29 Numerics library [numerics]

29.9 Basic linear algebra algorithms [linalg]

29.9.14 BLAS 2 algorithms [linalg.algs.blas2]

29.9.14.8 Symmetric and Hermitian rank-2 matrix updates [linalg.algs.blas2.rank2]

[Note 1:

These functions correspond to the BLAS functions xSYR2,xSPR2, xHER2 and xHPR2[bib].

— end note]

The following elements apply to all functions in [linalg.algs.blas2.rank2].

For any function F in this subclause with a parameter named t, an InMat template parameter, and a function parameter InMat E, t applies to accesses done through the parameter E.

F only accesses the triangle of E specified by t.

For accesses of diagonal elements E[i, i], F only uses the value real-if-needed(E[i, i]) if the name of F starts with hermitian.

For accesses E[i, j] outside the triangle specified by t, F only uses the value

(3.1)
conj-if-needed(E[j, i]) if the name of F starts with hermitian, or
(3.2)
E[j, i] if the name of F starts with symmetric.

Mandates:

(4.1)
If OutMat has layout_blas_packed layout, then the layout's Triangle template argument has the same type as the function's Triangle template argument;
(4.2)
If the function has an InMat template parameter and InMat has layout_blas_packed layout, then the layout's Triangle template argument has the same type as the function's Triangle template argument;
(4.3)
compatible-static-extents<decltype(A), decltype(A)>(0, 1) is true;
(4.4)
possibly-multipliable<decltype(A), decltype(x), decltype(y)>() is true; and
(4.5)
possibly-addable<decltype(A), decltype(E), decltype(A)>() is true for those overloads with an E parameter.

Preconditions:

(5.1)
A.extent(0) equals A.extent(1),
(5.2)
multipliable(A, x, y) is true, and
(5.3)
addable(A, E, A) is true for those overloads with an E parameter.

Complexity:

O (x.extent(0) \times y.extent(0))

🔗

template<in-vector InVec1, in-vector InVec2,
         possibly-packed-out-matrix OutMat, class Triangle>
  void symmetric_matrix_rank_2_update(InVec1 x, InVec2 y, OutMat A, Triangle t);
template<class ExecutionPolicy, in-vector InVec1, in-vector InVec2,
         possibly-packed-out-matrix OutMat, class Triangle>
  void symmetric_matrix_rank_2_update(ExecutionPolicy&& exec,
                                      InVec1 x, InVec2 y, OutMat A, Triangle t);

These functions perform an overwriting symmetric rank-2 update of the symmetric matrix A, taking into account the Triangle parameter that applies to A ([linalg.general]).

Effects: Computes

A = x y^{T} + y x^{T}

🔗

template<in-vector InVec1, in-vector InVec2, in-matrix InMat,
         possibly-packed-out-matrix OutMat, class Triangle>
  void symmetric_matrix_rank_2_update(InVec1 x, InVec2 y, InMat E, OutMat A, Triangle t);
template<class ExecutionPolicy, in-vector InVec1, in-vector InVec2, in-matrix InMat,
         possibly-packed-out-matrix OutMat, class Triangle>
  void symmetric_matrix_rank_2_update(ExecutionPolicy&& exec,
                                      InVec1 x, InVec2 y, InMat E, OutMat A, Triangle t);

These functions perform an updating symmetric rank-2 update of the symmetric matrix A using the symmetric matrix E, taking into account the Triangle parameter that applies to A and E ([linalg.general]).

Effects: Computes

A = E + x y^{T} + y x^{T}

Remarks: A may alias E.

🔗

template<in-vector InVec1, in-vector InVec2,
         possibly-packed-out-matrix OutMat, class Triangle>
  void hermitian_matrix_rank_2_update(InVec1 x, InVec2 y, OutMat A, Triangle t);
template<class ExecutionPolicy, in-vector InVec1, in-vector InVec2,
         possibly-packed-out-matrix OutMat, class Triangle>
  void hermitian_matrix_rank_2_update(ExecutionPolicy&& exec,
                                      InVec1 x, InVec2 y, OutMat A, Triangle t);

These functions perform an overwriting Hermitian rank-2 update of the Hermitian matrix A, taking into account the Triangle parameter that applies to A ([linalg.general]).

Effects: Computes

A = x y^{H} + y x^{H}

🔗

template<in-vector InVec1, in-vector InVec2, in-matrix InMat,
         possibly-packed-out-matrix OutMat, class Triangle>
  void hermitian_matrix_rank_2_update(InVec1 x, InVec2 y, InMat E, OutMat A, Triangle t);
template<class ExecutionPolicy, in-vector InVec1, in-vector InVec2, in-matrix InMat,
         possibly-packed-out-matrix OutMat, class Triangle>
  void hermitian_matrix_rank_2_update(ExecutionPolicy&& exec,
                                      InVec1 x, InVec2 y, InMat E, OutMat A, Triangle t);

These functions perform an updating Hermitian rank-2 update of the Hermitian matrix A using the Hermitian matrix E, taking into account the Triangle parameter that applies to A and E ([linalg.general]).

Effects: Computes

A = E + x y^{H} + y x^{H}

Remarks: A may alias E.