Module « scipy.linalg »

Fonction matrix_balance - module scipy.linalg

Signature de la fonction matrix_balance

def matrix_balance(A, permute=True, scale=True, separate=False, overwrite_a=False) 

Description

matrix_balance.__doc__

    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.