From Jason Turner

[linalg.algs.blas3.xxmm]

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+ #### Symmetric, Hermitian, and triangular matrix-matrix product <a id="linalg.algs.blas3.xxmm">[[linalg.algs.blas3.xxmm]]</a>
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+
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+ [*Note 1*: These functions correspond to the BLAS functions `xSYMM`,
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+ `xHEMM`, and `xTRMM`. — *end note*]
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+
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+ The following elements apply to all functions in
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+ [[linalg.algs.blas3.xxmm]] in addition to function-specific elements.
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+
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+ *Mandates:*
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+
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+ - `possibly-multipliable<decltype(A), decltype(B), decltype(C)>()` is
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+ `true`, and
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+ - `possibly-addable<decltype(E), decltype(E), decltype(C)>()` is `true`
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+ for those overloads that take an `E` parameter.
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+
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+ *Preconditions:*
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+
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+ - `multipliable(A, B, C)` is `true`, and
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+ - `addable(E, E, C)` is `true` for those overloads that take an `E`
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+ parameter.
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+
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+ *Complexity:* 𝑂(`A.extent(0)` × `A.extent(1)` × `B.extent(1)`).
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+
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+ ``` cpp
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+ template<in-matrix InMat1, class Triangle, in-matrix InMat2, out-matrix OutMat>
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+ void symmetric_matrix_product(InMat1 A, Triangle t, InMat2 B, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, class Triangle, in-matrix InMat2, out-matrix OutMat>
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+ void symmetric_matrix_product(ExecutionPolicy&& exec, InMat1 A, Triangle t, InMat2 B, OutMat C);
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+
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+ template<in-matrix InMat1, class Triangle, in-matrix InMat2, out-matrix OutMat>
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+ void hermitian_matrix_product(InMat1 A, Triangle t, InMat2 B, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, class Triangle, in-matrix InMat2, out-matrix OutMat>
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+ void hermitian_matrix_product(ExecutionPolicy&& exec, InMat1 A, Triangle t, InMat2 B, OutMat C);
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+
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+ template<in-matrix InMat1, class Triangle, class DiagonalStorage,
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+ in-matrix InMat2, out-matrix OutMat>
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+ void triangular_matrix_product(InMat1 A, Triangle t, DiagonalStorage d, InMat2 B, OutMat C);
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+ template<class ExecutionPolicy, in-matrix InMat1, class Triangle, class DiagonalStorage,
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+ in-matrix InMat2, out-matrix OutMat>
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+ void triangular_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, Triangle t, DiagonalStorage d, InMat2 B, OutMat C);
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+ ```
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+
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+ These functions perform a matrix-matrix multiply, taking into account
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+ the `Triangle` and `DiagonalStorage` (if applicable) parameters that
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+ apply to the symmetric, Hermitian, or triangular (respectively) matrix
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+ `A` [[linalg.general]].
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+
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+ *Mandates:*
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+
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+ - If `InMat1` has `layout_blas_packed` layout, then the layout’s
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+ `Triangle` template argument has the same type as the function’s
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+ `Triangle` template argument; and
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+ - *`compatible-static-extents`*`<InMat1, InMat1>(0, 1)` is `true`.
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+
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+ *Preconditions:* `A.extent(0) == A.extent(1)` is `true`.
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+
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+ *Effects:* Computes C = A B.
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+
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+ ``` cpp
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+ template<in-matrix InMat1, in-matrix InMat2, class Triangle, out-matrix OutMat>
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+ void symmetric_matrix_product(InMat1 A, InMat2 B, Triangle t, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, in-matrix InMat2, class Triangle, out-matrix OutMat>
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+ void symmetric_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, InMat2 B, Triangle t, OutMat C);
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+
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+ template<in-matrix InMat1, in-matrix InMat2, class Triangle, out-matrix OutMat>
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+ void hermitian_matrix_product(InMat1 A, InMat2 B, Triangle t, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, in-matrix InMat2, class Triangle, out-matrix OutMat>
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+ void hermitian_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, InMat2 B, Triangle t, OutMat C);
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+
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+ template<in-matrix InMat1, in-matrix InMat2, class Triangle, class DiagonalStorage,
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+ out-matrix OutMat>
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+ void triangular_matrix_product(InMat1 A, InMat2 B, Triangle t, DiagonalStorage d, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, in-matrix InMat2, class Triangle, class DiagonalStorage,
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+ out-matrix OutMat>
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+ void triangular_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, InMat2 B, Triangle t, DiagonalStorage d, OutMat C);
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+ ```
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+
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+ These functions perform a matrix-matrix multiply, taking into account
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+ the `Triangle` and `DiagonalStorage` (if applicable) parameters that
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+ apply to the symmetric, Hermitian, or triangular (respectively) matrix
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+ `B` [[linalg.general]].
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+
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+ *Mandates:*
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+
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+ - If `InMat2` has `layout_blas_packed` layout, then the layout’s
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+ `Triangle` template argument has the same type as the function’s
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+ `Triangle` template argument; and
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+ - *`compatible-static-extents`*`<InMat2, InMat2>(0, 1)` is `true`.
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+
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+ *Preconditions:* `B.extent(0) == B.extent(1)` is `true`.
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+
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+ *Effects:* Computes C = A B.
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+
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+ ``` cpp
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+ template<in-matrix InMat1, class Triangle, in-matrix InMat2, in-matrix InMat3,
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+ out-matrix OutMat>
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+ void symmetric_matrix_product(InMat1 A, Triangle t, InMat2 B, InMat3 E, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, class Triangle, in-matrix InMat2, in-matrix InMat3,
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+ out-matrix OutMat>
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+ void symmetric_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, Triangle t, InMat2 B, InMat3 E, OutMat C);
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+
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+ template<in-matrix InMat1, class Triangle, in-matrix InMat2, in-matrix InMat3,
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+ out-matrix OutMat>
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+ void hermitian_matrix_product(InMat1 A, Triangle t, InMat2 B, InMat3 E, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, class Triangle, in-matrix InMat2, in-matrix InMat3,
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+ out-matrix OutMat>
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+ void hermitian_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, Triangle t, InMat2 B, InMat3 E, OutMat C);
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+
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+ template<in-matrix InMat1, class Triangle, class DiagonalStorage,
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+ in-matrix InMat2, in-matrix InMat3, out-matrix OutMat>
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+ void triangular_matrix_product(InMat1 A, Triangle t, DiagonalStorage d, InMat2 B, InMat3 E,
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+ OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, class Triangle, class DiagonalStorage,
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+ in-matrix InMat2, in-matrix InMat3, out-matrix OutMat>
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+ void triangular_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, Triangle t, DiagonalStorage d, InMat2 B, InMat3 E,
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+ OutMat C);
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+ ```
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+
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+ These functions perform a potentially overwriting matrix-matrix
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+ multiply-add, taking into account the `Triangle` and `DiagonalStorage`
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+ (if applicable) parameters that apply to the symmetric, Hermitian, or
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+ triangular (respectively) matrix `A` [[linalg.general]].
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+
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+ *Mandates:*
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+
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+ - If `InMat1` has `layout_blas_packed` layout, then the layout’s
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+ `Triangle` template argument has the same type as the function’s
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+ `Triangle` template argument; and
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+ - *`compatible-static-extents`*`<InMat1, InMat1>(0, 1)` is `true`.
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+
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+ *Preconditions:* `A.extent(0) == A.extent(1)` is `true`.
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+
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+ *Effects:* Computes C = E + A B.
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+
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+ *Remarks:* `C` may alias `E`.
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+
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+ ``` cpp
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+ template<in-matrix InMat1, in-matrix InMat2, class Triangle, in-matrix InMat3,
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+ out-matrix OutMat>
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+ void symmetric_matrix_product(InMat1 A, InMat2 B, Triangle t, InMat3 E, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, in-matrix InMat2, class Triangle, in-matrix InMat3,
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+ out-matrix OutMat>
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+ void symmetric_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, InMat2 B, Triangle t, InMat3 E, OutMat C);
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+
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+ template<in-matrix InMat1, in-matrix InMat2, class Triangle, in-matrix InMat3,
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+ out-matrix OutMat>
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+ void hermitian_matrix_product(InMat1 A, InMat2 B, Triangle t, InMat3 E, OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, in-matrix InMat2, class Triangle, in-matrix InMat3,
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+ out-matrix OutMat>
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+ void hermitian_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, InMat2 B, Triangle t, InMat3 E, OutMat C);
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+
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+ template<in-matrix InMat1, in-matrix InMat2, class Triangle, class DiagonalStorage,
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+ in-matrix InMat3, out-matrix OutMat>
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+ void triangular_matrix_product(InMat1 A, InMat2 B, Triangle t, DiagonalStorage d, InMat3 E,
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+ OutMat C);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat1, in-matrix InMat2, class Triangle, class DiagonalStorage,
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+ in-matrix InMat3, out-matrix OutMat>
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+ void triangular_matrix_product(ExecutionPolicy&& exec,
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+ InMat1 A, InMat2 B, Triangle t, DiagonalStorage d, InMat3 E,
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+ OutMat C);
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+ ```
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+
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+ These functions perform a potentially overwriting matrix-matrix
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+ multiply-add, taking into account the `Triangle` and `DiagonalStorage`
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+ (if applicable) parameters that apply to the symmetric, Hermitian, or
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+ triangular (respectively) matrix `B` [[linalg.general]].
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+
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+ *Mandates:*
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+
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+ - If `InMat2` has `layout_blas_packed` layout, then the layout’s
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+ `Triangle` template argument has the same type as the function’s
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+ `Triangle` template argument; and
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+ - *`compatible-static-extents`*`<InMat2, InMat2>(0, 1)` is `true`.
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+
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+ *Preconditions:* `B.extent(0) == B.extent(1)` is `true`.
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+
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+ *Effects:* Computes C = E + A B.
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+
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+ *Remarks:* `C` may alias `E`.
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+