28 Numerics library [numerics]

28.9 Basic linear algebra algorithms [linalg]

28.9.15 BLAS 3 algorithms [linalg.algs.blas3]

28.9.15.6 Solve multiple triangular linear systems [linalg.algs.blas3.trsm]

[Note 1:

These functions correspond to the BLAS function xTRSM[bib].

— end note]

template<in-matrix InMat1, class Triangle, class DiagonalStorage,
         in-matrix InMat2, out-matrix OutMat, class BinaryDivideOp>
  void triangular_matrix_matrix_left_solve(InMat1 A, Triangle t, DiagonalStorage d,
                                           InMat2 B, OutMat X, BinaryDivideOp divide);
template<class ExecutionPolicy,
         in-matrix InMat1, class Triangle, class DiagonalStorage,
         in-matrix InMat2, out-matrix OutMat, class BinaryDivideOp>
  void triangular_matrix_matrix_left_solve(ExecutionPolicy&& exec,
                                           InMat1 A, Triangle t, DiagonalStorage d,
                                           InMat2 B, OutMat X, BinaryDivideOp divide);

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

Mandates:

(3.1)
If InMat1 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)
possibly-multipliable<InMat1, OutMat, InMat2>() is true; and
(3.3)
compatible-static-extents<InMat1, InMat1>(0, 1) is true.

Preconditions:

(4.1)
multipliable(A, X, B) is true, and
(4.2)
A.extent(0) == A.extent(1) is true.

Effects: Computes

X^{'}

such that

A X^{'} = B

, 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(0) \times X.extent(1) \times X.extent(1))

[Note 2:

Since the triangular matrix is on the left, the desired divide implementation in the case of noncommutative multiplication is mathematically equivalent to

y^{- 1} x

, where x is the first argument and y is the second argument, and

y^{- 1}

denotes the multiplicative inverse of y.

— end note]

🔗

template<in-matrix InMat1, class Triangle, class DiagonalStorage,
         in-matrix InMat2, out-matrix OutMat>
  void triangular_matrix_matrix_left_solve(InMat1 A, Triangle t, DiagonalStorage d,
                                           InMat2 B, OutMat X);

Effects: Equivalent to: triangular_matrix_matrix_left_solve(A, t, d, B, X, divides<void>{});

🔗

template<class ExecutionPolicy, in-matrix InMat1, class Triangle, class DiagonalStorage,
         in-matrix InMat2, out-matrix OutMat>
  void triangular_matrix_matrix_left_solve(ExecutionPolicy&& exec,
                                           InMat1 A, Triangle t, DiagonalStorage d,
                                           InMat2 B, OutMat X);

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

🔗

template<in-matrix InMat1, class Triangle, class DiagonalStorage,
         in-matrix InMat2, out-matrix OutMat, class BinaryDivideOp>
  void triangular_matrix_matrix_right_solve(InMat1 A, Triangle t, DiagonalStorage d,
                                            InMat2 B, OutMat X, BinaryDivideOp divide);
template<class ExecutionPolicy,
         in-matrix InMat1, class Triangle, class DiagonalStorage,
         in-matrix InMat2, out-matrix OutMat, class BinaryDivideOp>
  void triangular_matrix_matrix_right_solve(ExecutionPolicy&& exec,
                                            InMat1 A, Triangle t, DiagonalStorage d,
                                            InMat2 B, OutMat X, BinaryDivideOp divide);

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

Mandates:

(11.1)
If InMat1 has layout_blas_packed layout, then the layout's Triangle template argument has the same type as the function's Triangle template argument;
(11.2)
possibly-multipliable<OutMat, InMat1, InMat2>() is true; and
(11.3)
compatible-static-extents<InMat1, InMat1>(0,1) is true.

Preconditions:

(12.1)
multipliable(X, A, B) is true, and
(12.2)
A.extent(0) == A.extent(1) is true.

Effects: Computes

X^{'}

such that

X^{'} A = B

, 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( B.extent(0)

\cdot

B.extent(1)

\cdot

A.extent(1) )

[Note 3:

Since the triangular matrix is on the right, the desired divide implementation in the case of noncommutative multiplication is mathematically equivalent to

x y^{- 1}

, where x is the first argument and y is the second argument, and

y^{- 1}

denotes the multiplicative inverse of y.

— end note]

🔗

template<in-matrix InMat1, class Triangle, class DiagonalStorage,
         in-matrix InMat2, out-matrix OutMat>
  void triangular_matrix_matrix_right_solve(InMat1 A, Triangle t, DiagonalStorage d,
                                            InMat2 B, OutMat X);

Effects: Equivalent to: triangular_matrix_matrix_right_solve(A, t, d, B, X, divides<void>{});

🔗

template<class ExecutionPolicy, in-matrix InMat1, class Triangle, class DiagonalStorage,
         in-matrix InMat2, out-matrix OutMat>
  void triangular_matrix_matrix_right_solve(ExecutionPolicy&& exec,
                                            InMat1 A, Triangle t, DiagonalStorage d,
                                            InMat2 B, OutMat X);

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