28 Numerics library [numerics]

28.9 Basic linear algebra algorithms [linalg]

28.9.14 BLAS 2 algorithms [linalg.algs.blas2]

28.9.14.4 Triangular matrix-vector product [linalg.algs.blas2.trmv]

[Note 1:

These functions correspond to the BLAS functions xTRMV and xTPMV[bib].

— end note]

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

Mandates:

(3.1)
If InMat 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;
(3.3)
compatible-static-extents<decltype(A), decltype(y)>(0, 0) is true;
(3.4)
compatible-static-extents<decltype(A), decltype(x)>(0, 0) is true for those overloads that take an x parameter; and
(3.5)
compatible-static-extents<decltype(A), decltype(z)>(0, 0) is true for those overloads that take a z parameter.

Preconditions:

(4.1)
A.extent(0) equals A.extent(1),
(4.2)
A.extent(0) equals y.extent(0),
(4.3)
A.extent(0) equals x.extent(0) for those overloads that take an x parameter, and
(4.4)
A.extent(0) equals z.extent(0) for those overloads that take a z parameter.

🔗

template<in-matrix InMat, class Triangle, class DiagonalStorage, in-vector InVec,
         out-vector OutVec>
  void triangular_matrix_vector_product(InMat A, Triangle t, DiagonalStorage d, InVec x, OutVec y);
template<class ExecutionPolicy,
         in-matrix InMat, class Triangle, class DiagonalStorage, in-vector InVec,
         out-vector OutVec>
  void triangular_matrix_vector_product(ExecutionPolicy&& exec,
                                        InMat A, Triangle t, DiagonalStorage d, InVec x, OutVec y);

These functions perform an overwriting triangular matrix-vector product, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).

Effects: Computes

y = A x

Complexity:

O (x.extent(0) \times A.extent(1))

🔗

template<in-matrix InMat, class Triangle, class DiagonalStorage, inout-vector InOutVec>
  void triangular_matrix_vector_product(InMat A, Triangle t, DiagonalStorage d, InOutVec y);
template<class ExecutionPolicy,
         in-matrix InMat, class Triangle, class DiagonalStorage, inout-vector InOutVec>
  void triangular_matrix_vector_product(ExecutionPolicy&& exec,
                                        InMat A, Triangle t, DiagonalStorage d, InOutVec y);

These functions perform an in-place triangular matrix-vector product, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).

[Note 2:

Performing this operation in place hinders parallelization.

However, other ExecutionPolicy specific optimizations, such as vectorization, are still possible.

— end note]

Effects: Computes a vector

y^{'}

such that

y^{'} = A y

, and assigns each element of

y^{'}

to the corresponding element of y.

Complexity:

O (y.extent(0) \times A.extent(1))

🔗

template<in-matrix InMat, class Triangle, class DiagonalStorage,
         in-vector InVec1, in-vector InVec2, out-vector OutVec>
  void triangular_matrix_vector_product(InMat A, Triangle t, DiagonalStorage d,
                                        InVec1 x, InVec2 y, OutVec z);
template<class ExecutionPolicy, in-matrix InMat, class Triangle, class DiagonalStorage,
         in-vector InVec1, in-vector InVec2, out-vector OutVec>
  void triangular_matrix_vector_product(ExecutionPolicy&& exec,
                                        InMat A, Triangle t, DiagonalStorage d,
                                        InVec1 x, InVec2 y, OutVec z);

These functions perform an updating triangular matrix-vector product, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).

Effects: Computes

z = y + A x

Complexity:

O (x.extent(0) \times A.extent(1))

Remarks: z may alias y.