import numpy as np from numpy.linalg import LinAlgError from .blas import get_blas_funcs from .lapack import get_lapack_funcs from ._batched_linalg import _bandwidth from scipy._lib._util import _deprecate_dtypes __all__ = ['LinAlgError', 'LinAlgWarning', 'norm', 'bandwidth'] class LinAlgWarning(RuntimeWarning): """ The warning emitted when a linear algebra related operation is close to fail conditions of the algorithm or loss of accuracy is expected. """ pass def norm(a, ord=None, axis=None, keepdims=False, check_finite=True): """ 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 """ # Differs from numpy only in non-finite handling and the use of blas. if check_finite: a = np.asarray_chkfinite(a) else: a = np.asarray(a) _deprecate_dtypes('norm', a) if a.size and a.dtype.char in 'fdFD' and axis is None and not keepdims: if ord in (None, 2) and (a.ndim == 1): # use blas for fast and stable euclidean norm nrm2 = get_blas_funcs('nrm2', dtype=a.dtype, ilp64='preferred') return nrm2(a) if a.ndim == 2: # Use lapack for a couple fast matrix norms. # For some reason the *lange frobenius norm is slow. lange_args = None # Make sure this works if the user uses the axis keywords # to apply the norm to the transpose. if ord == 1: if np.isfortran(a): lange_args = '1', a elif np.isfortran(a.T): lange_args = 'i', a.T elif ord == np.inf: if np.isfortran(a): lange_args = 'i', a elif np.isfortran(a.T): lange_args = '1', a.T if lange_args: lange = get_lapack_funcs('lange', dtype=a.dtype, ilp64='preferred') return lange(*lange_args) # fall back to numpy in every other case return np.linalg.norm(a, ord=ord, axis=axis, keepdims=keepdims) def _datacopied(arr, original): """ Strict check for `arr` not sharing any data with `original`, under the assumption that arr = asarray(original) """ if arr is original: return False if not isinstance(original, np.ndarray) and hasattr(original, '__array__'): return False return arr.base is None def bandwidth(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) """ a = np.asarray(a) if a.ndim < 2: raise ValueError('Input array must be at least 2D.') if np.isdtype(a.dtype, (np.float16, np.longdouble, np.clongdouble)): raise TypeError(f'Input array with {a.dtype} dtype is not supported.') # Now that the problematic numeric types are tested, test for numeric or bool. elif not np.isdtype(a.dtype, ("numeric", "bool")): raise TypeError(f'Input array must have a numeric dtype, got {a.dtype}.') # Empty array bandwidth is defined to be zero. if a.size == 0: if a.ndim == 2: return (np.int64(0), np.int64(0)) batch_shape = a.shape[:-2] return ( np.zeros(batch_shape, dtype=np.int64), np.zeros(batch_shape, dtype=np.int64) ) return _bandwidth(a)