# scipy/scipy

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 from __future__ import division, print_function, absolute_import import numpy as np from numpy.linalg import LinAlgError from . import blas __all__ = ['LinAlgError', 'norm'] _nrm2_prefix = {'f': 's', 'F': 'sc', 'D': 'dz'} def norm(a, ord=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 : (M,) or (M, N) array_like Input array. ord : {non-zero int, inf, -inf, 'fro'}, optional Order of the norm (see table under ``Notes``). inf means numpy's `inf` object. Returns ------- norm : float Norm of the matrix or vector. 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]^{1/2}` 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 >>> 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) nan >>> norm(b, -2) 1.8570331885190563e-016 >>> norm(a, 3) 5.8480354764257312 >>> norm(a, -3) nan """ # Differs from numpy only in non-finite handling and the use of # blas a = np.asarray_chkfinite(a) if ord in (None, 2) and (a.ndim == 1) and (a.dtype.char in 'fdFD'): # use blas for fast and stable euclidean norm func_name = _nrm2_prefix.get(a.dtype.char, 'd') + 'nrm2' nrm2 = getattr(blas, func_name) return nrm2(a) return np.linalg.norm(a, ord=ord) def _datacopied(arr, original): """ Strict check for `arr` not sharing any data with `original`, under the assumption that arr = asarray(original) """ if arr is original: return False if not isinstance(original, np.ndarray) and hasattr(original, '__array__'): return False return arr.base is None
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