28 Numerics library [numerics]

28.9 Basic linear algebra algorithms [linalg]

28.9.3 General [linalg.general]

For the effects of all functions in [linalg], when the effects are described as “computes

R = E

” or “compute

R = E

” (for some R and mathematical expression E), the following apply:

(1.1)
E has the conventional mathematical meaning as written.
(1.2)
The pattern $x^{T}$ should be read as “the transpose of x.”
(1.3)
The pattern $x^{H}$ should be read as “the conjugate transpose of x.”
(1.4)
When R is the same name as a function parameter whose type is a template parameter with Out in its name, the intent is that the result of the computation is written to the elements of the function parameter R.

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 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, 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]