28 Numerics library [numerics]

28.9 Basic linear algebra algorithms [linalg]

28.9.13 BLAS 1 algorithms [linalg.algs.blas1]

28.9.13.9 Euclidean norm of a vector [linalg.algs.blas1.nrm2]

template<in-vector InVec, class Scalar>
  Scalar vector_two_norm(InVec v, Scalar init);
template<class ExecutionPolicy, in-vector InVec, class Scalar>
  Scalar vector_two_norm(ExecutionPolicy&& exec, InVec v, Scalar init);

[Note 1:

These functions correspond to the BLAS function xNRM2[bib].

— end note]

Mandates: Let a be abs-if-needed(declval<typename InVec::value_type>()).

Then, decltype(
init + a * a is convertible to Scalar.

Returns: The square root of the sum of the square of init and the squares of the absolute values of the elements of v.

[Note 2:

For init equal to zero, this is the Euclidean norm (also called 2-norm) of the vector v.

— end note]

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

[Note 3:

An implementation of this function for floating-point types T can use the scaled_sum_of_squares result from vector_sum_of_squares(x, {.scaling_factor=1.0, .scaled_sum_of_squares=init}).

— end note]

🔗

template<in-vector InVec>
  auto vector_two_norm(InVec v);
template<class ExecutionPolicy, in-vector InVec>
  auto vector_two_norm(ExecutionPolicy&& exec, InVec v);

Effects: Let a be abs-if-needed(declval<typename InVec::value_type>()).

Let T be decltype(a * a).

Then,

(5.1)
the one-parameter overload is equivalent to: return vector_two_norm(v, T{}); and
(5.2)
the two-parameter overload is equivalent to: return vector_two_norm(std::forward<ExecutionPolicy>(exec), v, T{});