the elementwise operations
+in the corresponding expansions provided above.
+
+Operations on floating-point matrices have the same rounding and floating-point
+environment behavior as ordinary floating-point operations in the expression's
+context. For the purposes of floating-point contraction, all calculations done
+as part of a matrix operation are considered intermediate operations, and their
+results need not be rounded to the format of the element type until the final
+result in the containing expression. This is subject to the normal restrictions
+on contraction, such as ``#pragma STDC FP_CONTRACT``.
+
+For the ``+=``, ``-=`` and ``*=`` operators the semantics match their expanded
+variants.
+
+Matrix Type Builtin Operations
+------------------------------
+
+Each matrix type supports a collection of builtin expressions that look like
+function calls but do not form an overload set. Here they are described as
+function declarations with rules for how to construct the argument list types
+and return type and the library description elements from
+[library.description.structure.specifications]/3 in the C++ standard.
+
+Definitions:
+
+* *M*, *M1*, *M2*, *M3* - Matrix types
+* *T* - Element type
+* *row*, *col* - Row and column arguments respectively.
+
+
+``M2 __builtin_matrix_transpose(M1 matrix)``
+
+**Remarks**: The return type is a cv-unqualified matrix type that has the same
+element type as ``M1`` and has the the same number of rows as ``M1`` has columns and
+the same number of columns as ``M1`` has rows.
+
+**Returns**: A matrix ``Res`` equivalent to the code below, where ``col`` refers to the
+number of columns of ``M``, and ``row`` to the number of rows of ``M``.
+
+**Effects**: Equivalent to:
+
+.. code-block:: c++
+
+ M Res;
+ for (int C = 0; C < col; ++C)
+ for (int R = 0; R < row; ++R)
+ Res[C][R] = matrix[R][C];
+
+
+``M __builtin_matrix_column_major_load(T *ptr, size_t row, size_t col, size_t columnStride)``
+
+**Mandates**: ``row`` and ``col`` shall be integral constants greater than 0.
+
+**Preconditions**: ``columnStride`` is greater than or equal to ``row``.
+
+**Remarks**: The return type is a cv-unqualified matrix type with an element
+type of the cv-unqualified version of ``T`` and a number of rows and columns equal
+to ``row`` and ``col`` respectively. The parameter ``columnStride`` is optional
+and if omitted ``row`` is used as ``columnStride``.
+
+**Returns**: A matrix ``Res`` equivalent to:
+
+.. code-block:: c++
+
+ M Res;
+ for (size_t C = 0; C < col; ++C) {
+ for (size_t R = 0; R < row; ++K)
+ Res[R][C] = ptr[R];
+ ptr += columnStride
+ }
+
+
+``void __builtin_matrix_column_major_store(M matrix, T *ptr, size_t columnStride)``
+
+**Preconditions**: ``columnStride`` is greater than or equal to the number of rows in ``M``.
+
+**Remarks**: The type ``T`` is the const-unqualified version of the matrix
+argument’s element type. The paramter ``columnStride`` is optional and if
+ommitted, the number of rows of ``M`` is used as ``columnStride``.
+
+**Effects**: Equivalent to:
+
+.. code-block:: c++
+
+ for (size_t C = 0; C < columns in M; ++C) {
+ for (size_t R = 0; R < rows in M; ++K)
+ ptr[R] = matrix[R][C];
+ ptr += columnStride
+ }
+
+
+TODOs
+-----
+
+TODO: Does it make sense to allow M::element_type, M::rows, and M::columns
+where M is a matrix type? We don’t support this anywhere else, but it’s
+convenient. The alternative is using template deduction to extract this
+information. Also add spelling for C.
+
+Future Work: Initialization syntax.
+
+
+Decisions for the Implementation in Clang
+=========================================
+
+This section details decisions taken for the implementation in Clang and is not
+part of the draft specification.
+
+The elements of a value of a matrix type are laid out in column-major order
+without padding.
+
+We propose to provide a Clang option to override this behavior and allow
+contraction of those operations (e.g. *-ffp-contract=matrix*).
+
+TODO: Specify how matrix values are passed to functions.