CpjlSrSSKrSSKJr SSKJr SSKJr SSKJ r SSKJ r J r S/r SrS S jrS rg) zSparse matrix norms. N)issparse)svds)convert_pydata_sparse_to_scipy)sqrtabsnormc~[RRU5n[RR U5$)N)sp_sputils_todatanplinalgr)xdatas U/var/www/html/pdf-tiff/venv/lib/python3.13/site-packages/scipy/sparse/linalg/_norm.py_sparse_frobenius_normrs) ;;  q !D 99>>$ c[USS9n[U5(d [S5eUcUS;a [U5$URS:XaUR 5nUc[ [UR55nO6[U[ 5(d!Sn[U5nX$:wa [U5eU4nURn[U5S:XGaUupxU*Us=::aU:aO OU*Us=::aU:d!O S U<S UR<3n [U 5eXv-X-:Xa [S 5eUS:Xa[US S SS9upn U S$US:Xa[eUS;a [U5$[ R""UR$[&5(aUR)[&SS9nUS :Xa&[+U5R-US9R/5$U[ R0:Xa&[+U5R-US9R/5$US:Xa&[+U5R-US9R35$U[ R0*:Xa&[+U5R-US9R35$[S5e[U5S :XGaUun U*U s=::aU:d!O S U<S UR<3n [U 5eUS:XaUR5U S9$[ R""UR$[&5(aUR)[&SS9nU[ R0:Xa![7[+U5R/U S95$U[ R0*:Xa![7[+U5R3U S95$US :Xa![7[+U5R-U S95$US;a9[7[9[+U5R;S5R-U S955$US - [ R:"[+U5R;U5R-U S9S U- 5$[S5e![an[U5UeSnAff=f![an[S5UeSnAff=f)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 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 The selected norm of `x`. 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], offsets=[0, 1], shape=(9, 10)) >>> norm(b, 2) 1.9753... csr) target_formatz*input is not sparse. use numpy.linalg.normN)Nfrofdiaz6'axis' must be None, an integer or a tuple of integersz Invalid axis z for an array with shape zDuplicate axes given.arpack)ksolverrngr)NrrF)copy)axisz Invalid norm order for matrices.)rNzInvalid norm order for vectors.z&Improper number of dimensions to norm.)rr TypeErrorrformattocsrtuplerangendim isinstanceintlenshape ValueErrorrNotImplementedErrorr can_castdtypefloatastypersummaxinfmin count_nonzero_ravelrpower) rordr"msgint_axisendrow_axiscol_axismessage_sas rrrs~ 'q>A A;;DEE |11%a(( xx5 GGI |U166]# e $ $F (4yH  C. { B 4yA~!x$"$")=2)=%dX-FqwwkRGW% % =HM )45 5 !81(=GA!Q4K "9% % $ $)!, , ;;qww & &U+A !8q6::8:,002 2 BFF]q6::8:,002 2 BYq6::8:,002 2 RVVG^q6::8:,002 2?@ @ Ta q 2 %dX-FqwwkRGW% % !8???* * ;;qww & &U+A "&&=#a&**!*,- - 266'>#a&**!*,- - !8#a&**!*,- - ) $s1v||A222:;< < G !GxxA S)--1-5q3w??ABBw (C.a ' (n G>?Q F Gs0 P+Q + Q5 QQ Q$ QQ$c[US5(aUR5R5$[US5(aURR5$UR5$)NtoarrayA)hasattrrGravelrH)Ms rr9r9sKq)yy{  "" Cssyy{wwyr)NN)__doc__numpyr scipy.sparserscipy.sparse.linalgrscipy.sparse._sputilsrsparser rr__all__rrr9rrrTs7!$@ ( lC^r