U ǻ|e@s`dZddlZddlmZddlmZddlmZddlm Z m Z m Z dgZ ddZ d d dZdS) zSparse matrix norms. N)issparse)svds)InfsqrtabsnormcCstj|}tj|S)N)sp_sputils_todatanplinalgr)xdatarV/var/www/website-v5/atlas_env/lib/python3.8/site-packages/scipy/sparse/linalg/_norm.py_sparse_frobenius_norms rc Cs~t|std|dkr(|dkr(t|S|}|dkr>d}n^t|tsd}z t|}Wn,tk r}zt||W5d}~XYnX||krt||f}d}t|dkr|\}}| |kr|krnn| |kr|ksntd||j f||||krtd|dkr:t |d d d \} } } | d S|d krJt n|d krpt |j |dj|ddS|tkrt |j |dj|ddS|dkrt |j |dj|ddS|t krt |j |dj|ddS|dkrt|Stdnxt|d krr|\} | | kr0|ksDntd||j f|tkr`t |j| d} n|t kr~t |j| d} n|d kr|d kj | d} n|d krt |j | d} n|dkrtt |dj | d} n^z |d Wn.tk r}ztd|W5d}~XYnXtt ||j | dd |} t| drR| St| drh| jS| SntddS)a Norm of a sparse matrix This function is able to return one of seven different matrix norms, depending on the value of the ``ord`` parameter. Parameters ---------- x : a sparse matrix Input sparse matrix. 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 Notes ----- Some of the ord are not implemented because some associated functions like, _multi_svd_norm, are not yet available for sparse matrix. This docstring is modified based on numpy.linalg.norm. https://github.com/numpy/numpy/blob/main/numpy/linalg/linalg.py The following norms can be calculated: ===== ============================ ord norm for sparse matrices ===== ============================ None Frobenius norm 'fro' Frobenius norm inf max(sum(abs(x), axis=1)) -inf min(sum(abs(x), axis=1)) 0 abs(x).sum(axis=axis) 1 max(sum(abs(x), axis=0)) -1 min(sum(abs(x), axis=0)) 2 Spectral norm (the largest singular value) -2 Not implemented other Not implemented ===== ============================ 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.sparse import * >>> import numpy as np >>> from scipy.sparse.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]]) >>> b = csr_matrix(b) >>> norm(b) 7.745966692414834 >>> norm(b, 'fro') 7.745966692414834 >>> norm(b, np.inf) 9 >>> norm(b, -np.inf) 2 >>> norm(b, 1) 7 >>> norm(b, -1) 6 The matrix 2-norm or the spectral norm is the largest singular value, computed approximately and with limitations. >>> b = diags([-1, 1], [0, 1], shape=(9, 10)) >>> norm(b, 2) 1.9753... z*input is not sparse. use numpy.linalg.normN)Nfrof)rz6'axis' must be None, an integer or a tuple of integersz*Invalid axis %r for an array with shape %rzDuplicate axes given.rlobpcg)ksolverr)axis)rr)Nrrz Invalid norm order for matrices.)rNzInvalid norm order for vectors.toarrayAz&Improper number of dimensions to norm.)r TypeErrorrtocsr isinstancetupleintlen ValueErrorshaperNotImplementedErrorrsummaxrminrpowerr hasattrrravelr) r ordrmsgZint_axisendrow_axiscol_axis_saMrrrrs^  2              "     )NN)__doc__numpyr scipy.sparserscipy.sparse.linalgrsparserrrr__all__rrrrrrs