4[gXdZddlZddlmZddlmZddlmZddlm Z m Z dgZ dZ d dZ y) zSparse matrix norms. N)issparse)svds)sqrtabsnormc~tjj|}tjj |S)N)sp_sputils_todatanplinalgr)xdatas T/var/www/html/bid-api/venv/lib/python3.12/site-packages/scipy/sparse/linalg/_norm.py_sparse_frobenius_normrs) ;;  q !D 99>>$ ct|s td||dvr t|S|j}|d}n1t |t s!d} t |}||k7r t||f}d}t|dk(r|\}}| |cxkr|krnn| |cxkr|ksnd|d|j} t| ||z||zk(r td |dk(rt|d d \} } } | d S|dk(rt|d k(r.t|j|j|dS|tj k(r.t|j|j|dS|dk(r.t|j|j#|dS|tj k(r.t|j|j#|dS|dvr t|Stdt|d k(r|\} | | cxkr|ksnd|d|j} t| |tj k(rt|j| } n|tj k(rt|j#| } n|d k(r|d k7j| } n|d k(rt|j| } n|dvr4t%t|j'dj| } nG |d ztj&t|j'|j| d |z } t)| dr| j+j-St)| dr| j.j-S| j-Std#t$r}t||d}~wwxYw#t$r}td|d}~wwxYw)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 z for an array with shape zDuplicate 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 isinstancetupleintlenshape ValueErrorrNotImplementedErrorrsummaxr infminrpowerhasattrrravelr)rordrmsgint_axisendrow_axiscol_axismessage_saMs rrrs| A;DEE |11%a((  A | e $F (4yH 8 C. { B 4yA~!(x$"$")=2)=%dX-FqwwkRGW% % b=HrM )45 5 !81(3GAq!Q4K BY% % AXq6::8:,00h0?D D BFF]q6::8:,00h0?D D BYq6::8:,00h0?D D RVVG^q6::8:,00h0?D D & &)!, ,?@ @ Ta q 2 %dX-FqwwkRGW% % "&&=A  "A RVVG^A  "A AXa ! $A AXA  "A I SV\\!_((a(01A KaQc*..A.6C@A 1i 99;$$& & Q_3399; 779 ABBA (C.a ' (l K !BCJ Ks0 OO O OO O8' O33O8)NN)__doc__numpyr scipy.sparserscipy.sparse.linalgrsparser rr__all__rrrrrCs.!$ ( nCr