[linalg.general]

Some of the functions and types in [linalg] distinguish between the “rows” and the “columns” of a matrix.

For a matrix A and a multidimensional index i, j in A.extents(),

(2.1)
row i of A is the set of elements A[i, k1] for all k1 such that i, k1 is in A.extents(); and
(2.2)
column j of A is the set of elements A[k0, j] for all k0 such that k0, j is in A.extents().

Some of the functions in [linalg] distinguish between the “upper triangle,” “lower triangle,” and “diagonal” of a matrix.

(3.1)
The diagonal is the set of all elements of A accessed by A[i,i] for 0 ≤ i < min(A.extent(0), A.extent(1)).
(3.2)
The upper triangle of a matrix A is the set of all elements of A accessed by A[i,j] with i ≤ j.

It includes the diagonal.
(3.3)
The lower triangle of A is the set of all elements of A accessed by A[i,j] with i ≥ j.

It includes the diagonal.

For any function F that takes a parameter named t, t applies to accesses done through the parameter preceding t in the parameter list of F.

Let m be such an access-modified function parameter.

F will only access the triangle of m specified by t.

For accesses of diagonal elements m[i, i], F will use the value real-if-needed(m[i, i]) if the name of F starts with hermitian.

For accesses m[i, j] outside the triangle specified by t, F will use the value

(4.1)
conj-if-needed(m[j, i]) if the name of F starts with hermitian,
(4.2)
m[j, i] if the name of F starts with symmetric, or
(4.3)
the additive identity if the name of F starts with triangular.

[Example 1:

Small vector product accessing only specified triangle.

It would not be a precondition violation for the non-accessed matrix element to be non-zero.

template<class Triangle> void triangular_matrix_vector_2x2_product( mdspan<const float, extents<int, 2> m, Triangle t, mdspan<const float, extents<int, 2> x, mdspan<float, extents<int, 2> y) { static_assert(is_same_v<Triangle, lower_triangle_t> || is_same_v<Triangle, upper_triangle_t>); if constexpr (is_same_v<Triangle, lower_triangle_t>) { y[0] = m[0,0] * x[0]; / + 0 * x[1] y[1] = m[1,0] * x[0] + m[1,1] * x[1]; } else { / upper_triangle_t y[0] = m[0,0] * x[0] + m[0,1] * x[1]; y[1] = /* 0 * x[0] + */ m[1,1] * x[1]; } } — end example]

For any function F that takes a parameter named d, d applies to accesses done through the previous-of-the-previous parameter of d in the parameter list of F.

Let m be such an access-modified function parameter.

If d specifies that an implicit unit diagonal is to be assumed, then

(5.1)
F will not access the diagonal of m; and
(5.2)
the algorithm will interpret m as if it has a unit diagonal, that is, a diagonal each of whose elements behaves as a two-sided multiplicative identity (even if m's value type does not have a two-sided multiplicative identity).

Otherwise, if d specifies that an explicit diagonal is to be assumed, then F will access the diagonal of m.

Within all the functions in [linalg], any calls to abs, conj, imag, and real are unqualified.

Two mdspan objects x and y alias each other, if they have the same extents e, and for every pack of integers i which is a multidimensional index in e, x[i..] and y[i..] refer to the same element.

[Note 1:

This means that x and y view the same elements in the same order.

— end note]

Two mdspan objects x and y overlap each other, if for some pack of integers i that is a multidimensional index in x.extents(), there exists a pack of integers j that is a multidimensional index in y.extents(), such that x[i..] and y[j..] refer to the same element.

[Note 2:

Aliasing is a special case of overlapping.

If x and y do not overlap, then they also do not alias each other.

— end note]