28 Numerics library [numerics]

28.9 Basic linear algebra algorithms [linalg]

28.9.14 BLAS 2 algorithms [linalg.algs.blas2]

28.9.14.7 Symmetric or Hermitian Rank-1 (outer product) update of a matrix [linalg.algs.blas2.symherrank1]

[Note 1:

These functions correspond to the BLAS functions xSYR, xSPR, xHER, and xHPR[bib].

They have overloads taking a scaling factor alpha, because it would be impossible to express the update

A = A - x x^{T}

otherwise.

— end note]

The following elements apply to all functions in [linalg.algs.blas2.symherrank1].

Mandates:

(3.1)
If InOutMat has layout_blas_packed layout, then the layout's Triangle template argument has the same type as the function's Triangle template argument;
(3.2)
compatible-static-extents<decltype(A), decltype(A)>(0, 1) is true; and
(3.3)
compatible-static-extents<decltype(A), decltype(x)>(0, 0) is true.

Preconditions:

(4.1)
A.extent(0) equals A.extent(1), and
(4.2)
A.extent(0) equals x.extent(0).

Complexity:

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

🔗

template<class Scalar, in-vector InVec, possibly-packed-inout-matrix InOutMat, class Triangle>
  void symmetric_matrix_rank_1_update(Scalar alpha, InVec x, InOutMat A, Triangle t);
template<class ExecutionPolicy,
         class Scalar, in-vector InVec, possibly-packed-inout-matrix InOutMat, class Triangle>
  void symmetric_matrix_rank_1_update(ExecutionPolicy&& exec,
                                      Scalar alpha, InVec x, InOutMat A, Triangle t);

These functions perform a symmetric rank-1 update of the symmetric matrix A, taking into account the Triangle parameter that applies to A ([linalg.general]).

Effects: Computes a matrix

A^{'}

such that

A^{'} = A + α x x^{T}

, where the scalar α is alpha, and assigns each element of

A^{'}

to the corresponding element of A.

🔗

template<in-vector InVec, possibly-packed-inout-matrix InOutMat, class Triangle>
  void symmetric_matrix_rank_1_update(InVec x, InOutMat A, Triangle t);
template<class ExecutionPolicy,
         in-vector InVec, possibly-packed-inout-matrix InOutMat, class Triangle>
  void symmetric_matrix_rank_1_update(ExecutionPolicy&& exec, InVec x, InOutMat A, Triangle t);

These functions perform a symmetric rank-1 update of the symmetric matrix A, taking into account the Triangle parameter that applies to A ([linalg.general]).

Effects: Computes a matrix

A^{'}

such that

A^{'} = A + x x^{T}

and assigns each element of

A^{'}

to the corresponding element of A.

🔗

template<class Scalar, in-vector InVec, possibly-packed-inout-matrix InOutMat, class Triangle>
  void hermitian_matrix_rank_1_update(Scalar alpha, InVec x, InOutMat A, Triangle t);
template<class ExecutionPolicy,
         class Scalar, in-vector InVec, possibly-packed-inout-matrix InOutMat, class Triangle>
  void hermitian_matrix_rank_1_update(ExecutionPolicy&& exec,
                                      Scalar alpha, InVec x, InOutMat A, Triangle t);

These functions perform a Hermitian rank-1 update of the Hermitian matrix A, taking into account the Triangle parameter that applies to A ([linalg.general]).

Effects: Computes

A^{'}

such that

A^{'} = A + α x x^{H}

, where the scalar α is alpha, and assigns each element of

A^{'}

to the corresponding element of A.

🔗

template<in-vector InVec, possibly-packed-inout-matrix InOutMat, class Triangle>
  void hermitian_matrix_rank_1_update(InVec x, InOutMat A, Triangle t);
template<class ExecutionPolicy,
         in-vector InVec, possibly-packed-inout-matrix InOutMat, class Triangle>
  void hermitian_matrix_rank_1_update(ExecutionPolicy&& exec, InVec x, InOutMat A, Triangle t);

These functions perform a Hermitian rank-1 update of the Hermitian matrix A, taking into account the Triangle parameter that applies to A ([linalg.general]).

Effects: Computes a matrix

A^{'}

such that

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

and assigns each element of

A^{'}

to the corresponding element of A.