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 ---------- a : (M,) or (M, N) array_like Input array. If `axis` is None, `a` must be 1D or 2D. ord : non-zero int, inf, -inf, 'fro', optional Order of the norm (see table under ``Notes``). inf means NumPy's `inf` object axis : nt, 2-tuple of ints, None, optional If `axis` is an integer, it specifies the axis of `a` 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 `a` is 1-D) or a matrix norm (when `a` is 2-D) is returned. keepdims : bool, optional If this is set to True, the axes which are normed over are left in the result as dimensions with size one. With this option the result will broadcast correctly against the original `a`. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.
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 1_:
:math:`||A||_F = \sum_{i,j} abs(a_{i,j})^2^
/2
`
The ``axis`` and ``keepdims`` arguments are passed directly to ``numpy.linalg.norm`` and are only usable if they are supported by the version of numpy in use.
References ---------- .. 1 G. H. Golub and C. F. Van Loan, *Matrix Computations*, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples -------- >>> from scipy.linalg import norm >>> a = np.arange(9) - 4.0 >>> 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.])
>>> norm(a) 7.745966692414834 >>> norm(b) 7.745966692414834 >>> norm(b, 'fro') 7.745966692414834 >>> norm(a, np.inf) 4 >>> norm(b, np.inf) 9 >>> norm(a, -np.inf) 0 >>> norm(b, -np.inf) 2
>>> norm(a, 1) 20 >>> norm(b, 1) 7 >>> norm(a, -1) -4.6566128774142013e-010 >>> norm(b, -1) 6 >>> norm(a, 2) 7.745966692414834 >>> norm(b, 2) 7.3484692283495345
>>> norm(a, -2) 0 >>> norm(b, -2) 1.8570331885190563e-016 >>> norm(a, 3) 5.8480354764257312 >>> norm(a, -3) 0