linalg: Matrix norms

[perazz]

Address #820 in joint effort with @jalvesz.
Compute matrix norms using LAPACK *LANGE backend.
Array $A$, real or complex, has general rank n>=2.

For higher ranks n>2, user may specify two dimensions dims(2) the norms should be computed along. dims == [1,2] is otherwise assumed by default.

Proposed implementation

• na = mnorm(a [, order=1] [, dim=dim] [, err=err]): function interface

Key facts

• The implementations supported by *LANGE are currently provided:

  • 1, '1': 1-norm or "maximum column sum", $\max_j \sum_i{ \left|a_{i,j}\right| }$
  • 2, '2': 2-norm or "maximum singular value", maxval(svdvals(a))
  • 'Euclidean','Frobenius','fro' (or order not provided): Frobenius norm, $\sqrt{\sum_{i,j}{ a_{i,j}^2}} $
  • 'inf', huge(0): $\infty$-norm or "maximum row sum", $\max_i \sum_j{ \left|a_{i,j}\right| }$

• order can either be an integer (1, huge(0)) or a character input ('1', '2', 'inf', 'euclidean', ...). For similarity with NumPy/SciPy, this is an optional argument. If not provided, the Frobenius norm is computed.

Progress

• base implementation
• tests
• documentation
• submodule
• examples

Prior art

• Numpy: linalg.matrix_norm(x, /, *, keepdims=False, ord='fro')
• Scipy: norm(a, ord=None, axis=None, keepdims=False, check_finite=True)

cc: @jvdp1 @jalvesz @loiseaujc