29 Numerics library [numerics]

29.9 Basic linear algebra algorithms [linalg]

29.9.13 BLAS 1 algorithms [linalg.algs.blas1]

29.9.13.7 Dot product of two vectors [linalg.algs.blas1.dot]

[Note 1:

The functions in this section correspond to the BLAS functions xDOT, xDOTU, and xDOTC[bib].

— end note]

The following elements apply to all functions in [linalg.algs.blas1.dot].

Mandates: compatible-static-extents<InVec1, InVec2>(0, 0) is true.

Preconditions: v1.extent(0) equals v2.extent(0).

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template<in-vector InVec1, in-vector InVec2, class Scalar>
  Scalar dot(InVec1 v1, InVec2 v2, Scalar init);
template<class ExecutionPolicy, in-vector InVec1, in-vector InVec2, class Scalar>
  Scalar dot(ExecutionPolicy&& exec,
             InVec1 v1, InVec2 v2, Scalar init);

These functions compute a non-conjugated dot product with an explicitly specified result type.

Returns: Let N be v1.extent(0).

(6.1)
init if N is zero;
(6.2)
otherwise, GENERALIZED_SUM(plus<>(), init, v1[0]*v2[0], …, v1[N-1]*v2[N-1]).

Remarks: If InVec1::value_type, InVec2::value_type, and Scalar are all floating-point types or specializations of complex, and if Scalar has higher precision than InVec1::value_type or InVec2::value_type, then intermediate terms in the sum use Scalar's precision or greater.

🔗

  template<in-vector InVec1, in-vector InVec2>
    auto dot(InVec1 v1, InVec2 v2);
  template<class ExecutionPolicy, in-vector InVec1, in-vector InVec2>
    auto dot(ExecutionPolicy&& exec,
             InVec1 v1, InVec2 v2);

These functions compute a non-conjugated dot product with a default result type.

Effects: Let T be decltype(declval<typename InVec1::value_type>() * declval<typename InVec2::value_type>()).

Then,

(9.1)
the two-parameter overload is equivalent to: return dot(v1, v2, T{}); and
(9.2)
the three-parameter overload is equivalent to: return dot(std::forward<ExecutionPolicy>(exec), v1, v2, T{});

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template<in-vector InVec1, in-vector InVec2, class Scalar>
  Scalar dotc(InVec1 v1, InVec2 v2, Scalar init);
template<class ExecutionPolicy, in-vector InVec1, in-vector InVec2, class Scalar>
  Scalar dotc(ExecutionPolicy&& exec,
              InVec1 v1, InVec2 v2, Scalar init);

These functions compute a conjugated dot product with an explicitly specified result type.

Effects:

(11.1)
The three-parameter overload is equivalent to: return dot(conjugated(v1), v2, init); and
(11.2)
the four-parameter overload is equivalent to: return dot(std::forward<ExecutionPolicy>(exec), conjugated(v1), v2, init);

🔗

template<in-vector InVec1, in-vector InVec2>
  auto dotc(InVec1 v1, InVec2 v2);
template<class ExecutionPolicy, in-vector InVec1, in-vector InVec2>
  auto dotc(ExecutionPolicy&& exec,
            InVec1 v1, InVec2 v2);

These functions compute a conjugated dot product with a default result type.

Effects: Let T be decltype(conj-if-needed(declval<typename InVec1::value_type>()) * declval<typename InVec2::value_type>()).

Then,

(13.1)
the two-parameter overload is equivalent to: return dotc(v1, v2, T{}); and
(13.2)
the three-parameter overload is equivalent to return dotc(std::forward<ExecutionPolicy>(exec), v1, v2, T{});