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[linalg.algs.blas2.gemv]

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+ #### General matrix-vector product <a id="linalg.algs.blas2.gemv">[[linalg.algs.blas2.gemv]]</a>
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+
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+ [*Note 1*: These functions correspond to the BLAS function
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+ `xGEMV`. — *end note*]
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+
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+ The following elements apply to all functions in
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+ [[linalg.algs.blas2.gemv]].
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+
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+ *Mandates:*
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+
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+ - `possibly-multipliable<decltype(A), decltype(x), decltype(y)>()` is
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+ `true`, and
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+ - `possibly-addable<decltype(x), decltype(y), decltype(z)>()` is `true`
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+ for those overloads that take a `z` parameter.
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+
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+ *Preconditions:*
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+
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+ - `multipliable(A,x,y)` is `true`, and
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+ - `addable(x,y,z)` is `true` for those overloads that take a `z`
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+ parameter.
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+
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+ *Complexity:* 𝑂(`x.extent(0)` × `A.extent(1)`).
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+
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+ ``` cpp
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+ template<in-matrix InMat, in-vector InVec, out-vector OutVec>
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+ void matrix_vector_product(InMat A, InVec x, OutVec y);
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+ template<class ExecutionPolicy, in-matrix InMat, in-vector InVec, out-vector OutVec>
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+ void matrix_vector_product(ExecutionPolicy&& exec, InMat A, InVec x, OutVec y);
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+ ```
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+
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+ These functions perform an overwriting matrix-vector product.
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+
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+ *Effects:* Computes y = A x.
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+
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+ [*Example 1*:
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+
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+ ``` cpp
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+ constexpr size_t num_rows = 5;
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+ constexpr size_t num_cols = 6;
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+
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+ // y = 3.0 * A * x
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+ void scaled_matvec_1(mdspan<double, extents<size_t, num_rows, num_cols>> A,
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+ mdspan<double, extents<size_t, num_cols>> x, mdspan<double, extents<size_t, num_rows>> y) {
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+ matrix_vector_product(scaled(3.0, A), x, y);
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+ }
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+
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+ // z = 7.0 times the transpose of A, times y
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+ void scaled_transposed_matvec(mdspan<double, extents<size_t, num_rows, num_cols>> A,
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+ mdspan<double, extents<size_t, num_rows>> y, mdspan<double, extents<size_t, num_cols>> z) {
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+ matrix_vector_product(scaled(7.0, transposed(A)), y, z);
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+ }
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+ ```
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+
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+ — *end example*]
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+
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+ ``` cpp
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+ template<in-matrix InMat, in-vector InVec1, in-vector InVec2, out-vector OutVec>
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+ void matrix_vector_product(InMat A, InVec1 x, InVec2 y, OutVec z);
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+ template<class ExecutionPolicy,
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+ in-matrix InMat, in-vector InVec1, in-vector InVec2, out-vector OutVec>
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+ void matrix_vector_product(ExecutionPolicy&& exec,
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+ InMat A, InVec1 x, InVec2 y, OutVec z);
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+ ```
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+
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+ These functions perform an updating matrix-vector product.
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+
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+ *Effects:* Computes z = y + A x.
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+
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+ *Remarks:* `z` may alias `y`.
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+
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+ [*Example 2*:
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+
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+ ``` cpp
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+ // y = 3.0 * A * x + 2.0 * y
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+ void scaled_matvec_2(mdspan<double, extents<size_t, num_rows, num_cols>> A,
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+ mdspan<double, extents<size_t, num_cols>> x, mdspan<double, extents<size_t, num_rows>> y) {
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+ matrix_vector_product(scaled(3.0, A), x, scaled(2.0, y), y);
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+ }
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+ ```
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+
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+ — *end example*]
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+