28 Numerics library [numerics]

28.9 Basic linear algebra algorithms [linalg]

28.9.14 BLAS 2 algorithms [linalg.algs.blas2]

28.9.14.5 Solve a triangular linear system [linalg.algs.blas2.trsv]

[Note 1:

These functions correspond to the BLAS functions xTRSV and xTPSV[bib].

— end note]

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

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(b)>(0, 0) is true; and
(3.4)
compatible-static-extents<decltype(A), decltype(x)>(0, 0) is true for those overloads that take an x parameter.

Preconditions:

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

🔗

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

These functions perform a triangular solve, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).

Effects: Computes a vector

x^{'}

such that

b = A x^{'}

, and assigns each element of

x^{'}

to the corresponding element of x.

If no such

x^{'}

exists, then the elements of x are valid but unspecified.

Complexity:

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

🔗

template<in-matrix InMat, class Triangle, class DiagonalStorage,
         in-vector InVec, out-vector OutVec>
  void triangular_matrix_vector_solve(InMat A, Triangle t, DiagonalStorage d, InVec b, OutVec x);

Effects: Equivalent to: triangular_matrix_vector_solve(A, t, d, b, x, divides<void>{});

🔗

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

Effects: Equivalent to: triangular_matrix_vector_solve(std::forward<ExecutionPolicy>(exec), A, t, d, b, x, divides<void>{});

🔗

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

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

[Note 2:

Performing triangular solve in place hinders parallelization.

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

— end note]

Effects: Computes a vector

x^{'}

such that

b = A x^{'}

, and assigns each element of

x^{'}

to the corresponding element of b.

If no such

x^{'}

exists, then the elements of b are valid but unspecified.

Complexity:

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

🔗

template<in-matrix InMat, class Triangle, class DiagonalStorage, inout-vector InOutVec>
  void triangular_matrix_vector_solve(InMat A, Triangle t, DiagonalStorage d, InOutVec b);

Effects: Equivalent to: triangular_matrix_vector_solve(A, t, d, b, divides<void>{});

🔗

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

Effects: Equivalent to: triangular_matrix_vector_solve(std::forward<ExecutionPolicy>(exec), A, t, d, b, divides<void>{});