Multivariate Pattern Analysis in Python

# mvpa2.atlases.warehouse.norm

`mvpa2.atlases.warehouse.``norm`(x, ord=None, axis=None)

  Matrix or vector norm.

  This function is able to return one of seven different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the `ord` parameter.

  Parameters:

  x : array_like

  > Input array. If `axis` is None, `x` must be 1-D or 2-D.

  ord : {non-zero int, inf, -inf, ‘fro’}, optional

  > Order of the norm (see table under `Notes`). inf means numpy’s `inf` object.

  axis : {int, 2-tuple of ints, None}, optional

  > If `axis` is an integer, it specifies the axis of `x` along which to compute the vector norms. If `axis` is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed. If `axis` is None then either a vector norm (when `x` is 1-D) or a matrix norm (when `x` is 2-D) is returned.

  Returns:

  n : float or ndarray

  > Norm of the matrix or vector(s).

  Notes

  For values of `ord <= 0`, the result is, strictly speaking, not a mathematical ‘norm’, but it may still be useful for various numerical purposes.

  The following norms can be calculated:

  ord | norm for matrices | norm for vectors
  --- | --- | ---
  None | Frobenius norm | 2-norm
  ‘fro’ | Frobenius norm | –
  inf | max(sum(abs(x), axis=1)) | max(abs(x))
  -inf | min(sum(abs(x), axis=1)) | min(abs(x))
  0 | – | sum(x != 0)
  1 | max(sum(abs(x), axis=0)) | as below
  -1 | min(sum(abs(x), axis=0)) | as below
  2 | 2-norm (largest sing. value) | as below
  -2 | smallest singular value | as below
  other | – | sum(abs(x)**ord)**(1./ord)

  The Frobenius norm is given by [R6]:

  > ||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}

  References

  [R6] | (1, 2) G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

  Examples

      >>> from numpy import linalg as LA
      >>> a = np.arange(9) - 4
      >>> a
      array([-4, -3, -2, -1,  0,  1,  2,  3,  4])
      >>> b = a.reshape((3, 3))
      >>> b
      array([[-4, -3, -2],
             [-1,  0,  1],
             [ 2,  3,  4]])

      >>> LA.norm(a)
      7.745966692414834
      >>> LA.norm(b)
      7.745966692414834
      >>> LA.norm(b, 'fro')
      7.745966692414834
      >>> LA.norm(a, np.inf)
      4
      >>> LA.norm(b, np.inf)
      9
      >>> LA.norm(a, -np.inf)
      0
      >>> LA.norm(b, -np.inf)
      2

      >>> LA.norm(a, 1)
      20
      >>> LA.norm(b, 1)
      7
      >>> LA.norm(a, -1)
      -4.6566128774142013e-010
      >>> LA.norm(b, -1)
      6
      >>> LA.norm(a, 2)
      7.745966692414834
      >>> LA.norm(b, 2)
      7.3484692283495345

      >>> LA.norm(a, -2)
      nan
      >>> LA.norm(b, -2)
      1.8570331885190563e-016
      >>> LA.norm(a, 3)
      5.8480354764257312
      >>> LA.norm(a, -3)
      nan

  Using the `axis` argument to compute vector norms:

      >>> c = np.array([[ 1, 2, 3],
      ...               [-1, 1, 4]])
      >>> LA.norm(c, axis=0)
      array([ 1.41421356,  2.23606798,  5.        ])
      >>> LA.norm(c, axis=1)
      array([ 3.74165739,  4.24264069])
      >>> LA.norm(c, ord=1, axis=1)
      array([6, 6])

  Using the `axis` argument to compute matrix norms:

      >>> m = np.arange(8).reshape(2,2,2)
      >>> LA.norm(m, axis=(1,2))
      array([  3.74165739,  11.22497216])
      >>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :])
      (3.7416573867739413, 11.224972160321824)