28 Numerics library [numerics]

28.9 Basic linear algebra algorithms [linalg]

28.9.13 BLAS 1 algorithms [linalg.algs.blas1]

28.9.13.12 Frobenius norm of a matrix [linalg.algs.blas1.matfrobnorm]

[Note 1:

These functions exist in the BLAS standard[bib] but are not part of the reference implementation.

— end note]

template<in-matrix InMat, class Scalar>
  Scalar matrix_frob_norm(InMat A, Scalar init);
template<class ExecutionPolicy, in-matrix InMat, class Scalar>
  Scalar matrix_frob_norm(ExecutionPolicy&& exec, InMat A, Scalar init);

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

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

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

[Note 2:

For init equal to zero, this is the Frobenius norm of the matrix A.

— end note]

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

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template<in-matrix InMat>
  auto matrix_frob_norm(InMat A);
template<class ExecutionPolicy, in-matrix InMat>
  auto matrix_frob_norm(ExecutionPolicy&& exec, InMat A);

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

Let T be decltype(a * a).

Then,

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