29 Numerics library [numerics]

29.9 Basic linear algebra algorithms [linalg]

29.9.15 BLAS 3 algorithms [linalg.algs.blas3]

29.9.15.1 General matrix-matrix product [linalg.algs.blas3.gemm]

29.9.15.2 Symmetric, Hermitian, and triangular matrix-matrix product [linalg.algs.blas3.xxmm]

29.9.15.3 In-place triangular matrix-matrix product [linalg.algs.blas3.trmm]

29.9.15.4 Rank-k update of a symmetric or Hermitian matrix [linalg.algs.blas3.rankk]

29.9.15.5 Rank-2k update of a symmetric or Hermitian matrix [linalg.algs.blas3.rank2k]

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

29.9.15.7 Solve multiple triangular linear systems in-place [linalg.algs.blas3.inplacetrsm]

29.9.15.1 General matrix-matrix product [linalg.algs.blas3.gemm]

1
#
[Note 1: 
These functions correspond to the BLAS function xGEMM[bib].
— end note]
2
#
The following elements apply to all functions in [linalg.algs.blas3.gemm] in addition to function-specific elements.
3
#
Mandates: possibly-multipliable<decltype(A), decltype(B), decltype(C)>() is true.
4
#
Preconditions: multipliable(A, B, C) is true.
5
#
Complexity: .
🔗
template<in-matrix InMat1, in-matrix InMat2, out-matrix OutMat> void matrix_product(InMat1 A, InMat2 B, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, out-matrix OutMat> void matrix_product(ExecutionPolicy&& exec, InMat1 A, InMat2 B, OutMat C);
6
#
Effects: Computes .
🔗
template<in-matrix InMat1, in-matrix InMat2, in-matrix InMat3, out-matrix OutMat> void matrix_product(InMat1 A, InMat2 B, InMat3 E, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, in-matrix InMat3, out-matrix OutMat> void matrix_product(ExecutionPolicy&& exec, InMat1 A, InMat2 B, InMat3 E, OutMat C);
7
#
Mandates: possibly-addable<InMat3, InMat3, OutMat>() is true.
8
#
Preconditions: addable(E, E, C) is true.
9
#
Effects: Computes .
10
#
Remarks: C may alias E.

29.9.15.2 Symmetric, Hermitian, and triangular matrix-matrix product [linalg.algs.blas3.xxmm]

1
#
[Note 1: 
These functions correspond to the BLAS functions xSYMM, xHEMM, and xTRMM[bib].
— end note]
2
#
The following elements apply to all functions in [linalg.algs.blas3.xxmm] in addition to function-specific elements.
3
#
Mandates:
  • (3.1)
    possibly-multipliable<decltype(A), decltype(B), decltype(C)>() is true, and
  • (3.2)
    possibly-addable<decltype(E), decltype(E), decltype(C)>() is true for those overloads that take an E parameter.
4
#
Preconditions:
  • (4.1)
    multipliable(A, B, C) is true, and
  • (4.2)
    addable(E, E, C) is true for those overloads that take an E parameter.
5
#
Complexity: .
🔗
template<in-matrix InMat1, class Triangle, in-matrix InMat2, out-matrix OutMat> void symmetric_matrix_product(InMat1 A, Triangle t, InMat2 B, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, class Triangle, in-matrix InMat2, out-matrix OutMat> void symmetric_matrix_product(ExecutionPolicy&& exec, InMat1 A, Triangle t, InMat2 B, OutMat C); template<in-matrix InMat1, class Triangle, in-matrix InMat2, out-matrix OutMat> void hermitian_matrix_product(InMat1 A, Triangle t, InMat2 B, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, class Triangle, in-matrix InMat2, out-matrix OutMat> void hermitian_matrix_product(ExecutionPolicy&& exec, InMat1 A, Triangle t, InMat2 B, OutMat C); template<in-matrix InMat1, class Triangle, class DiagonalStorage, in-matrix InMat2, out-matrix OutMat> void triangular_matrix_product(InMat1 A, Triangle t, DiagonalStorage d, InMat2 B, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, class Triangle, class DiagonalStorage, in-matrix InMat2, out-matrix OutMat> void triangular_matrix_product(ExecutionPolicy&& exec, InMat1 A, Triangle t, DiagonalStorage d, InMat2 B, OutMat C);
6
#
These functions perform a matrix-matrix multiply, taking into account the Triangle and DiagonalStorage (if applicable) parameters that apply to the symmetric, Hermitian, or triangular (respectively) matrix A ([linalg.general]).
7
#
Mandates:
  • (7.1)
    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; and
  • (7.2)
    compatible-static-extents<InMat1, InMat1>(0, 1) is true.
8
#
Preconditions: A.extent(0) == A.extent(1) is true.
9
#
Effects: Computes .
🔗
template<in-matrix InMat1, in-matrix InMat2, class Triangle, out-matrix OutMat> void symmetric_matrix_product(InMat1 A, InMat2 B, Triangle t, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, class Triangle, out-matrix OutMat> void symmetric_matrix_product(ExecutionPolicy&& exec, InMat1 A, InMat2 B, Triangle t, OutMat C); template<in-matrix InMat1, in-matrix InMat2, class Triangle, out-matrix OutMat> void hermitian_matrix_product(InMat1 A, InMat2 B, Triangle t, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, class Triangle, out-matrix OutMat> void hermitian_matrix_product(ExecutionPolicy&& exec, InMat1 A, InMat2 B, Triangle t, OutMat C); template<in-matrix InMat1, in-matrix InMat2, class Triangle, class DiagonalStorage, out-matrix OutMat> void triangular_matrix_product(InMat1 A, InMat2 B, Triangle t, DiagonalStorage d, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, class Triangle, class DiagonalStorage, out-matrix OutMat> void triangular_matrix_product(ExecutionPolicy&& exec, InMat1 A, InMat2 B, Triangle t, DiagonalStorage d, OutMat C);
10
#
These functions perform a matrix-matrix multiply, taking into account the Triangle and DiagonalStorage (if applicable) parameters that apply to the symmetric, Hermitian, or triangular (respectively) matrix B ([linalg.general]).
11
#
Mandates:
  • (11.1)
    If InMat2 has layout_blas_packed layout, then the layout's Triangle template argument has the same type as the function's Triangle template argument; and
  • (11.2)
    compatible-static-extents<InMat2, InMat2>(0, 1) is true.
12
#
Preconditions: B.extent(0) == B.extent(1) is true.
13
#
Effects: Computes .
🔗
template<in-matrix InMat1, class Triangle, in-matrix InMat2, in-matrix InMat3, out-matrix OutMat> void symmetric_matrix_product(InMat1 A, Triangle t, InMat2 B, InMat3 E, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, class Triangle, in-matrix InMat2, in-matrix InMat3, out-matrix OutMat> void symmetric_matrix_product(ExecutionPolicy&& exec, InMat1 A, Triangle t, InMat2 B, InMat3 E, OutMat C); template<in-matrix InMat1, class Triangle, in-matrix InMat2, in-matrix InMat3, out-matrix OutMat> void hermitian_matrix_product(InMat1 A, Triangle t, InMat2 B, InMat3 E, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, class Triangle, in-matrix InMat2, in-matrix InMat3, out-matrix OutMat> void hermitian_matrix_product(ExecutionPolicy&& exec, InMat1 A, Triangle t, InMat2 B, InMat3 E, OutMat C); template<in-matrix InMat1, class Triangle, class DiagonalStorage, in-matrix InMat2, in-matrix InMat3, out-matrix OutMat> void triangular_matrix_product(InMat1 A, Triangle t, DiagonalStorage d, InMat2 B, InMat3 E, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, class Triangle, class DiagonalStorage, in-matrix InMat2, in-matrix InMat3, out-matrix OutMat> void triangular_matrix_product(ExecutionPolicy&& exec, InMat1 A, Triangle t, DiagonalStorage d, InMat2 B, InMat3 E, OutMat C);
14
#
These functions perform a potentially overwriting matrix-matrix multiply-add, taking into account the Triangle and DiagonalStorage (if applicable) parameters that apply to the symmetric, Hermitian, or triangular (respectively) matrix A ([linalg.general]).
15
#
Mandates:
  • (15.1)
    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; and
  • (15.2)
    compatible-static-extents<InMat1, InMat1>(0, 1) is true.
16
#
Preconditions: A.extent(0) == A.extent(1) is true.
17
#
Effects: Computes .
18
#
Remarks: C may alias E.
🔗
template<in-matrix InMat1, in-matrix InMat2, class Triangle, in-matrix InMat3, out-matrix OutMat> void symmetric_matrix_product(InMat1 A, InMat2 B, Triangle t, InMat3 E, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, class Triangle, in-matrix InMat3, out-matrix OutMat> void symmetric_matrix_product(ExecutionPolicy&& exec, InMat1 A, InMat2 B, Triangle t, InMat3 E, OutMat C); template<in-matrix InMat1, in-matrix InMat2, class Triangle, in-matrix InMat3, out-matrix OutMat> void hermitian_matrix_product(InMat1 A, InMat2 B, Triangle t, InMat3 E, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, class Triangle, in-matrix InMat3, out-matrix OutMat> void hermitian_matrix_product(ExecutionPolicy&& exec, InMat1 A, InMat2 B, Triangle t, InMat3 E, OutMat C); template<in-matrix InMat1, in-matrix InMat2, class Triangle, class DiagonalStorage, in-matrix InMat3, out-matrix OutMat> void triangular_matrix_product(InMat1 A, InMat2 B, Triangle t, DiagonalStorage d, InMat3 E, OutMat C); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, class Triangle, class DiagonalStorage, in-matrix InMat3, out-matrix OutMat> void triangular_matrix_product(ExecutionPolicy&& exec, InMat1 A, InMat2 B, Triangle t, DiagonalStorage d, InMat3 E, OutMat C);
19
#
These functions perform a potentially overwriting matrix-matrix multiply-add, taking into account the Triangle and DiagonalStorage (if applicable) parameters that apply to the symmetric, Hermitian, or triangular (respectively) matrix B ([linalg.general]).
20
#
Mandates:
  • (20.1)
    If InMat2 has layout_blas_packed layout, then the layout's Triangle template argument has the same type as the function's Triangle template argument; and
  • (20.2)
    compatible-static-extents<InMat2, InMat2>(0, 1) is true.
21
#
Preconditions: B.extent(0) == B.extent(1) is true.
22
#
Effects: Computes .
23
#
Remarks: C may alias E.

29.9.15.3 In-place triangular matrix-matrix product [linalg.algs.blas3.trmm]

1
#
These functions perform an in-place matrix-matrix multiply, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).
[Note 1: 
These functions correspond to the BLAS function xTRMM[bib].
— end note]
🔗
template<in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat> void triangular_matrix_left_product(InMat A, Triangle t, DiagonalStorage d, InOutMat C); template<class ExecutionPolicy, in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat> void triangular_matrix_left_product(ExecutionPolicy&& exec, InMat A, Triangle t, DiagonalStorage d, InOutMat C);
2
#
Mandates:
  • (2.1)
    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;
  • (2.2)
    possibly-multipliable<InMat, InOutMat, InOutMat>() is true; and
  • (2.3)
    compatible-static-extents<InMat, InMat>(0, 1) is true.
3
#
Preconditions:
  • (3.1)
    multipliable(A, C, C) is true, and
  • (3.2)
    A.extent(0) == A.extent(1) is true.
4
#
Effects: Computes a matrix such that and assigns each element of to the corresponding element of C.
5
#
Complexity: .
🔗
template<in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat> void triangular_matrix_right_product(InMat A, Triangle t, DiagonalStorage d, InOutMat C); template<class ExecutionPolicy, in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat> void triangular_matrix_right_product(ExecutionPolicy&& exec, InMat A, Triangle t, DiagonalStorage d, InOutMat C);
6
#
Mandates:
  • (6.1)
    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;
  • (6.2)
    possibly-multipliable<InOutMat, InMat, InOutMat>() is true; and
  • (6.3)
    compatible-static-extents<InMat, InMat>(0, 1) is true.
7
#
Preconditions:
  • (7.1)
    multipliable(C, A, C) is true, and
  • (7.2)
    A.extent(0) == A.extent(1) is true.
8
#
Effects: Computes a matrix such that and assigns each element of to the corresponding element of C.
9
#
Complexity: .

29.9.15.4 Rank-k update of a symmetric or Hermitian matrix [linalg.algs.blas3.rankk]

1
#
[Note 1: 
These functions correspond to the BLAS functions xSYRK and xHERK[bib].
— end note]
2
#
The following elements apply to all functions in [linalg.algs.blas3.rankk].
3
#
For any function F in this subclause with a parameter named t, an InMat2 template parameter, and a function parameter InMat2 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.
4
#
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 InMat2 template parameter and if InMat2 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)
    possibly-multipliable<decltype(A), decltype(transposed(A)), decltype(C)>() is true; and
  • (4.4)
    possibly-addable<decltype(C), decltype(E), decltype(C)>() is true for those overloads with an E parameter.
5
#
Preconditions:
  • (5.1)
    multipliable(A, transposed(A), C) is true; and
    [Note 2: 
    This implies that C is square.
    — end note]
  • (5.2)
    addable(C, E, C) is true for those overloads with an E parameter.
6
#
Complexity: .
7
#
Remarks: C may alias E for those overloads with an E parameter.
🔗
template<scalar Scalar, in-matrix InMat, possibly-packed-out-matrix OutMat, class Triangle> void symmetric_matrix_rank_k_update(Scalar alpha, InMat A, OutMat C, Triangle t); template<class ExecutionPolicy, scalar Scalar, in-matrix InMat, possibly-packed-out-matrix OutMat, class Triangle> void symmetric_matrix_rank_k_update(ExecutionPolicy&& exec, Scalar alpha, InMat A, OutMat C, Triangle t);
8
#
Effects: Computes , where the scalar α is alpha.
🔗
template<scalar Scalar, in-matrix InMat, possibly-packed-out-matrix OutMat, class Triangle> void hermitian_matrix_rank_k_update(Scalar alpha, InMat A, OutMat C, Triangle t); template<class ExecutionPolicy, scalar Scalar, in-matrix InMat, possibly-packed-out-matrix OutMat, class Triangle> void hermitian_matrix_rank_k_update(ExecutionPolicy&& exec, Scalar alpha, InMat A, OutMat C, Triangle t);
9
#
Effects: Computes , where the scalar α is real-if-needed(alpha).
🔗
template<scalar Scalar, in-matrix InMat1, in-matrix InMat2, possibly-packed-out-matrix OutMat, class Triangle> void symmetric_matrix_rank_k_update(Scalar alpha, InMat1 A, InMat2 E, OutMat C, Triangle t); template<class ExecutionPolicy, scalar Scalar, in-matrix InMat1, in-matrix InMat2, possibly-packed-out-matrix OutMat, class Triangle> void symmetric_matrix_rank_k_update(ExecutionPolicy&& exec, Scalar alpha, InMat1 A, InMat2 E, OutMat C, Triangle t);
10
#
Effects: Computes , where the scalar α is alpha.
🔗
template<scalar Scalar, in-matrix InMat1, in-matrix InMat2, possibly-packed-out-matrix OutMat, class Triangle> void hermitian_matrix_rank_k_update(Scalar alpha, InMat1 A, InMat2 E, OutMat C, Triangle t); template<class ExecutionPolicy, scalar Scalar, in-matrix InMat1, in-matrix InMat2, possibly-packed-out-matrix OutMat, class Triangle> void hermitian_matrix_rank_k_update(ExecutionPolicy&& exec, Scalar alpha, InMat1 A, InMat2 E, OutMat C, Triangle t);
11
#
Effects: Computes , where the scalar α is real-if-needed(alpha).

29.9.15.5 Rank-2k update of a symmetric or Hermitian matrix [linalg.algs.blas3.rank2k]

1
#
[Note 1: 
These functions correspond to the BLAS functions xSYR2K and xHER2K[bib].
— end note]
2
#
The following elements apply to all functions in [linalg.algs.blas3.rank2k].
3
#
For any function F in this subclause with a parameter named t, an InMat3 template parameter, and a function parameter InMat3 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.
4
#
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 InMat3 template parameter and if InMat3 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)
    possibly-multipliable<decltype(A), decltype(transposed(B)), decltype(C)>() is true;
  • (4.4)
    possibly-multipliable<decltype(B), decltype(transposed(A)), decltype(C)>() is true; and
  • (4.5)
    possibly-addable<decltype(C), decltype(E), decltype(C)>() is true for those overloads with an E parameter.
5
#
Preconditions:
  • (5.1)
    multipliable(A, transposed(B), C) is true,
  • (5.2)
    multipliable(B, transposed(A), C) is true, and
    [Note 2: 
    This and the previous imply that C is square.
    — end note]
  • (5.3)
    addable(C, E, C) is true for those overloads with an E parameter.
6
#
Complexity: .
7
#
Remarks: C may alias E for those overloads with an E parameter.
🔗
template<in-matrix InMat1, in-matrix InMat2, possibly-packed-out-matrix OutMat, class Triangle> void symmetric_matrix_rank_2k_update(InMat1 A, InMat2 B, OutMat C, Triangle t); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, possibly-packed-out-matrix OutMat, class Triangle> void symmetric_matrix_rank_2k_update(ExecutionPolicy&& exec, InMat1 A, InMat2 B, OutMat C, Triangle t);
8
#
Effects: Computes .
🔗
template<in-matrix InMat1, in-matrix InMat2, possibly-packed-out-matrix OutMat, class Triangle> void hermitian_matrix_rank_2k_update(InMat1 A, InMat2 B, OutMat C, Triangle t); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, possibly-packed-out-matrix OutMat, class Triangle> void hermitian_matrix_rank_2k_update(ExecutionPolicy&& exec, InMat1 A, InMat2 B, OutMat C, Triangle t);
9
#
Effects: Computes .
🔗
template<in-matrix InMat1, in-matrix InMat2, in-matrix InMat3, possibly-packed-out-matrix OutMat, class Triangle> void symmetric_matrix_rank_2k_update(InMat1 A, InMat2 B, InMat3 E, OutMat C, Triangle t); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, in-matrix InMat3, possibly-packed-out-matrix OutMat, class Triangle> void symmetric_matrix_rank_2k_update(ExecutionPolicy&& exec, InMat1 A, InMat2 B, InMat3 E, OutMat C, Triangle t);
Computes .
🔗
template<in-matrix InMat1, in-matrix InMat2, in-matrix InMat3, possibly-packed-out-matrix OutMat, class Triangle> void hermitian_matrix_rank_2k_update(InMat1 A, InMat2 B, InMat3 E, OutMat C, Triangle t); template<class ExecutionPolicy, in-matrix InMat1, in-matrix InMat2, in-matrix InMat3, possibly-packed-out-matrix OutMat, class Triangle> void hermitian_matrix_rank_2k_update(ExecutionPolicy&& exec, InMat1 A, InMat2 B, InMat3 E, OutMat C, Triangle t);
10
#
Effects: Computes .

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

1
#
[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);
2
#
These functions perform multiple matrix solves, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).
3
#
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.
4
#
Preconditions:
  • (4.1)
    multipliable(A, X, B) is true, and
  • (4.2)
    A.extent(0) == A.extent(1) is true.
5
#
Effects: Computes such that , and assigns each element of to the corresponding element of X.
If no such exists, then the elements of X are valid but unspecified.
6
#
Complexity: .
7
#
[Note 2: 
Since the triangular matrix is on the left, the desired divide implementation in the case of noncommutative multiplication is mathematically equivalent to , where x is the first argument and y is the second argument, and 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);
8
#
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);
9
#
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);
10
#
These functions perform multiple matrix solves, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).
11
#
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.
12
#
Preconditions:
  • (12.1)
    multipliable(X, A, B) is true, and
  • (12.2)
    A.extent(0) == A.extent(1) is true.
13
#
Effects: Computes such that , and assigns each element of to the corresponding element of X.
If no such exists, then the elements of X are valid but unspecified.
14
#
Complexity: O( B.extent(0) B.extent(1) 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 , where x is the first argument and y is the second argument, and 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);
15
#
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);
16
#
Effects: Equivalent to: triangular_matrix_matrix_right_solve(std::forward<ExecutionPolicy>(exec), A, t, d, B, X, divides<void>{});

29.9.15.7 Solve multiple triangular linear systems in-place [linalg.algs.blas3.inplacetrsm]

1
#
[Note 1: 
These functions correspond to the BLAS function xTRSM[bib].
— end note]
🔗
template<in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat, class BinaryDivideOp> void triangular_matrix_matrix_left_solve(InMat A, Triangle t, DiagonalStorage d, InOutMat B, BinaryDivideOp divide); template<class ExecutionPolicy, in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat, class BinaryDivideOp> void triangular_matrix_matrix_left_solve(ExecutionPolicy&& exec, InMat A, Triangle t, DiagonalStorage d, InOutMat B, BinaryDivideOp divide);
2
#
These functions perform multiple in-place matrix solves, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).
[Note 2: 
This algorithm makes it possible to compute factorizations like Cholesky and LU in place.
Performing triangular solve in place hinders parallelization.
However, other ExecutionPolicy specific optimizations, such as vectorization, are still possible.
— end note]
3
#
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)
    possibly-multipliable<InMat, InOutMat, InOutMat>() is true; and
  • (3.3)
    compatible-static-extents<InMat, InMat>(0, 1) is true.
4
#
Preconditions:
  • (4.1)
    multipliable(A, B, B) is true, and
  • (4.2)
    A.extent(0) == A.extent(1) is true.
5
#
Effects: Computes such that , and assigns each element of to the corresponding element of B.
If so such exists, then the elements of B are valid but unspecified.
6
#
Complexity: .
🔗
template<in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat> void triangular_matrix_matrix_left_solve(InMat A, Triangle t, DiagonalStorage d, InOutMat B);
7
#
Effects: Equivalent to: triangular_matrix_matrix_left_solve(A, t, d, B, divides<void>{});
🔗
template<class ExecutionPolicy, in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat> void triangular_matrix_matrix_left_solve(ExecutionPolicy&& exec, InMat A, Triangle t, DiagonalStorage d, InOutMat B);
8
#
Effects: Equivalent to: triangular_matrix_matrix_left_solve(std::forward<ExecutionPolicy>(exec), A, t, d, B, divides<void>{});
🔗
template<in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat, class BinaryDivideOp> void triangular_matrix_matrix_right_solve(InMat A, Triangle t, DiagonalStorage d, InOutMat B, BinaryDivideOp divide); template<class ExecutionPolicy, in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat, class BinaryDivideOp> void triangular_matrix_matrix_right_solve(ExecutionPolicy&& exec, InMat A, Triangle t, DiagonalStorage d, InOutMat B, BinaryDivideOp divide);
9
#
These functions perform multiple in-place matrix solves, taking into account the Triangle and DiagonalStorage parameters that apply to the triangular matrix A ([linalg.general]).
[Note 3: 
This algorithm makes it possible to compute factorizations like Cholesky and LU in place.
Performing triangular solve in place hinders parallelization.
However, other ExecutionPolicy specific optimizations, such as vectorization, are still possible.
— end note]
10
#
Mandates:
  • (10.1)
    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;
  • (10.2)
    possibly-multipliable<InOutMat, InMat, InOutMat>() is true; and
  • (10.3)
    compatible-static-extents<InMat, InMat>(0, 1) is true.
11
#
Preconditions:
  • (11.1)
    multipliable(B, A, B) is true, and
  • (11.2)
    A.extent(0) == A.extent(1) is true.
12
#
Effects: Computes such that , and assigns each element of to the corresponding element of B.
If so such exists, then the elements of B are valid but unspecified.
13
#
Complexity: .
🔗
template<in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat> void triangular_matrix_matrix_right_solve(InMat A, Triangle t, DiagonalStorage d, InOutMat B);
14
#
Effects: Equivalent to: triangular_matrix_matrix_right_solve(A, t, d, B, divides<void>{});
🔗
template<class ExecutionPolicy, in-matrix InMat, class Triangle, class DiagonalStorage, inout-matrix InOutMat> void triangular_matrix_matrix_right_solve(ExecutionPolicy&& exec, InMat A, Triangle t, DiagonalStorage d, InOutMat B);
15
#
Effects: Equivalent to: triangular_matrix_matrix_right_solve(std::forward<ExecutionPolicy>(exec), A, t, d, B, divides<void>{});