(phfSrSSKrSSKJr SSKJr SSKJr SSKJ r SSKJ r J r S/r SrS S jrg) zSparse matrix norms. N)issparse)svds)convert_pydata_sparse_to_scipy)sqrtabsnormc~[RRU5n[RR U5$)N)sp_sputils_todatanplinalgr)xdatas L/var/www/html/venv/lib/python3.13/site-packages/scipy/sparse/linalg/_norm.py_sparse_frobenius_normrs) ;;  q !D 99>>$ cP[USS9n[U5(d [S5eUcUS;a [U5$UR 5nUc[ [ UR55nO6[U[ 5(d!Sn[U5nX$:wa [U5eU4nURn[U5S:XGagUupxU*Us=::aU:aO OU*Us=::aU:d!O SU<S UR<3n [U 5eXv-X-:Xa [S 5eUS:Xa[US S S 9upn U S$US:Xa[eUS :Xa&[U5R!US9R#5$U[$R&:Xa&[U5R!US9R#5$US:Xa&[U5R!US9R)5$U[$R&*:Xa&[U5R!US9R)5$US;a [U5$[S5e[U5S :XGaUun U*U s=::aU:d!O SU<S UR<3n [U 5eU[$R&:Xa[U5R#U S9n OU[$R&*:Xa[U5R)U S9n OUS:XaUS:gR!U S9n OUS :Xa[U5R!U S9n O|US;a1[+[U5R-S5R!U S95n OEUS - [$R,"[U5R-U5R!U S9S U- 5n [/U S5(aU R15R35$[/U S5(aU R4R35$U R35$[S5e![an[U5UeSnAff=f![an[S5UeSnAff=f)az 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 array Input sparse array. 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 array. 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 arrays ===== ============================ 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 csr_array, diags_array >>> 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_array(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_array([-1, 1], [0, 1], shape=(9, 10)) >>> norm(b, 2) 1.9753... csr) target_formatz*input is not sparse. use numpy.linalg.normN)Nfrofz6'axis' must be None, an integer or a tuple of integersz Invalid axis z for an array with shape zDuplicate axes given.lobpcg)ksolverr)axis)Nrrz Invalid norm order for matrices.)rNzInvalid norm order for vectors.toarrayAz&Improper number of dimensions to norm.)rr TypeErrorrtocsrtuplerangendim isinstanceintlenshape ValueErrorrNotImplementedErrorrsummaxr infminrpowerhasattrr!ravelr")rordrmsgint_axisendrow_axiscol_axismessage_saMs rrrs| 'q>A A;;DEE |11%a((  A |U166]# e $ $F (4yH  C. { B 4yA~!x$"$")=2)=%dX-FqwwkRGW% % =HM )45 5 !81(3GA!Q4K BY% % AXq6::8:,002 2 BFF]q6::8:,002 2 BYq6::8:,002 2 RVVG^q6::8:,002 2 & &)!, ,?@ @ 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 O,3P , P6 PP P% P  P%)NN)__doc__numpyr scipy.sparserscipy.sparse.linalgrscipy.sparse._sputilsrsparser rr__all__rrrrrIs1!$@ ( oCr