^j$zddlZddlmZddlmZddlmZddlm Z ddl m Z gdZ Gd d e Zdd Zd Zd Zy)N) LinAlgError)get_blas_funcs)get_lapack_funcs) _bandwidth)_deprecate_dtypes)r LinAlgWarningnorm bandwidthceZdZdZy)r z The warning emitted when a linear algebra related operation is close to fail conditions of the algorithm or loss of accuracy is expected. N)__name__ __module__ __qualname____doc__K/opt/ringagent/.cad-venv/lib/python3.12/site-packages/scipy/linalg/_misc.pyr r s  rr c&|rtj|}ntj|}td||jr)|j j dvr| |s |dvr/|jdk(r td|j d}||S|jd k(rd}|dk(rHtj|rd |f}ntj|jrid |jf}nZ|tjk(rGtj|rd |f}n-tj|jrd |jf}|rtd |j d}||Stjj|||| S)a Matrix or vector norm. This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ``ord`` parameter. For tensors with rank different from 1 or 2, only `ord=None` is supported. Parameters ---------- a : array_like Input array. If `axis` is None, `a` must be 1-D or 2-D, unless `ord` is None. If both `axis` and `ord` are None, the 2-norm of ``a.ravel`` will be returned. ord : {int, inf, -inf, 'fro', 'nuc', None}, 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 `a` 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 `a` is 1-D) or a matrix norm (when `a` is 2-D) is returned. keepdims : bool, optional If this is set to True, the axes which are normed over are left in the result as dimensions with size one. With this option the result will broadcast correctly against the original `a`. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- n : float or ndarray Norm of the matrix or vector(s). 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 -- 'nuc' nuclear norm -- inf max(sum(abs(a), axis=1)) max(abs(a)) -inf min(sum(abs(a), axis=1)) min(abs(a)) 0 -- sum(a != 0) 1 max(sum(abs(a), axis=0)) as below -1 min(sum(abs(a), axis=0)) as below 2 2-norm (largest sing. value) as below -2 smallest singular value as below other -- sum(abs(a)**ord)**(1./ord) ===== ============================ ========================== The Frobenius norm is given by [1]_: :math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}` The nuclear norm is the sum of the singular values. Both the Frobenius and nuclear norm orders are only defined for matrices. References ---------- .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15 Examples -------- >>> import numpy as np >>> from scipy.linalg import norm >>> a = np.arange(9) - 4.0 >>> 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) 4.0 >>> norm(b, np.inf) 9.0 >>> norm(a, -np.inf) 0.0 >>> norm(b, -np.inf) 2.0 >>> norm(a, 1) 20.0 >>> norm(b, 1) 7.0 >>> norm(a, -1) -4.6566128774142013e-010 >>> norm(b, -1) 6.0 >>> norm(a, 2) 7.745966692414834 >>> norm(b, 2) 7.3484692283495345 >>> norm(a, -2) 0.0 >>> norm(b, -2) 1.8570331885190563e-016 >>> norm(a, 3) 5.8480354764257312 >>> norm(a, -3) 0.0 r fdFDN)Nrnrm2 preferred)dtypeilp64r1ilange)ordaxiskeepdims)npasarray_chkfiniteasarrayrsizercharndimr isfortranTinfrlinalgr )arrr check_finiter lange_argsrs rr r s=@   # JJqMfa vv!'',,&(T\( ) 1!&{KD7N 66Q;Jax<<?!$aJ\\!##&!$accJ<<?!$aJ\\!##&!$accJ({Sj)) 99>>!4(> CCrcv||uryt|tjs t|dry|jduS)z} Strict check for `arr` not sharing any data with `original`, under the assumption that arr = asarray(original) F __array__N) isinstancer!ndarrayhasattrbase)arroriginals r _datacopiedr6s7  h h ++0N 88t rctj|}|jdkr tdtj|j tj tjtjfrtd|j dtj|j dstd|j d|jdk(r|jdk(r*tjdtjdfS|jd d }tj|tj tj|tj fSt|S) a~Return the lower and upper bandwidth of a numeric array. Parameters ---------- a : (..., N, M) array_like Input array of at least 2 dimensions. Returns ------- lower : np.int64 | nDArray[np.int64] Lower bandwidth. a scalar ``np.int64`` is assigned per 2D slice of the input array of last two dimensions. A value of 0 means the slice is upper triangular; ``N - 1`` means the lower part is full. If the input array is 2D then a scalar int64 is returned. upper : np.int64 | nDArray[np.int64] Upper bandwidth. Same shape rules as `lower`. A value of 0 means the slice is lower triangular; ``M - 1`` means the upper part is full. If the input array is 2D then a scalar int64 is returned. Raises ------ TypeError If the dtype of the array is not supported, in particular, for NumPy float16, float128 and complex256 and other NumPy non-numeric types. Notes ----- This helper function simply runs over the array looking for the nonzero entries whether there exists a banded structure in the array or not. Hence, the performance depends on the density of nonzero entries and also memory-layout. Fortran- or C- contiguous arrays are handled best and otherwise suffers from extra random memory access cost. The strategy is to look for only untested band elements in the upper and lower triangular parts separately; depending on the memory layout we scan row-wise or column-wise. Moreover, say we are scanning rows and in the 6th row, 4th entry is nonzero then, on the succeeding rows the horizontal search is done only up to that band entries since we know that band is occupied. Therefore, a completely dense matrix scan cost is in the order of n. Examples -------- >>> import numpy as np >>> from scipy.linalg import bandwidth >>> A = np.array([[3., 0., 0., 0., 0.], ... [0., 4., 0., 0., 0.], ... [0., 0., 5., 1., 0.], ... [8., 0., 0., 6., 2.], ... [0., 9., 0., 0., 7.]]) >>> bandwidth(A) (3, 1) rz Input array must be at least 2D.zInput array with z dtype is not supported.)numericboolz+Input array must have a numeric dtype, got .rN)r)r!r#r& ValueErrorisdtyperfloat16 longdouble clongdouble TypeErrorr$int64shapezerosr)r+ batch_shapes rr r sn 1 Avvz;<< zz!''BJJ r~~FG+AGG94LMNNZZ!4 5EaggYaPQQ vv{ 66Q;HHQK!- -ggcrl HH[ 1 HH[ 1  a=r)NNFT)numpyr! numpy.linalgrblasrlapackr_batched_linalgrscipy._lib._utilr__all__RuntimeWarningr r r6r rrrrNs;$ $'. ? N bDJ Mr