Proposed resolution:

This wording is relative to N5032.

1. Modify [linalg.algs.blas1.nrm2] as indicated:

   [Drafting note: As a drive-by fix the missing closing parenthesis of the decltype form has been added.]

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

   -1- [Note 1: These functions correspond to the BLAS function xNRM2[17]. — end note]

   -2- Mandates: `InVec::value_type` and `Scalar` are either a floating-point type, or a specialization of `complex`. Let `a` be abs-if-needed(declval<typename InVec::value_type>()) and let `init_abs` be abs-if-needed(init). Then, decltype(init_abs * init_abs + a * a) is convertible to `Scalar`.

   -3- Returns: The square root of the sum of the square of `init` and the squares of the absolute values of the elements of `v`.whose terms are the following:

   • (3.1) — the square of the absolute value of `init`, and

   • (3.2) — 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]

   -4- Remarks: If `Scalar` has higher precision than `InVec::value_type`, then intermediate terms in the sum use `Scalar`'s precision or greater.

2. Modify [linalg.algs.blas1.matfrobnorm] as indicated:

   [Drafting note: As a drive-by fix one spurious `InVec` has been changed to `InMat`]

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

   -2- Mandates: InVecInMat::value_type and `Scalar` are either a floating-point type, or a specialization of `complex`. Let `a` be abs-if-needed(declval<typename InMat::value_type>()) and let `init_abs` be abs-if-needed(init). Then, decltype(init_abs * init_abs + a * a) is convertible to `Scalar`.

   -3- Returns: The square root of the sum of the squares of `init` and the absolute values of the elements of `A`.whose terms are the following:

   • (3.1) — the square of the absolute value of `init`, and

   • (3.2) — the squares of 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]

   -4- Remarks: If `Scalar` has higher precision than `InMat::value_type`, then intermediate terms in the sum use `Scalar`'s precision or greater.

This pull request discussion points out two issues with the current wording of `vector_two_norm`:

1. If `Scalar` is the complex number `1 + i`, then the current wording would make `vector_two_norm` return sqrt(2i + nonnegative_real_number). This would have nonzero imaginary part.

2. If `Scalar` is the real number `-1.0`, then the current wording would make `vector_two_norm` return sqrt(-1.0 + nonnegative_real_number). If nonnegative_real_number is less than `1`, then the result would be imaginary.

The analogous issues would also apply to `matrix_frob_norm`.

It's acceptable for the return type `Scalar` to be `complex`. However, the point of letting `init` be nonzero is to support computing the 2-norm of a vector (or the Frobenius norm of a matrix) in multiple steps. The value of `init` should thus always represent a valid 2-norm (or Frobenius norm) result.

There are two ways to fix this:

1. Impose a precondition on `init`, that it have nonnegative real part and zero imaginary part.

2. Change the Returns element to take the absolute value of `init`. (The [linalg] clause generally uses "absolute value" to mean the magnitude of a complex number, or the absolute value of a non-complex number.)

We offer (2) as the Proposed Fix, as it is consistent with the approach we took for fixing LWG 4136.