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1832 lines
63 KiB
Python
1832 lines
63 KiB
Python
#
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# Author: Pearu Peterson, March 2002
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#
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# w/ additions by Travis Oliphant, March 2002
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# and Jake Vanderplas, August 2012
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from warnings import warn
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import numpy as np
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from numpy import atleast_1d, atleast_2d
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from .flinalg import get_flinalg_funcs
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from .lapack import get_lapack_funcs, _compute_lwork
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from .misc import LinAlgError, _datacopied, LinAlgWarning
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from .decomp import _asarray_validated
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from . import decomp, decomp_svd
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from ._solve_toeplitz import levinson
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__all__ = ['solve', 'solve_triangular', 'solveh_banded', 'solve_banded',
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'solve_toeplitz', 'solve_circulant', 'inv', 'det', 'lstsq',
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'pinv', 'pinv2', 'pinvh', 'matrix_balance', 'matmul_toeplitz']
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# Linear equations
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def _solve_check(n, info, lamch=None, rcond=None):
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""" Check arguments during the different steps of the solution phase """
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if info < 0:
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raise ValueError('LAPACK reported an illegal value in {}-th argument'
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'.'.format(-info))
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elif 0 < info:
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raise LinAlgError('Matrix is singular.')
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if lamch is None:
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return
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E = lamch('E')
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if rcond < E:
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warn('Ill-conditioned matrix (rcond={:.6g}): '
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'result may not be accurate.'.format(rcond),
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LinAlgWarning, stacklevel=3)
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def solve(a, b, sym_pos=False, lower=False, overwrite_a=False,
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overwrite_b=False, debug=None, check_finite=True, assume_a='gen',
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transposed=False):
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"""
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Solves the linear equation set ``a * x = b`` for the unknown ``x``
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for square ``a`` matrix.
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If the data matrix is known to be a particular type then supplying the
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corresponding string to ``assume_a`` key chooses the dedicated solver.
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The available options are
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=================== ========
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generic matrix 'gen'
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symmetric 'sym'
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hermitian 'her'
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positive definite 'pos'
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=================== ========
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If omitted, ``'gen'`` is the default structure.
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The datatype of the arrays define which solver is called regardless
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of the values. In other words, even when the complex array entries have
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precisely zero imaginary parts, the complex solver will be called based
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on the data type of the array.
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Parameters
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----------
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a : (N, N) array_like
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Square input data
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b : (N, NRHS) array_like
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Input data for the right hand side.
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sym_pos : bool, optional
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Assume `a` is symmetric and positive definite. This key is deprecated
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and assume_a = 'pos' keyword is recommended instead. The functionality
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is the same. It will be removed in the future.
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lower : bool, optional
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If True, only the data contained in the lower triangle of `a`. Default
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is to use upper triangle. (ignored for ``'gen'``)
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overwrite_a : bool, optional
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Allow overwriting data in `a` (may enhance performance).
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Default is False.
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overwrite_b : bool, optional
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Allow overwriting data in `b` (may enhance performance).
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Default is False.
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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assume_a : str, optional
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Valid entries are explained above.
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transposed: bool, optional
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If True, ``a^T x = b`` for real matrices, raises `NotImplementedError`
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for complex matrices (only for True).
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Returns
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-------
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x : (N, NRHS) ndarray
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The solution array.
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Raises
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------
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ValueError
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If size mismatches detected or input a is not square.
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LinAlgError
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If the matrix is singular.
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LinAlgWarning
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If an ill-conditioned input a is detected.
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NotImplementedError
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If transposed is True and input a is a complex matrix.
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Examples
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--------
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Given `a` and `b`, solve for `x`:
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>>> a = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]])
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>>> b = np.array([2, 4, -1])
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>>> from scipy import linalg
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>>> x = linalg.solve(a, b)
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>>> x
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array([ 2., -2., 9.])
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>>> np.dot(a, x) == b
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array([ True, True, True], dtype=bool)
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Notes
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-----
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If the input b matrix is a 1-D array with N elements, when supplied
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together with an NxN input a, it is assumed as a valid column vector
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despite the apparent size mismatch. This is compatible with the
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numpy.dot() behavior and the returned result is still 1-D array.
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The generic, symmetric, Hermitian and positive definite solutions are
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obtained via calling ?GESV, ?SYSV, ?HESV, and ?POSV routines of
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LAPACK respectively.
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"""
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# Flags for 1-D or N-D right-hand side
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b_is_1D = False
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a1 = atleast_2d(_asarray_validated(a, check_finite=check_finite))
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b1 = atleast_1d(_asarray_validated(b, check_finite=check_finite))
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n = a1.shape[0]
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overwrite_a = overwrite_a or _datacopied(a1, a)
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overwrite_b = overwrite_b or _datacopied(b1, b)
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if a1.shape[0] != a1.shape[1]:
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raise ValueError('Input a needs to be a square matrix.')
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if n != b1.shape[0]:
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# Last chance to catch 1x1 scalar a and 1-D b arrays
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if not (n == 1 and b1.size != 0):
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raise ValueError('Input b has to have same number of rows as '
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'input a')
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# accommodate empty arrays
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if b1.size == 0:
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return np.asfortranarray(b1.copy())
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# regularize 1-D b arrays to 2D
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if b1.ndim == 1:
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if n == 1:
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b1 = b1[None, :]
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else:
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b1 = b1[:, None]
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b_is_1D = True
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# Backwards compatibility - old keyword.
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if sym_pos:
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assume_a = 'pos'
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if assume_a not in ('gen', 'sym', 'her', 'pos'):
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raise ValueError('{} is not a recognized matrix structure'
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''.format(assume_a))
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# Deprecate keyword "debug"
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if debug is not None:
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warn('Use of the "debug" keyword is deprecated '
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'and this keyword will be removed in future '
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'versions of SciPy.', DeprecationWarning, stacklevel=2)
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# Get the correct lamch function.
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# The LAMCH functions only exists for S and D
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# So for complex values we have to convert to real/double.
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if a1.dtype.char in 'fF': # single precision
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lamch = get_lapack_funcs('lamch', dtype='f')
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else:
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lamch = get_lapack_funcs('lamch', dtype='d')
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# Currently we do not have the other forms of the norm calculators
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# lansy, lanpo, lanhe.
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# However, in any case they only reduce computations slightly...
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lange = get_lapack_funcs('lange', (a1,))
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# Since the I-norm and 1-norm are the same for symmetric matrices
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# we can collect them all in this one call
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# Note however, that when issuing 'gen' and form!='none', then
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# the I-norm should be used
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if transposed:
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trans = 1
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norm = 'I'
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if np.iscomplexobj(a1):
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raise NotImplementedError('scipy.linalg.solve can currently '
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'not solve a^T x = b or a^H x = b '
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'for complex matrices.')
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else:
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trans = 0
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norm = '1'
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anorm = lange(norm, a1)
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# Generalized case 'gesv'
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if assume_a == 'gen':
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gecon, getrf, getrs = get_lapack_funcs(('gecon', 'getrf', 'getrs'),
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(a1, b1))
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lu, ipvt, info = getrf(a1, overwrite_a=overwrite_a)
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_solve_check(n, info)
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x, info = getrs(lu, ipvt, b1,
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trans=trans, overwrite_b=overwrite_b)
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_solve_check(n, info)
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rcond, info = gecon(lu, anorm, norm=norm)
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# Hermitian case 'hesv'
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elif assume_a == 'her':
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hecon, hesv, hesv_lw = get_lapack_funcs(('hecon', 'hesv',
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'hesv_lwork'), (a1, b1))
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lwork = _compute_lwork(hesv_lw, n, lower)
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lu, ipvt, x, info = hesv(a1, b1, lwork=lwork,
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lower=lower,
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overwrite_a=overwrite_a,
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overwrite_b=overwrite_b)
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_solve_check(n, info)
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rcond, info = hecon(lu, ipvt, anorm)
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# Symmetric case 'sysv'
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elif assume_a == 'sym':
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sycon, sysv, sysv_lw = get_lapack_funcs(('sycon', 'sysv',
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'sysv_lwork'), (a1, b1))
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lwork = _compute_lwork(sysv_lw, n, lower)
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lu, ipvt, x, info = sysv(a1, b1, lwork=lwork,
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lower=lower,
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overwrite_a=overwrite_a,
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overwrite_b=overwrite_b)
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_solve_check(n, info)
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rcond, info = sycon(lu, ipvt, anorm)
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# Positive definite case 'posv'
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else:
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pocon, posv = get_lapack_funcs(('pocon', 'posv'),
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(a1, b1))
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lu, x, info = posv(a1, b1, lower=lower,
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overwrite_a=overwrite_a,
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overwrite_b=overwrite_b)
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_solve_check(n, info)
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rcond, info = pocon(lu, anorm)
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_solve_check(n, info, lamch, rcond)
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if b_is_1D:
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x = x.ravel()
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return x
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def solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False,
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overwrite_b=False, debug=None, check_finite=True):
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"""
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Solve the equation `a x = b` for `x`, assuming a is a triangular matrix.
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Parameters
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----------
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a : (M, M) array_like
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A triangular matrix
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b : (M,) or (M, N) array_like
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Right-hand side matrix in `a x = b`
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lower : bool, optional
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Use only data contained in the lower triangle of `a`.
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Default is to use upper triangle.
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trans : {0, 1, 2, 'N', 'T', 'C'}, optional
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Type of system to solve:
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======== =========
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trans system
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======== =========
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0 or 'N' a x = b
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1 or 'T' a^T x = b
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2 or 'C' a^H x = b
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======== =========
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unit_diagonal : bool, optional
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If True, diagonal elements of `a` are assumed to be 1 and
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will not be referenced.
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overwrite_b : bool, optional
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Allow overwriting data in `b` (may enhance performance)
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Returns
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-------
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x : (M,) or (M, N) ndarray
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Solution to the system `a x = b`. Shape of return matches `b`.
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Raises
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------
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LinAlgError
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If `a` is singular
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Notes
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-----
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.. versionadded:: 0.9.0
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Examples
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--------
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Solve the lower triangular system a x = b, where::
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[3 0 0 0] [4]
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a = [2 1 0 0] b = [2]
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[1 0 1 0] [4]
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[1 1 1 1] [2]
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>>> from scipy.linalg import solve_triangular
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>>> a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 1]])
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>>> b = np.array([4, 2, 4, 2])
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>>> x = solve_triangular(a, b, lower=True)
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>>> x
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array([ 1.33333333, -0.66666667, 2.66666667, -1.33333333])
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>>> a.dot(x) # Check the result
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array([ 4., 2., 4., 2.])
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"""
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# Deprecate keyword "debug"
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if debug is not None:
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warn('Use of the "debug" keyword is deprecated '
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'and this keyword will be removed in the future '
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'versions of SciPy.', DeprecationWarning, stacklevel=2)
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a1 = _asarray_validated(a, check_finite=check_finite)
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b1 = _asarray_validated(b, check_finite=check_finite)
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if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
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raise ValueError('expected square matrix')
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if a1.shape[0] != b1.shape[0]:
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raise ValueError('shapes of a {} and b {} are incompatible'
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.format(a1.shape, b1.shape))
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overwrite_b = overwrite_b or _datacopied(b1, b)
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if debug:
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print('solve:overwrite_b=', overwrite_b)
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trans = {'N': 0, 'T': 1, 'C': 2}.get(trans, trans)
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trtrs, = get_lapack_funcs(('trtrs',), (a1, b1))
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if a1.flags.f_contiguous or trans == 2:
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x, info = trtrs(a1, b1, overwrite_b=overwrite_b, lower=lower,
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trans=trans, unitdiag=unit_diagonal)
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else:
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# transposed system is solved since trtrs expects Fortran ordering
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x, info = trtrs(a1.T, b1, overwrite_b=overwrite_b, lower=not lower,
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trans=not trans, unitdiag=unit_diagonal)
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if info == 0:
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return x
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if info > 0:
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raise LinAlgError("singular matrix: resolution failed at diagonal %d" %
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(info-1))
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raise ValueError('illegal value in %dth argument of internal trtrs' %
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(-info))
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def solve_banded(l_and_u, ab, b, overwrite_ab=False, overwrite_b=False,
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debug=None, check_finite=True):
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"""
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Solve the equation a x = b for x, assuming a is banded matrix.
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The matrix a is stored in `ab` using the matrix diagonal ordered form::
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ab[u + i - j, j] == a[i,j]
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Example of `ab` (shape of a is (6,6), `u` =1, `l` =2)::
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* a01 a12 a23 a34 a45
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a00 a11 a22 a33 a44 a55
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a10 a21 a32 a43 a54 *
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a20 a31 a42 a53 * *
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Parameters
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----------
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(l, u) : (integer, integer)
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Number of non-zero lower and upper diagonals
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ab : (`l` + `u` + 1, M) array_like
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Banded matrix
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b : (M,) or (M, K) array_like
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Right-hand side
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overwrite_ab : bool, optional
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Discard data in `ab` (may enhance performance)
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overwrite_b : bool, optional
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Discard data in `b` (may enhance performance)
|
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
|
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
|
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Returns
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-------
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x : (M,) or (M, K) ndarray
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The solution to the system a x = b. Returned shape depends on the
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shape of `b`.
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Examples
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--------
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Solve the banded system a x = b, where::
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[5 2 -1 0 0] [0]
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[1 4 2 -1 0] [1]
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a = [0 1 3 2 -1] b = [2]
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[0 0 1 2 2] [2]
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[0 0 0 1 1] [3]
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There is one nonzero diagonal below the main diagonal (l = 1), and
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two above (u = 2). The diagonal banded form of the matrix is::
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[* * -1 -1 -1]
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ab = [* 2 2 2 2]
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[5 4 3 2 1]
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[1 1 1 1 *]
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>>> from scipy.linalg import solve_banded
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>>> ab = np.array([[0, 0, -1, -1, -1],
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... [0, 2, 2, 2, 2],
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... [5, 4, 3, 2, 1],
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... [1, 1, 1, 1, 0]])
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>>> b = np.array([0, 1, 2, 2, 3])
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>>> x = solve_banded((1, 2), ab, b)
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>>> x
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array([-2.37288136, 3.93220339, -4. , 4.3559322 , -1.3559322 ])
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"""
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# Deprecate keyword "debug"
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if debug is not None:
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warn('Use of the "debug" keyword is deprecated '
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'and this keyword will be removed in the future '
|
|
'versions of SciPy.', DeprecationWarning, stacklevel=2)
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a1 = _asarray_validated(ab, check_finite=check_finite, as_inexact=True)
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b1 = _asarray_validated(b, check_finite=check_finite, as_inexact=True)
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# Validate shapes.
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if a1.shape[-1] != b1.shape[0]:
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raise ValueError("shapes of ab and b are not compatible.")
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(nlower, nupper) = l_and_u
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if nlower + nupper + 1 != a1.shape[0]:
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raise ValueError("invalid values for the number of lower and upper "
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"diagonals: l+u+1 (%d) does not equal ab.shape[0] "
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"(%d)" % (nlower + nupper + 1, ab.shape[0]))
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overwrite_b = overwrite_b or _datacopied(b1, b)
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if a1.shape[-1] == 1:
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b2 = np.array(b1, copy=(not overwrite_b))
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b2 /= a1[1, 0]
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return b2
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if nlower == nupper == 1:
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overwrite_ab = overwrite_ab or _datacopied(a1, ab)
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gtsv, = get_lapack_funcs(('gtsv',), (a1, b1))
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du = a1[0, 1:]
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d = a1[1, :]
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dl = a1[2, :-1]
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du2, d, du, x, info = gtsv(dl, d, du, b1, overwrite_ab, overwrite_ab,
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overwrite_ab, overwrite_b)
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else:
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gbsv, = get_lapack_funcs(('gbsv',), (a1, b1))
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a2 = np.zeros((2*nlower + nupper + 1, a1.shape[1]), dtype=gbsv.dtype)
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a2[nlower:, :] = a1
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lu, piv, x, info = gbsv(nlower, nupper, a2, b1, overwrite_ab=True,
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overwrite_b=overwrite_b)
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if info == 0:
|
|
return x
|
|
if info > 0:
|
|
raise LinAlgError("singular matrix")
|
|
raise ValueError('illegal value in %d-th argument of internal '
|
|
'gbsv/gtsv' % -info)
|
|
|
|
|
|
def solveh_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False,
|
|
check_finite=True):
|
|
"""
|
|
Solve equation a x = b. a is Hermitian positive-definite banded matrix.
|
|
|
|
The matrix a is stored in `ab` either in lower diagonal or upper
|
|
diagonal ordered form:
|
|
|
|
ab[u + i - j, j] == a[i,j] (if upper form; i <= j)
|
|
ab[ i - j, j] == a[i,j] (if lower form; i >= j)
|
|
|
|
Example of `ab` (shape of a is (6, 6), `u` =2)::
|
|
|
|
upper form:
|
|
* * a02 a13 a24 a35
|
|
* a01 a12 a23 a34 a45
|
|
a00 a11 a22 a33 a44 a55
|
|
|
|
lower form:
|
|
a00 a11 a22 a33 a44 a55
|
|
a10 a21 a32 a43 a54 *
|
|
a20 a31 a42 a53 * *
|
|
|
|
Cells marked with * are not used.
|
|
|
|
Parameters
|
|
----------
|
|
ab : (`u` + 1, M) array_like
|
|
Banded matrix
|
|
b : (M,) or (M, K) array_like
|
|
Right-hand side
|
|
overwrite_ab : bool, optional
|
|
Discard data in `ab` (may enhance performance)
|
|
overwrite_b : bool, optional
|
|
Discard data in `b` (may enhance performance)
|
|
lower : bool, optional
|
|
Is the matrix in the lower form. (Default is upper form)
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrices contain 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
|
|
-------
|
|
x : (M,) or (M, K) ndarray
|
|
The solution to the system a x = b. Shape of return matches shape
|
|
of `b`.
|
|
|
|
Examples
|
|
--------
|
|
Solve the banded system A x = b, where::
|
|
|
|
[ 4 2 -1 0 0 0] [1]
|
|
[ 2 5 2 -1 0 0] [2]
|
|
A = [-1 2 6 2 -1 0] b = [2]
|
|
[ 0 -1 2 7 2 -1] [3]
|
|
[ 0 0 -1 2 8 2] [3]
|
|
[ 0 0 0 -1 2 9] [3]
|
|
|
|
>>> from scipy.linalg import solveh_banded
|
|
|
|
`ab` contains the main diagonal and the nonzero diagonals below the
|
|
main diagonal. That is, we use the lower form:
|
|
|
|
>>> ab = np.array([[ 4, 5, 6, 7, 8, 9],
|
|
... [ 2, 2, 2, 2, 2, 0],
|
|
... [-1, -1, -1, -1, 0, 0]])
|
|
>>> b = np.array([1, 2, 2, 3, 3, 3])
|
|
>>> x = solveh_banded(ab, b, lower=True)
|
|
>>> x
|
|
array([ 0.03431373, 0.45938375, 0.05602241, 0.47759104, 0.17577031,
|
|
0.34733894])
|
|
|
|
|
|
Solve the Hermitian banded system H x = b, where::
|
|
|
|
[ 8 2-1j 0 0 ] [ 1 ]
|
|
H = [2+1j 5 1j 0 ] b = [1+1j]
|
|
[ 0 -1j 9 -2-1j] [1-2j]
|
|
[ 0 0 -2+1j 6 ] [ 0 ]
|
|
|
|
In this example, we put the upper diagonals in the array `hb`:
|
|
|
|
>>> hb = np.array([[0, 2-1j, 1j, -2-1j],
|
|
... [8, 5, 9, 6 ]])
|
|
>>> b = np.array([1, 1+1j, 1-2j, 0])
|
|
>>> x = solveh_banded(hb, b)
|
|
>>> x
|
|
array([ 0.07318536-0.02939412j, 0.11877624+0.17696461j,
|
|
0.10077984-0.23035393j, -0.00479904-0.09358128j])
|
|
|
|
"""
|
|
a1 = _asarray_validated(ab, check_finite=check_finite)
|
|
b1 = _asarray_validated(b, check_finite=check_finite)
|
|
# Validate shapes.
|
|
if a1.shape[-1] != b1.shape[0]:
|
|
raise ValueError("shapes of ab and b are not compatible.")
|
|
|
|
overwrite_b = overwrite_b or _datacopied(b1, b)
|
|
overwrite_ab = overwrite_ab or _datacopied(a1, ab)
|
|
|
|
if a1.shape[0] == 2:
|
|
ptsv, = get_lapack_funcs(('ptsv',), (a1, b1))
|
|
if lower:
|
|
d = a1[0, :].real
|
|
e = a1[1, :-1]
|
|
else:
|
|
d = a1[1, :].real
|
|
e = a1[0, 1:].conj()
|
|
d, du, x, info = ptsv(d, e, b1, overwrite_ab, overwrite_ab,
|
|
overwrite_b)
|
|
else:
|
|
pbsv, = get_lapack_funcs(('pbsv',), (a1, b1))
|
|
c, x, info = pbsv(a1, b1, lower=lower, overwrite_ab=overwrite_ab,
|
|
overwrite_b=overwrite_b)
|
|
if info > 0:
|
|
raise LinAlgError("%dth leading minor not positive definite" % info)
|
|
if info < 0:
|
|
raise ValueError('illegal value in %dth argument of internal '
|
|
'pbsv' % -info)
|
|
return x
|
|
|
|
|
|
def solve_toeplitz(c_or_cr, b, check_finite=True):
|
|
"""Solve a Toeplitz system using Levinson Recursion
|
|
|
|
The Toeplitz matrix has constant diagonals, with c as its first column
|
|
and r as its first row. If r is not given, ``r == conjugate(c)`` is
|
|
assumed.
|
|
|
|
Parameters
|
|
----------
|
|
c_or_cr : array_like or tuple of (array_like, array_like)
|
|
The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the
|
|
actual shape of ``c``, it will be converted to a 1-D array. If not
|
|
supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
|
|
real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
|
|
of the Toeplitz matrix is ``[c[0], r[1:]]``. Whatever the actual shape
|
|
of ``r``, it will be converted to a 1-D array.
|
|
b : (M,) or (M, K) array_like
|
|
Right-hand side in ``T x = b``.
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrices contain only finite numbers.
|
|
Disabling may give a performance gain, but may result in problems
|
|
(result entirely NaNs) if the inputs do contain infinities or NaNs.
|
|
|
|
Returns
|
|
-------
|
|
x : (M,) or (M, K) ndarray
|
|
The solution to the system ``T x = b``. Shape of return matches shape
|
|
of `b`.
|
|
|
|
See Also
|
|
--------
|
|
toeplitz : Toeplitz matrix
|
|
|
|
Notes
|
|
-----
|
|
The solution is computed using Levinson-Durbin recursion, which is faster
|
|
than generic least-squares methods, but can be less numerically stable.
|
|
|
|
Examples
|
|
--------
|
|
Solve the Toeplitz system T x = b, where::
|
|
|
|
[ 1 -1 -2 -3] [1]
|
|
T = [ 3 1 -1 -2] b = [2]
|
|
[ 6 3 1 -1] [2]
|
|
[10 6 3 1] [5]
|
|
|
|
To specify the Toeplitz matrix, only the first column and the first
|
|
row are needed.
|
|
|
|
>>> c = np.array([1, 3, 6, 10]) # First column of T
|
|
>>> r = np.array([1, -1, -2, -3]) # First row of T
|
|
>>> b = np.array([1, 2, 2, 5])
|
|
|
|
>>> from scipy.linalg import solve_toeplitz, toeplitz
|
|
>>> x = solve_toeplitz((c, r), b)
|
|
>>> x
|
|
array([ 1.66666667, -1. , -2.66666667, 2.33333333])
|
|
|
|
Check the result by creating the full Toeplitz matrix and
|
|
multiplying it by `x`. We should get `b`.
|
|
|
|
>>> T = toeplitz(c, r)
|
|
>>> T.dot(x)
|
|
array([ 1., 2., 2., 5.])
|
|
|
|
"""
|
|
# If numerical stability of this algorithm is a problem, a future
|
|
# developer might consider implementing other O(N^2) Toeplitz solvers,
|
|
# such as GKO (https://www.jstor.org/stable/2153371) or Bareiss.
|
|
|
|
r, c, b, dtype, b_shape = _validate_args_for_toeplitz_ops(
|
|
c_or_cr, b, check_finite, keep_b_shape=True)
|
|
|
|
# Form a 1-D array of values to be used in the matrix, containing a
|
|
# reversed copy of r[1:], followed by c.
|
|
vals = np.concatenate((r[-1:0:-1], c))
|
|
if b is None:
|
|
raise ValueError('illegal value, `b` is a required argument')
|
|
|
|
if b.ndim == 1:
|
|
x, _ = levinson(vals, np.ascontiguousarray(b))
|
|
else:
|
|
x = np.column_stack([levinson(vals, np.ascontiguousarray(b[:, i]))[0]
|
|
for i in range(b.shape[1])])
|
|
x = x.reshape(*b_shape)
|
|
|
|
return x
|
|
|
|
|
|
def _get_axis_len(aname, a, axis):
|
|
ax = axis
|
|
if ax < 0:
|
|
ax += a.ndim
|
|
if 0 <= ax < a.ndim:
|
|
return a.shape[ax]
|
|
raise ValueError("'%saxis' entry is out of bounds" % (aname,))
|
|
|
|
|
|
def solve_circulant(c, b, singular='raise', tol=None,
|
|
caxis=-1, baxis=0, outaxis=0):
|
|
"""Solve C x = b for x, where C is a circulant matrix.
|
|
|
|
`C` is the circulant matrix associated with the vector `c`.
|
|
|
|
The system is solved by doing division in Fourier space. The
|
|
calculation is::
|
|
|
|
x = ifft(fft(b) / fft(c))
|
|
|
|
where `fft` and `ifft` are the fast Fourier transform and its inverse,
|
|
respectively. For a large vector `c`, this is *much* faster than
|
|
solving the system with the full circulant matrix.
|
|
|
|
Parameters
|
|
----------
|
|
c : array_like
|
|
The coefficients of the circulant matrix.
|
|
b : array_like
|
|
Right-hand side matrix in ``a x = b``.
|
|
singular : str, optional
|
|
This argument controls how a near singular circulant matrix is
|
|
handled. If `singular` is "raise" and the circulant matrix is
|
|
near singular, a `LinAlgError` is raised. If `singular` is
|
|
"lstsq", the least squares solution is returned. Default is "raise".
|
|
tol : float, optional
|
|
If any eigenvalue of the circulant matrix has an absolute value
|
|
that is less than or equal to `tol`, the matrix is considered to be
|
|
near singular. If not given, `tol` is set to::
|
|
|
|
tol = abs_eigs.max() * abs_eigs.size * np.finfo(np.float64).eps
|
|
|
|
where `abs_eigs` is the array of absolute values of the eigenvalues
|
|
of the circulant matrix.
|
|
caxis : int
|
|
When `c` has dimension greater than 1, it is viewed as a collection
|
|
of circulant vectors. In this case, `caxis` is the axis of `c` that
|
|
holds the vectors of circulant coefficients.
|
|
baxis : int
|
|
When `b` has dimension greater than 1, it is viewed as a collection
|
|
of vectors. In this case, `baxis` is the axis of `b` that holds the
|
|
right-hand side vectors.
|
|
outaxis : int
|
|
When `c` or `b` are multidimensional, the value returned by
|
|
`solve_circulant` is multidimensional. In this case, `outaxis` is
|
|
the axis of the result that holds the solution vectors.
|
|
|
|
Returns
|
|
-------
|
|
x : ndarray
|
|
Solution to the system ``C x = b``.
|
|
|
|
Raises
|
|
------
|
|
LinAlgError
|
|
If the circulant matrix associated with `c` is near singular.
|
|
|
|
See Also
|
|
--------
|
|
circulant : circulant matrix
|
|
|
|
Notes
|
|
-----
|
|
For a 1-D vector `c` with length `m`, and an array `b`
|
|
with shape ``(m, ...)``,
|
|
|
|
solve_circulant(c, b)
|
|
|
|
returns the same result as
|
|
|
|
solve(circulant(c), b)
|
|
|
|
where `solve` and `circulant` are from `scipy.linalg`.
|
|
|
|
.. versionadded:: 0.16.0
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.linalg import solve_circulant, solve, circulant, lstsq
|
|
|
|
>>> c = np.array([2, 2, 4])
|
|
>>> b = np.array([1, 2, 3])
|
|
>>> solve_circulant(c, b)
|
|
array([ 0.75, -0.25, 0.25])
|
|
|
|
Compare that result to solving the system with `scipy.linalg.solve`:
|
|
|
|
>>> solve(circulant(c), b)
|
|
array([ 0.75, -0.25, 0.25])
|
|
|
|
A singular example:
|
|
|
|
>>> c = np.array([1, 1, 0, 0])
|
|
>>> b = np.array([1, 2, 3, 4])
|
|
|
|
Calling ``solve_circulant(c, b)`` will raise a `LinAlgError`. For the
|
|
least square solution, use the option ``singular='lstsq'``:
|
|
|
|
>>> solve_circulant(c, b, singular='lstsq')
|
|
array([ 0.25, 1.25, 2.25, 1.25])
|
|
|
|
Compare to `scipy.linalg.lstsq`:
|
|
|
|
>>> x, resid, rnk, s = lstsq(circulant(c), b)
|
|
>>> x
|
|
array([ 0.25, 1.25, 2.25, 1.25])
|
|
|
|
A broadcasting example:
|
|
|
|
Suppose we have the vectors of two circulant matrices stored in an array
|
|
with shape (2, 5), and three `b` vectors stored in an array with shape
|
|
(3, 5). For example,
|
|
|
|
>>> c = np.array([[1.5, 2, 3, 0, 0], [1, 1, 4, 3, 2]])
|
|
>>> b = np.arange(15).reshape(-1, 5)
|
|
|
|
We want to solve all combinations of circulant matrices and `b` vectors,
|
|
with the result stored in an array with shape (2, 3, 5). When we
|
|
disregard the axes of `c` and `b` that hold the vectors of coefficients,
|
|
the shapes of the collections are (2,) and (3,), respectively, which are
|
|
not compatible for broadcasting. To have a broadcast result with shape
|
|
(2, 3), we add a trivial dimension to `c`: ``c[:, np.newaxis, :]`` has
|
|
shape (2, 1, 5). The last dimension holds the coefficients of the
|
|
circulant matrices, so when we call `solve_circulant`, we can use the
|
|
default ``caxis=-1``. The coefficients of the `b` vectors are in the last
|
|
dimension of the array `b`, so we use ``baxis=-1``. If we use the
|
|
default `outaxis`, the result will have shape (5, 2, 3), so we'll use
|
|
``outaxis=-1`` to put the solution vectors in the last dimension.
|
|
|
|
>>> x = solve_circulant(c[:, np.newaxis, :], b, baxis=-1, outaxis=-1)
|
|
>>> x.shape
|
|
(2, 3, 5)
|
|
>>> np.set_printoptions(precision=3) # For compact output of numbers.
|
|
>>> x
|
|
array([[[-0.118, 0.22 , 1.277, -0.142, 0.302],
|
|
[ 0.651, 0.989, 2.046, 0.627, 1.072],
|
|
[ 1.42 , 1.758, 2.816, 1.396, 1.841]],
|
|
[[ 0.401, 0.304, 0.694, -0.867, 0.377],
|
|
[ 0.856, 0.758, 1.149, -0.412, 0.831],
|
|
[ 1.31 , 1.213, 1.603, 0.042, 1.286]]])
|
|
|
|
Check by solving one pair of `c` and `b` vectors (cf. ``x[1, 1, :]``):
|
|
|
|
>>> solve_circulant(c[1], b[1, :])
|
|
array([ 0.856, 0.758, 1.149, -0.412, 0.831])
|
|
|
|
"""
|
|
c = np.atleast_1d(c)
|
|
nc = _get_axis_len("c", c, caxis)
|
|
b = np.atleast_1d(b)
|
|
nb = _get_axis_len("b", b, baxis)
|
|
if nc != nb:
|
|
raise ValueError('Shapes of c {} and b {} are incompatible'
|
|
.format(c.shape, b.shape))
|
|
|
|
fc = np.fft.fft(np.rollaxis(c, caxis, c.ndim), axis=-1)
|
|
abs_fc = np.abs(fc)
|
|
if tol is None:
|
|
# This is the same tolerance as used in np.linalg.matrix_rank.
|
|
tol = abs_fc.max(axis=-1) * nc * np.finfo(np.float64).eps
|
|
if tol.shape != ():
|
|
tol.shape = tol.shape + (1,)
|
|
else:
|
|
tol = np.atleast_1d(tol)
|
|
|
|
near_zeros = abs_fc <= tol
|
|
is_near_singular = np.any(near_zeros)
|
|
if is_near_singular:
|
|
if singular == 'raise':
|
|
raise LinAlgError("near singular circulant matrix.")
|
|
else:
|
|
# Replace the small values with 1 to avoid errors in the
|
|
# division fb/fc below.
|
|
fc[near_zeros] = 1
|
|
|
|
fb = np.fft.fft(np.rollaxis(b, baxis, b.ndim), axis=-1)
|
|
|
|
q = fb / fc
|
|
|
|
if is_near_singular:
|
|
# `near_zeros` is a boolean array, same shape as `c`, that is
|
|
# True where `fc` is (near) zero. `q` is the broadcasted result
|
|
# of fb / fc, so to set the values of `q` to 0 where `fc` is near
|
|
# zero, we use a mask that is the broadcast result of an array
|
|
# of True values shaped like `b` with `near_zeros`.
|
|
mask = np.ones_like(b, dtype=bool) & near_zeros
|
|
q[mask] = 0
|
|
|
|
x = np.fft.ifft(q, axis=-1)
|
|
if not (np.iscomplexobj(c) or np.iscomplexobj(b)):
|
|
x = x.real
|
|
if outaxis != -1:
|
|
x = np.rollaxis(x, -1, outaxis)
|
|
return x
|
|
|
|
|
|
# matrix inversion
|
|
def inv(a, overwrite_a=False, check_finite=True):
|
|
"""
|
|
Compute the inverse of a matrix.
|
|
|
|
Parameters
|
|
----------
|
|
a : array_like
|
|
Square matrix to be inverted.
|
|
overwrite_a : bool, optional
|
|
Discard data in `a` (may improve performance). Default is False.
|
|
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
|
|
-------
|
|
ainv : ndarray
|
|
Inverse of the matrix `a`.
|
|
|
|
Raises
|
|
------
|
|
LinAlgError
|
|
If `a` is singular.
|
|
ValueError
|
|
If `a` is not square, or not 2D.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy import linalg
|
|
>>> a = np.array([[1., 2.], [3., 4.]])
|
|
>>> linalg.inv(a)
|
|
array([[-2. , 1. ],
|
|
[ 1.5, -0.5]])
|
|
>>> np.dot(a, linalg.inv(a))
|
|
array([[ 1., 0.],
|
|
[ 0., 1.]])
|
|
|
|
"""
|
|
a1 = _asarray_validated(a, check_finite=check_finite)
|
|
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
|
|
raise ValueError('expected square matrix')
|
|
overwrite_a = overwrite_a or _datacopied(a1, a)
|
|
# XXX: I found no advantage or disadvantage of using finv.
|
|
# finv, = get_flinalg_funcs(('inv',),(a1,))
|
|
# if finv is not None:
|
|
# a_inv,info = finv(a1,overwrite_a=overwrite_a)
|
|
# if info==0:
|
|
# return a_inv
|
|
# if info>0: raise LinAlgError, "singular matrix"
|
|
# if info<0: raise ValueError('illegal value in %d-th argument of '
|
|
# 'internal inv.getrf|getri'%(-info))
|
|
getrf, getri, getri_lwork = get_lapack_funcs(('getrf', 'getri',
|
|
'getri_lwork'),
|
|
(a1,))
|
|
lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
|
|
if info == 0:
|
|
lwork = _compute_lwork(getri_lwork, a1.shape[0])
|
|
|
|
# XXX: the following line fixes curious SEGFAULT when
|
|
# benchmarking 500x500 matrix inverse. This seems to
|
|
# be a bug in LAPACK ?getri routine because if lwork is
|
|
# minimal (when using lwork[0] instead of lwork[1]) then
|
|
# all tests pass. Further investigation is required if
|
|
# more such SEGFAULTs occur.
|
|
lwork = int(1.01 * lwork)
|
|
inv_a, info = getri(lu, piv, lwork=lwork, overwrite_lu=1)
|
|
if info > 0:
|
|
raise LinAlgError("singular matrix")
|
|
if info < 0:
|
|
raise ValueError('illegal value in %d-th argument of internal '
|
|
'getrf|getri' % -info)
|
|
return inv_a
|
|
|
|
|
|
# Determinant
|
|
|
|
def det(a, overwrite_a=False, check_finite=True):
|
|
"""
|
|
Compute the determinant of a matrix
|
|
|
|
The determinant of a square matrix is a value derived arithmetically
|
|
from the coefficients of the matrix.
|
|
|
|
The determinant for a 3x3 matrix, for example, is computed as follows::
|
|
|
|
a b c
|
|
d e f = A
|
|
g h i
|
|
|
|
det(A) = a*e*i + b*f*g + c*d*h - c*e*g - b*d*i - a*f*h
|
|
|
|
Parameters
|
|
----------
|
|
a : (M, M) array_like
|
|
A square matrix.
|
|
overwrite_a : bool, optional
|
|
Allow overwriting data in a (may enhance performance).
|
|
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
|
|
-------
|
|
det : float or complex
|
|
Determinant of `a`.
|
|
|
|
Notes
|
|
-----
|
|
The determinant is computed via LU factorization, LAPACK routine z/dgetrf.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy import linalg
|
|
>>> a = np.array([[1,2,3], [4,5,6], [7,8,9]])
|
|
>>> linalg.det(a)
|
|
0.0
|
|
>>> a = np.array([[0,2,3], [4,5,6], [7,8,9]])
|
|
>>> linalg.det(a)
|
|
3.0
|
|
|
|
"""
|
|
a1 = _asarray_validated(a, check_finite=check_finite)
|
|
if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
|
|
raise ValueError('expected square matrix')
|
|
overwrite_a = overwrite_a or _datacopied(a1, a)
|
|
fdet, = get_flinalg_funcs(('det',), (a1,))
|
|
a_det, info = fdet(a1, overwrite_a=overwrite_a)
|
|
if info < 0:
|
|
raise ValueError('illegal value in %d-th argument of internal '
|
|
'det.getrf' % -info)
|
|
return a_det
|
|
|
|
|
|
# Linear Least Squares
|
|
def lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False,
|
|
check_finite=True, lapack_driver=None):
|
|
"""
|
|
Compute least-squares solution to equation Ax = b.
|
|
|
|
Compute a vector x such that the 2-norm ``|b - A x|`` is minimized.
|
|
|
|
Parameters
|
|
----------
|
|
a : (M, N) array_like
|
|
Left-hand side array
|
|
b : (M,) or (M, K) array_like
|
|
Right hand side array
|
|
cond : float, optional
|
|
Cutoff for 'small' singular values; used to determine effective
|
|
rank of a. Singular values smaller than
|
|
``rcond * largest_singular_value`` are considered zero.
|
|
overwrite_a : bool, optional
|
|
Discard data in `a` (may enhance performance). Default is False.
|
|
overwrite_b : bool, optional
|
|
Discard data in `b` (may enhance performance). Default is False.
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrices contain 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.
|
|
lapack_driver : str, optional
|
|
Which LAPACK driver is used to solve the least-squares problem.
|
|
Options are ``'gelsd'``, ``'gelsy'``, ``'gelss'``. Default
|
|
(``'gelsd'``) is a good choice. However, ``'gelsy'`` can be slightly
|
|
faster on many problems. ``'gelss'`` was used historically. It is
|
|
generally slow but uses less memory.
|
|
|
|
.. versionadded:: 0.17.0
|
|
|
|
Returns
|
|
-------
|
|
x : (N,) or (N, K) ndarray
|
|
Least-squares solution. Return shape matches shape of `b`.
|
|
residues : (K,) ndarray or float
|
|
Square of the 2-norm for each column in ``b - a x``, if ``M > N`` and
|
|
``ndim(A) == n`` (returns a scalar if b is 1-D). Otherwise a
|
|
(0,)-shaped array is returned.
|
|
rank : int
|
|
Effective rank of `a`.
|
|
s : (min(M, N),) ndarray or None
|
|
Singular values of `a`. The condition number of a is
|
|
``abs(s[0] / s[-1])``.
|
|
|
|
Raises
|
|
------
|
|
LinAlgError
|
|
If computation does not converge.
|
|
|
|
ValueError
|
|
When parameters are not compatible.
|
|
|
|
See Also
|
|
--------
|
|
scipy.optimize.nnls : linear least squares with non-negativity constraint
|
|
|
|
Notes
|
|
-----
|
|
When ``'gelsy'`` is used as a driver, `residues` is set to a (0,)-shaped
|
|
array and `s` is always ``None``.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.linalg import lstsq
|
|
>>> import matplotlib.pyplot as plt
|
|
|
|
Suppose we have the following data:
|
|
|
|
>>> x = np.array([1, 2.5, 3.5, 4, 5, 7, 8.5])
|
|
>>> y = np.array([0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6])
|
|
|
|
We want to fit a quadratic polynomial of the form ``y = a + b*x**2``
|
|
to this data. We first form the "design matrix" M, with a constant
|
|
column of 1s and a column containing ``x**2``:
|
|
|
|
>>> M = x[:, np.newaxis]**[0, 2]
|
|
>>> M
|
|
array([[ 1. , 1. ],
|
|
[ 1. , 6.25],
|
|
[ 1. , 12.25],
|
|
[ 1. , 16. ],
|
|
[ 1. , 25. ],
|
|
[ 1. , 49. ],
|
|
[ 1. , 72.25]])
|
|
|
|
We want to find the least-squares solution to ``M.dot(p) = y``,
|
|
where ``p`` is a vector with length 2 that holds the parameters
|
|
``a`` and ``b``.
|
|
|
|
>>> p, res, rnk, s = lstsq(M, y)
|
|
>>> p
|
|
array([ 0.20925829, 0.12013861])
|
|
|
|
Plot the data and the fitted curve.
|
|
|
|
>>> plt.plot(x, y, 'o', label='data')
|
|
>>> xx = np.linspace(0, 9, 101)
|
|
>>> yy = p[0] + p[1]*xx**2
|
|
>>> plt.plot(xx, yy, label='least squares fit, $y = a + bx^2$')
|
|
>>> plt.xlabel('x')
|
|
>>> plt.ylabel('y')
|
|
>>> plt.legend(framealpha=1, shadow=True)
|
|
>>> plt.grid(alpha=0.25)
|
|
>>> plt.show()
|
|
|
|
"""
|
|
a1 = _asarray_validated(a, check_finite=check_finite)
|
|
b1 = _asarray_validated(b, check_finite=check_finite)
|
|
if len(a1.shape) != 2:
|
|
raise ValueError('Input array a should be 2D')
|
|
m, n = a1.shape
|
|
if len(b1.shape) == 2:
|
|
nrhs = b1.shape[1]
|
|
else:
|
|
nrhs = 1
|
|
if m != b1.shape[0]:
|
|
raise ValueError('Shape mismatch: a and b should have the same number'
|
|
' of rows ({} != {}).'.format(m, b1.shape[0]))
|
|
if m == 0 or n == 0: # Zero-sized problem, confuses LAPACK
|
|
x = np.zeros((n,) + b1.shape[1:], dtype=np.common_type(a1, b1))
|
|
if n == 0:
|
|
residues = np.linalg.norm(b1, axis=0)**2
|
|
else:
|
|
residues = np.empty((0,))
|
|
return x, residues, 0, np.empty((0,))
|
|
|
|
driver = lapack_driver
|
|
if driver is None:
|
|
driver = lstsq.default_lapack_driver
|
|
if driver not in ('gelsd', 'gelsy', 'gelss'):
|
|
raise ValueError('LAPACK driver "%s" is not found' % driver)
|
|
|
|
lapack_func, lapack_lwork = get_lapack_funcs((driver,
|
|
'%s_lwork' % driver),
|
|
(a1, b1))
|
|
real_data = True if (lapack_func.dtype.kind == 'f') else False
|
|
|
|
if m < n:
|
|
# need to extend b matrix as it will be filled with
|
|
# a larger solution matrix
|
|
if len(b1.shape) == 2:
|
|
b2 = np.zeros((n, nrhs), dtype=lapack_func.dtype)
|
|
b2[:m, :] = b1
|
|
else:
|
|
b2 = np.zeros(n, dtype=lapack_func.dtype)
|
|
b2[:m] = b1
|
|
b1 = b2
|
|
|
|
overwrite_a = overwrite_a or _datacopied(a1, a)
|
|
overwrite_b = overwrite_b or _datacopied(b1, b)
|
|
|
|
if cond is None:
|
|
cond = np.finfo(lapack_func.dtype).eps
|
|
|
|
if driver in ('gelss', 'gelsd'):
|
|
if driver == 'gelss':
|
|
lwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
|
|
v, x, s, rank, work, info = lapack_func(a1, b1, cond, lwork,
|
|
overwrite_a=overwrite_a,
|
|
overwrite_b=overwrite_b)
|
|
|
|
elif driver == 'gelsd':
|
|
if real_data:
|
|
lwork, iwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
|
|
x, s, rank, info = lapack_func(a1, b1, lwork,
|
|
iwork, cond, False, False)
|
|
else: # complex data
|
|
lwork, rwork, iwork = _compute_lwork(lapack_lwork, m, n,
|
|
nrhs, cond)
|
|
x, s, rank, info = lapack_func(a1, b1, lwork, rwork, iwork,
|
|
cond, False, False)
|
|
if info > 0:
|
|
raise LinAlgError("SVD did not converge in Linear Least Squares")
|
|
if info < 0:
|
|
raise ValueError('illegal value in %d-th argument of internal %s'
|
|
% (-info, lapack_driver))
|
|
resids = np.asarray([], dtype=x.dtype)
|
|
if m > n:
|
|
x1 = x[:n]
|
|
if rank == n:
|
|
resids = np.sum(np.abs(x[n:])**2, axis=0)
|
|
x = x1
|
|
return x, resids, rank, s
|
|
|
|
elif driver == 'gelsy':
|
|
lwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
|
|
jptv = np.zeros((a1.shape[1], 1), dtype=np.int32)
|
|
v, x, j, rank, info = lapack_func(a1, b1, jptv, cond,
|
|
lwork, False, False)
|
|
if info < 0:
|
|
raise ValueError("illegal value in %d-th argument of internal "
|
|
"gelsy" % -info)
|
|
if m > n:
|
|
x1 = x[:n]
|
|
x = x1
|
|
return x, np.array([], x.dtype), rank, None
|
|
|
|
|
|
lstsq.default_lapack_driver = 'gelsd'
|
|
|
|
|
|
def pinv(a, cond=None, rcond=None, return_rank=False, check_finite=True):
|
|
"""
|
|
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
|
|
|
|
Calculate a generalized inverse of a matrix using a least-squares
|
|
solver.
|
|
|
|
Parameters
|
|
----------
|
|
a : (M, N) array_like
|
|
Matrix to be pseudo-inverted.
|
|
cond, rcond : float, optional
|
|
Cutoff factor for 'small' singular values. In `lstsq`,
|
|
singular values less than ``cond*largest_singular_value`` will be
|
|
considered as zero. If both are omitted, the default value
|
|
``max(M, N) * eps`` is passed to `lstsq` where ``eps`` is the
|
|
corresponding machine precision value of the datatype of ``a``.
|
|
|
|
.. versionchanged:: 1.3.0
|
|
Previously the default cutoff value was just `eps` without the
|
|
factor ``max(M, N)``.
|
|
|
|
return_rank : bool, optional
|
|
if True, return the effective rank of the matrix
|
|
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
|
|
-------
|
|
B : (N, M) ndarray
|
|
The pseudo-inverse of matrix `a`.
|
|
rank : int
|
|
The effective rank of the matrix. Returned if return_rank == True
|
|
|
|
Raises
|
|
------
|
|
LinAlgError
|
|
If computation does not converge.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy import linalg
|
|
>>> a = np.random.randn(9, 6)
|
|
>>> B = linalg.pinv(a)
|
|
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
|
|
True
|
|
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
|
|
True
|
|
|
|
"""
|
|
a = _asarray_validated(a, check_finite=check_finite)
|
|
# If a is sufficiently tall it is cheaper to compute using the transpose
|
|
trans = a.shape[0] / a.shape[1] >= 1.1
|
|
b = np.eye(a.shape[1] if trans else a.shape[0], dtype=a.dtype)
|
|
|
|
if rcond is not None:
|
|
cond = rcond
|
|
|
|
if cond is None:
|
|
cond = max(a.shape) * np.spacing(a.real.dtype.type(1))
|
|
|
|
x, resids, rank, s = lstsq(a.T if trans else a, b,
|
|
cond=cond, check_finite=False)
|
|
|
|
if return_rank:
|
|
return (x.T if trans else x), rank
|
|
else:
|
|
return x.T if trans else x
|
|
|
|
|
|
def pinv2(a, cond=None, rcond=None, return_rank=False, check_finite=True):
|
|
"""
|
|
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
|
|
|
|
Calculate a generalized inverse of a matrix using its
|
|
singular-value decomposition and including all 'large' singular
|
|
values.
|
|
|
|
Parameters
|
|
----------
|
|
a : (M, N) array_like
|
|
Matrix to be pseudo-inverted.
|
|
cond, rcond : float or None
|
|
Cutoff for 'small' singular values; singular values smaller than this
|
|
value are considered as zero. If both are omitted, the default value
|
|
``max(M,N)*largest_singular_value*eps`` is used where ``eps`` is the
|
|
machine precision value of the datatype of ``a``.
|
|
|
|
.. versionchanged:: 1.3.0
|
|
Previously the default cutoff value was just ``eps*f`` where ``f``
|
|
was ``1e3`` for single precision and ``1e6`` for double precision.
|
|
|
|
return_rank : bool, optional
|
|
If True, return the effective rank of the matrix.
|
|
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
|
|
-------
|
|
B : (N, M) ndarray
|
|
The pseudo-inverse of matrix `a`.
|
|
rank : int
|
|
The effective rank of the matrix. Returned if `return_rank` is True.
|
|
|
|
Raises
|
|
------
|
|
LinAlgError
|
|
If SVD computation does not converge.
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy import linalg
|
|
>>> a = np.random.randn(9, 6)
|
|
>>> B = linalg.pinv2(a)
|
|
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
|
|
True
|
|
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
|
|
True
|
|
|
|
"""
|
|
a = _asarray_validated(a, check_finite=check_finite)
|
|
u, s, vh = decomp_svd.svd(a, full_matrices=False, check_finite=False)
|
|
|
|
if rcond is not None:
|
|
cond = rcond
|
|
if cond in [None, -1]:
|
|
t = u.dtype.char.lower()
|
|
cond = np.max(s) * max(a.shape) * np.finfo(t).eps
|
|
|
|
rank = np.sum(s > cond)
|
|
|
|
u = u[:, :rank]
|
|
u /= s[:rank]
|
|
B = np.transpose(np.conjugate(np.dot(u, vh[:rank])))
|
|
|
|
if return_rank:
|
|
return B, rank
|
|
else:
|
|
return B
|
|
|
|
|
|
def pinvh(a, cond=None, rcond=None, lower=True, return_rank=False,
|
|
check_finite=True):
|
|
"""
|
|
Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix.
|
|
|
|
Calculate a generalized inverse of a Hermitian or real symmetric matrix
|
|
using its eigenvalue decomposition and including all eigenvalues with
|
|
'large' absolute value.
|
|
|
|
Parameters
|
|
----------
|
|
a : (N, N) array_like
|
|
Real symmetric or complex hermetian matrix to be pseudo-inverted
|
|
cond, rcond : float or None
|
|
Cutoff for 'small' singular values; singular values smaller than this
|
|
value are considered as zero. If both are omitted, the default
|
|
``max(M,N)*largest_eigenvalue*eps`` is used where ``eps`` is the
|
|
machine precision value of the datatype of ``a``.
|
|
|
|
.. versionchanged:: 1.3.0
|
|
Previously the default cutoff value was just ``eps*f`` where ``f``
|
|
was ``1e3`` for single precision and ``1e6`` for double precision.
|
|
|
|
lower : bool, optional
|
|
Whether the pertinent array data is taken from the lower or upper
|
|
triangle of `a`. (Default: lower)
|
|
return_rank : bool, optional
|
|
If True, return the effective rank of the matrix.
|
|
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
|
|
-------
|
|
B : (N, N) ndarray
|
|
The pseudo-inverse of matrix `a`.
|
|
rank : int
|
|
The effective rank of the matrix. Returned if `return_rank` is True.
|
|
|
|
Raises
|
|
------
|
|
LinAlgError
|
|
If eigenvalue does not converge
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy.linalg import pinvh
|
|
>>> a = np.random.randn(9, 6)
|
|
>>> a = np.dot(a, a.T)
|
|
>>> B = pinvh(a)
|
|
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
|
|
True
|
|
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
|
|
True
|
|
|
|
"""
|
|
a = _asarray_validated(a, check_finite=check_finite)
|
|
s, u = decomp.eigh(a, lower=lower, check_finite=False)
|
|
|
|
if rcond is not None:
|
|
cond = rcond
|
|
if cond in [None, -1]:
|
|
t = u.dtype.char.lower()
|
|
cond = np.max(np.abs(s)) * max(a.shape) * np.finfo(t).eps
|
|
|
|
# For Hermitian matrices, singular values equal abs(eigenvalues)
|
|
above_cutoff = (abs(s) > cond)
|
|
psigma_diag = 1.0 / s[above_cutoff]
|
|
u = u[:, above_cutoff]
|
|
|
|
B = np.dot(u * psigma_diag, np.conjugate(u).T)
|
|
|
|
if return_rank:
|
|
return B, len(psigma_diag)
|
|
else:
|
|
return B
|
|
|
|
|
|
def matrix_balance(A, permute=True, scale=True, separate=False,
|
|
overwrite_a=False):
|
|
"""
|
|
Compute a diagonal similarity transformation for row/column balancing.
|
|
|
|
The balancing tries to equalize the row and column 1-norms by applying
|
|
a similarity transformation such that the magnitude variation of the
|
|
matrix entries is reflected to the scaling matrices.
|
|
|
|
Moreover, if enabled, the matrix is first permuted to isolate the upper
|
|
triangular parts of the matrix and, again if scaling is also enabled,
|
|
only the remaining subblocks are subjected to scaling.
|
|
|
|
The balanced matrix satisfies the following equality
|
|
|
|
.. math::
|
|
|
|
B = T^{-1} A T
|
|
|
|
The scaling coefficients are approximated to the nearest power of 2
|
|
to avoid round-off errors.
|
|
|
|
Parameters
|
|
----------
|
|
A : (n, n) array_like
|
|
Square data matrix for the balancing.
|
|
permute : bool, optional
|
|
The selector to define whether permutation of A is also performed
|
|
prior to scaling.
|
|
scale : bool, optional
|
|
The selector to turn on and off the scaling. If False, the matrix
|
|
will not be scaled.
|
|
separate : bool, optional
|
|
This switches from returning a full matrix of the transformation
|
|
to a tuple of two separate 1-D permutation and scaling arrays.
|
|
overwrite_a : bool, optional
|
|
This is passed to xGEBAL directly. Essentially, overwrites the result
|
|
to the data. It might increase the space efficiency. See LAPACK manual
|
|
for details. This is False by default.
|
|
|
|
Returns
|
|
-------
|
|
B : (n, n) ndarray
|
|
Balanced matrix
|
|
T : (n, n) ndarray
|
|
A possibly permuted diagonal matrix whose nonzero entries are
|
|
integer powers of 2 to avoid numerical truncation errors.
|
|
scale, perm : (n,) ndarray
|
|
If ``separate`` keyword is set to True then instead of the array
|
|
``T`` above, the scaling and the permutation vectors are given
|
|
separately as a tuple without allocating the full array ``T``.
|
|
|
|
Notes
|
|
-----
|
|
|
|
This algorithm is particularly useful for eigenvalue and matrix
|
|
decompositions and in many cases it is already called by various
|
|
LAPACK routines.
|
|
|
|
The algorithm is based on the well-known technique of [1]_ and has
|
|
been modified to account for special cases. See [2]_ for details
|
|
which have been implemented since LAPACK v3.5.0. Before this version
|
|
there are corner cases where balancing can actually worsen the
|
|
conditioning. See [3]_ for such examples.
|
|
|
|
The code is a wrapper around LAPACK's xGEBAL routine family for matrix
|
|
balancing.
|
|
|
|
.. versionadded:: 0.19.0
|
|
|
|
Examples
|
|
--------
|
|
>>> from scipy import linalg
|
|
>>> x = np.array([[1,2,0], [9,1,0.01], [1,2,10*np.pi]])
|
|
|
|
>>> y, permscale = linalg.matrix_balance(x)
|
|
>>> np.abs(x).sum(axis=0) / np.abs(x).sum(axis=1)
|
|
array([ 3.66666667, 0.4995005 , 0.91312162])
|
|
|
|
>>> np.abs(y).sum(axis=0) / np.abs(y).sum(axis=1)
|
|
array([ 1.2 , 1.27041742, 0.92658316]) # may vary
|
|
|
|
>>> permscale # only powers of 2 (0.5 == 2^(-1))
|
|
array([[ 0.5, 0. , 0. ], # may vary
|
|
[ 0. , 1. , 0. ],
|
|
[ 0. , 0. , 1. ]])
|
|
|
|
References
|
|
----------
|
|
.. [1] : B.N. Parlett and C. Reinsch, "Balancing a Matrix for
|
|
Calculation of Eigenvalues and Eigenvectors", Numerische Mathematik,
|
|
Vol.13(4), 1969, :doi:`10.1007/BF02165404`
|
|
|
|
.. [2] : R. James, J. Langou, B.R. Lowery, "On matrix balancing and
|
|
eigenvector computation", 2014, :arxiv:`1401.5766`
|
|
|
|
.. [3] : D.S. Watkins. A case where balancing is harmful.
|
|
Electron. Trans. Numer. Anal, Vol.23, 2006.
|
|
|
|
"""
|
|
|
|
A = np.atleast_2d(_asarray_validated(A, check_finite=True))
|
|
|
|
if not np.equal(*A.shape):
|
|
raise ValueError('The data matrix for balancing should be square.')
|
|
|
|
gebal = get_lapack_funcs(('gebal'), (A,))
|
|
B, lo, hi, ps, info = gebal(A, scale=scale, permute=permute,
|
|
overwrite_a=overwrite_a)
|
|
|
|
if info < 0:
|
|
raise ValueError('xGEBAL exited with the internal error '
|
|
'"illegal value in argument number {}.". See '
|
|
'LAPACK documentation for the xGEBAL error codes.'
|
|
''.format(-info))
|
|
|
|
# Separate the permutations from the scalings and then convert to int
|
|
scaling = np.ones_like(ps, dtype=float)
|
|
scaling[lo:hi+1] = ps[lo:hi+1]
|
|
|
|
# gebal uses 1-indexing
|
|
ps = ps.astype(int, copy=False) - 1
|
|
n = A.shape[0]
|
|
perm = np.arange(n)
|
|
|
|
# LAPACK permutes with the ordering n --> hi, then 0--> lo
|
|
if hi < n:
|
|
for ind, x in enumerate(ps[hi+1:][::-1], 1):
|
|
if n-ind == x:
|
|
continue
|
|
perm[[x, n-ind]] = perm[[n-ind, x]]
|
|
|
|
if lo > 0:
|
|
for ind, x in enumerate(ps[:lo]):
|
|
if ind == x:
|
|
continue
|
|
perm[[x, ind]] = perm[[ind, x]]
|
|
|
|
if separate:
|
|
return B, (scaling, perm)
|
|
|
|
# get the inverse permutation
|
|
iperm = np.empty_like(perm)
|
|
iperm[perm] = np.arange(n)
|
|
|
|
return B, np.diag(scaling)[iperm, :]
|
|
|
|
|
|
def _validate_args_for_toeplitz_ops(c_or_cr, b, check_finite, keep_b_shape,
|
|
enforce_square=True):
|
|
"""Validate arguments and format inputs for toeplitz functions
|
|
|
|
Parameters
|
|
----------
|
|
c_or_cr : array_like or tuple of (array_like, array_like)
|
|
The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the
|
|
actual shape of ``c``, it will be converted to a 1-D array. If not
|
|
supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
|
|
real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
|
|
of the Toeplitz matrix is ``[c[0], r[1:]]``. Whatever the actual shape
|
|
of ``r``, it will be converted to a 1-D array.
|
|
b : (M,) or (M, K) array_like
|
|
Right-hand side in ``T x = b``.
|
|
check_finite : bool
|
|
Whether to check that the input matrices contain only finite numbers.
|
|
Disabling may give a performance gain, but may result in problems
|
|
(result entirely NaNs) if the inputs do contain infinities or NaNs.
|
|
keep_b_shape: bool
|
|
Whether to convert a (M,) dimensional b into a (M, 1) dimensional
|
|
matrix.
|
|
enforce_square: bool, optional
|
|
If True (default), this verifies that the Toeplitz matrix is square.
|
|
|
|
Returns
|
|
-------
|
|
r : array
|
|
1d array corresponding to the first row of the Toeplitz matrix.
|
|
c: array
|
|
1d array corresponding to the first column of the Toeplitz matrix.
|
|
b: array
|
|
(M,), (M, 1) or (M, K) dimensional array, post validation,
|
|
corresponding to ``b``.
|
|
dtype: numpy datatype
|
|
``dtype`` stores the datatype of ``r``, ``c`` and ``b``. If any of
|
|
``r``, ``c`` or ``b`` are complex, ``dtype`` is ``np.complex128``,
|
|
otherwise, it is ``np.float``.
|
|
b_shape: tuple
|
|
Shape of ``b`` after passing it through ``_asarray_validated``.
|
|
|
|
"""
|
|
|
|
if isinstance(c_or_cr, tuple):
|
|
c, r = c_or_cr
|
|
c = _asarray_validated(c, check_finite=check_finite).ravel()
|
|
r = _asarray_validated(r, check_finite=check_finite).ravel()
|
|
else:
|
|
c = _asarray_validated(c_or_cr, check_finite=check_finite).ravel()
|
|
r = c.conjugate()
|
|
|
|
if b is None:
|
|
raise ValueError('`b` must be an array, not None.')
|
|
|
|
b = _asarray_validated(b, check_finite=check_finite)
|
|
b_shape = b.shape
|
|
|
|
is_not_square = r.shape[0] != c.shape[0]
|
|
if (enforce_square and is_not_square) or b.shape[0] != r.shape[0]:
|
|
raise ValueError('Incompatible dimensions.')
|
|
|
|
is_cmplx = np.iscomplexobj(r) or np.iscomplexobj(c) or np.iscomplexobj(b)
|
|
dtype = np.complex128 if is_cmplx else np.double
|
|
r, c, b = (np.asarray(i, dtype=dtype) for i in (r, c, b))
|
|
|
|
if b.ndim == 1 and not keep_b_shape:
|
|
b = b.reshape(-1, 1)
|
|
elif b.ndim != 1:
|
|
b = b.reshape(b.shape[0], -1)
|
|
|
|
return r, c, b, dtype, b_shape
|
|
|
|
|
|
def matmul_toeplitz(c_or_cr, x, check_finite=False, workers=None):
|
|
"""Efficient Toeplitz Matrix-Matrix Multiplication using FFT
|
|
|
|
This function returns the matrix multiplication between a Toeplitz
|
|
matrix and a dense matrix.
|
|
|
|
The Toeplitz matrix has constant diagonals, with c as its first column
|
|
and r as its first row. If r is not given, ``r == conjugate(c)`` is
|
|
assumed.
|
|
|
|
Parameters
|
|
----------
|
|
c_or_cr : array_like or tuple of (array_like, array_like)
|
|
The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the
|
|
actual shape of ``c``, it will be converted to a 1-D array. If not
|
|
supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
|
|
real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
|
|
of the Toeplitz matrix is ``[c[0], r[1:]]``. Whatever the actual shape
|
|
of ``r``, it will be converted to a 1-D array.
|
|
x : (M,) or (M, K) array_like
|
|
Matrix with which to multiply.
|
|
check_finite : bool, optional
|
|
Whether to check that the input matrices contain only finite numbers.
|
|
Disabling may give a performance gain, but may result in problems
|
|
(result entirely NaNs) if the inputs do contain infinities or NaNs.
|
|
workers : int, optional
|
|
To pass to scipy.fft.fft and ifft. Maximum number of workers to use
|
|
for parallel computation. If negative, the value wraps around from
|
|
``os.cpu_count()``. See scipy.fft.fft for more details.
|
|
|
|
Returns
|
|
-------
|
|
T @ x : (M,) or (M, K) ndarray
|
|
The result of the matrix multiplication ``T @ x``. Shape of return
|
|
matches shape of `x`.
|
|
|
|
See Also
|
|
--------
|
|
toeplitz : Toeplitz matrix
|
|
solve_toeplitz : Solve a Toeplitz system using Levinson Recursion
|
|
|
|
Notes
|
|
-----
|
|
The Toeplitz matrix is embedded in a circulant matrix and the FFT is used
|
|
to efficiently calculate the matrix-matrix product.
|
|
|
|
Because the computation is based on the FFT, integer inputs will
|
|
result in floating point outputs. This is unlike NumPy's `matmul`,
|
|
which preserves the data type of the input.
|
|
|
|
This is partly based on the implementation that can be found in [1]_,
|
|
licensed under the MIT license. More information about the method can be
|
|
found in reference [2]_. References [3]_ and [4]_ have more reference
|
|
implementations in Python.
|
|
|
|
.. versionadded:: 1.6.0
|
|
|
|
References
|
|
----------
|
|
.. [1] Jacob R Gardner, Geoff Pleiss, David Bindel, Kilian
|
|
Q Weinberger, Andrew Gordon Wilson, "GPyTorch: Blackbox Matrix-Matrix
|
|
Gaussian Process Inference with GPU Acceleration" with contributions
|
|
from Max Balandat and Ruihan Wu. Available online:
|
|
https://github.com/cornellius-gp/gpytorch
|
|
|
|
.. [2] J. Demmel, P. Koev, and X. Li, "A Brief Survey of Direct Linear
|
|
Solvers". In Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der
|
|
Vorst, editors. Templates for the Solution of Algebraic Eigenvalue
|
|
Problems: A Practical Guide. SIAM, Philadelphia, 2000. Available at:
|
|
http://www.netlib.org/utk/people/JackDongarra/etemplates/node384.html
|
|
|
|
.. [3] R. Scheibler, E. Bezzam, I. Dokmanic, Pyroomacoustics: A Python
|
|
package for audio room simulations and array processing algorithms,
|
|
Proc. IEEE ICASSP, Calgary, CA, 2018.
|
|
https://github.com/LCAV/pyroomacoustics/blob/pypi-release/
|
|
pyroomacoustics/adaptive/util.py
|
|
|
|
.. [4] Marano S, Edwards B, Ferrari G and Fah D (2017), "Fitting
|
|
Earthquake Spectra: Colored Noise and Incomplete Data", Bulletin of
|
|
the Seismological Society of America., January, 2017. Vol. 107(1),
|
|
pp. 276-291.
|
|
|
|
Examples
|
|
--------
|
|
Multiply the Toeplitz matrix T with matrix x::
|
|
|
|
[ 1 -1 -2 -3] [1 10]
|
|
T = [ 3 1 -1 -2] x = [2 11]
|
|
[ 6 3 1 -1] [2 11]
|
|
[10 6 3 1] [5 19]
|
|
|
|
To specify the Toeplitz matrix, only the first column and the first
|
|
row are needed.
|
|
|
|
>>> c = np.array([1, 3, 6, 10]) # First column of T
|
|
>>> r = np.array([1, -1, -2, -3]) # First row of T
|
|
>>> x = np.array([[1, 10], [2, 11], [2, 11], [5, 19]])
|
|
|
|
>>> from scipy.linalg import toeplitz, matmul_toeplitz
|
|
>>> matmul_toeplitz((c, r), x)
|
|
array([[-20., -80.],
|
|
[ -7., -8.],
|
|
[ 9., 85.],
|
|
[ 33., 218.]])
|
|
|
|
Check the result by creating the full Toeplitz matrix and
|
|
multiplying it by ``x``.
|
|
|
|
>>> toeplitz(c, r) @ x
|
|
array([[-20, -80],
|
|
[ -7, -8],
|
|
[ 9, 85],
|
|
[ 33, 218]])
|
|
|
|
The full matrix is never formed explicitly, so this routine
|
|
is suitable for very large Toeplitz matrices.
|
|
|
|
>>> n = 1000000
|
|
>>> matmul_toeplitz([1] + [0]*(n-1), np.ones(n))
|
|
array([1., 1., 1., ..., 1., 1., 1.])
|
|
|
|
"""
|
|
|
|
from ..fft import fft, ifft, rfft, irfft
|
|
|
|
r, c, x, dtype, x_shape = _validate_args_for_toeplitz_ops(
|
|
c_or_cr, x, check_finite, keep_b_shape=False, enforce_square=False)
|
|
n, m = x.shape
|
|
|
|
T_nrows = len(c)
|
|
T_ncols = len(r)
|
|
p = T_nrows + T_ncols - 1 # equivalent to len(embedded_col)
|
|
|
|
embedded_col = np.concatenate((c, r[-1:0:-1]))
|
|
|
|
if np.iscomplexobj(embedded_col) or np.iscomplexobj(x):
|
|
fft_mat = fft(embedded_col, axis=0, workers=workers).reshape(-1, 1)
|
|
fft_x = fft(x, n=p, axis=0, workers=workers)
|
|
|
|
mat_times_x = ifft(fft_mat*fft_x, axis=0,
|
|
workers=workers)[:T_nrows, :]
|
|
else:
|
|
# Real inputs; using rfft is faster
|
|
fft_mat = rfft(embedded_col, axis=0, workers=workers).reshape(-1, 1)
|
|
fft_x = rfft(x, n=p, axis=0, workers=workers)
|
|
|
|
mat_times_x = irfft(fft_mat*fft_x, axis=0,
|
|
workers=workers, n=p)[:T_nrows, :]
|
|
|
|
return_shape = (T_nrows,) if len(x_shape) == 1 else (T_nrows, m)
|
|
return mat_times_x.reshape(*return_shape)
|