Python numpy.asarray_chkfinite() Examples

def _cholesky(a, lower=False, overwrite_a=False, clean=True,
                check_finite=True):
    """Common code for cholesky() and cho_factor()."""

    if check_finite:
        a1 = asarray_chkfinite(a)
    else:
        a1 = asarray(a)
    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)
    potrf, = get_lapack_funcs(('potrf',), (a1,))
    c, info = potrf(a1, lower=lower, overwrite_a=overwrite_a, clean=clean)
    if info > 0:
        raise LinAlgError("%d-th leading minor not positive definite" % info)
    if info < 0:
        raise ValueError('illegal value in %d-th argument of internal potrf'
                                                                    % -info)
    return c, lower 

def __init__(self, inverse_array, lower=True, make_triangular=True):
        """
        Args:
            inverse_array (array): 2D containing values of *inverse* of this
                matrix, with the inverse of a lower (upper) triangular matrix
                being itself lower (upper) triangular. Any values above (below)
                diagonal are ignored for lower (upper) triangular matrices i.e.
                when `lower == True` (`lower == False`).
            lower (bool): Whether the matrix is lower-triangular (`True`) or
                upper-triangular (`False`).
            make_triangular (bool): Whether to ensure `inverse_array` is
                triangular by explicitly zeroing entries in upper triangle if
                `lower == True` and in lower triangle if `lower == False`.
        """
        inverse_array = np.asarray_chkfinite(inverse_array)
        inverse_array = (
            _make_array_triangular(inverse_array, lower) if make_triangular
            else inverse_array)
        super().__init__(inverse_array.shape, _inverse_array=inverse_array)
        self._lower = lower 

def _cholesky(a, lower=False, overwrite_a=False, clean=True,
                check_finite=True):
    """Common code for cholesky() and cho_factor()."""

    if check_finite:
        a1 = asarray_chkfinite(a)
    else:
        a1 = asarray(a)
    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)
    potrf, = get_lapack_funcs(('potrf',), (a1,))
    c, info = potrf(a1, lower=lower, overwrite_a=overwrite_a, clean=clean)
    if info > 0:
        raise LinAlgError("%d-th leading minor not positive definite" % info)
    if info < 0:
        raise ValueError('illegal value in %d-th argument of internal potrf'
                                                                    % -info)
    return c, lower 

def ackley(x, a=20, b=0.2, c=2*np.pi):

    x = np.asarray_chkfinite(x)  # ValueError if any NaN or Inf
    n = len(x)
    s1 = np.sum(x**2, axis=0)
    s2 = np.sum(np.cos(c * x), axis=0)
    return -a*np.exp(-b*np.sqrt(s1 / n)) - np.exp(s2 / n) + a + np.exp(1) 

def fit(self, X):
        self.X_ = np.asarray_chkfinite(X)
        self.F_, self.l_, self.omega_ = compute_rdmd(
            self.X_, rank=self.rank, dt=self.dt, oversample=self.oversample,
            n_subspace=self.n_subspace, modes=self.modes, order=self.order,
            random_state=self.random_state)
        return self 

def pinv(a, cond=None, rcond=None):
    """Compute the (Moore-Penrose) pseudo-inverse of a matrix.

    Calculate a generalized inverse of a matrix using a least-squares
    solver.

    Parameters
    ----------
    a : array, shape (M, N)
        Matrix to be pseudo-inverted
    cond, rcond : float
        Cutoff for 'small' singular values in the least-squares solver.
        Singular values smaller than rcond*largest_singular_value are
        considered zero.

    Returns
    -------
    B : array, shape (N, M)

    Raises LinAlgError if computation does not converge

    Examples
    --------
    >>> from numpy import *
    >>> a = random.randn(9, 6)
    >>> B = linalg.pinv(a)
    >>> allclose(a, dot(a, dot(B, a)))
    True
    >>> allclose(B, dot(B, dot(a, B)))
    True

    """
    a = asarray_chkfinite(a)
    b = numpy.identity(a.shape[0], dtype=a.dtype)
    if rcond is not None:
        cond = rcond
    return lstsq(a, b, cond=cond)[0] 

def fit(self, X, y=None):
        '''Fits DMD'''
        self.X_ = np.asarray_chkfinite(X)
        self.F_, self.l_, self.omega_ = compute_dmd(
            self.X_, rank=self.rank, dt=self.dt, modes=self.modes,
            order=self.order)
        return self 

def fast_johnson_lindenstrauss(A, l, axis=1, random_state=None):
    """

    Given an m x n matrix A, and an integer l, this scheme computes an m x l
    orthonormal matrix Q whose range approximates the range of A

    """
    random_state = check_random_state(random_state)

    A = np.asarray_chkfinite(A)
    if A.ndim != 2:
        raise ValueError('A must be a 2D array, not %dD' % A.ndim)

    if axis not in (0, 1):
        raise ValueError('If supplied, axis must be in (0, 1)')

    # TODO: Find name for sketch and put in _sketches
    # construct gaussian random matrix
    diag = random_state.choice((-1, 1), size=A.shape[axis]).astype(A.dtype)

    if axis == 0:
        diag = diag[:, np.newaxis]

    # discrete fourier transform of AD (or DA)
    FDA = fftpack.dct(A * diag, axis=axis, norm='ortho')

    # randomly sample axis
    return randomized_uniform_sampling(
        FDA, l, axis=axis, random_state=random_state) 

def check_and_set_idx(ids, idx, prefix):
    """ Reconciles passed-in IDs and indices and returns indices, as well as unique IDs
    in the order specified by the indices.  If only IDs supplied, returns the sort-arg
    as the index.  If only indices supplied, returns None for IDs.  If both supplied,
    checks that the correspondence is unique and returns unique IDs in the sort order of
    the associated index.
    :param np.ndarray ids: array of IDs
    :param np.ndarray[int] idx: array of indices
    :param str prefix: variable name (for error logging)
    :return: unique IDs and indices (passed in or derived from the IDs)
    :rtype: np.ndarray, np.ndarray
    """
    if ids is None and idx is None:
        raise ValueError('Both {}_ids and {}_idx cannot be None'.format(prefix, prefix))
    if ids is None:
        return None, np.asarray_chkfinite(idx)
    if idx is None:
        return np.unique(ids, return_inverse=True)
    else:
        ids = np.asarray(ids)
        idx = np.asarray_chkfinite(idx)
        if len(idx) != len(ids):
            raise ValueError('{}_ids ({}) and {}_idx ({}) must have the same length'.format(
                prefix, len(ids), prefix, len(idx)))
        uniq_idx, idx_sort_index = np.unique(idx, return_index=True)
        # make sure each unique index corresponds to a unique id
        if not all(len(set(ids[idx == i])) == 1 for i in uniq_idx):
            raise ValueError("Each index must correspond to a unique {}_id".format(prefix))
        return ids[idx_sort_index], idx 

def pinv(a, cond=None, rcond=None):
    """Compute the (Moore-Penrose) pseudo-inverse of a matrix.

    Calculate a generalized inverse of a matrix using a least-squares
    solver.

    Parameters
    ----------
    a : array, shape (M, N)
        Matrix to be pseudo-inverted
    cond, rcond : float
        Cutoff for 'small' singular values in the least-squares solver.
        Singular values smaller than rcond*largest_singular_value are
        considered zero.

    Returns
    -------
    B : array, shape (N, M)

    Raises LinAlgError if computation does not converge

    Examples
    --------
    >>> from numpy import *
    >>> a = random.randn(9, 6)
    >>> B = linalg.pinv(a)
    >>> allclose(a, dot(a, dot(B, a)))
    True
    >>> allclose(B, dot(B, dot(a, B)))
    True

    """
    a = asarray_chkfinite(a)
    b = numpy.identity(a.shape[0], dtype=a.dtype)
    if rcond is not None:
        cond = rcond
    return lstsq(a, b, cond=cond)[0] 

def _fractal_mfdfa_q(q=2):
    # TODO: Add log calculator for q ≈ 0

    # Fractal powers as floats
    q = np.asarray_chkfinite(q, dtype=np.float)

    # Ensure q≈0 is removed, since it does not converge. Limit set at |q| < 0.1
    q = q[(q < -0.1) + (q > 0.1)]

    # Reshape q to perform np.float_power
    q = q.reshape(-1, 1)
    return q 

def _cholesky(a, lower=False, overwrite_a=False, clean=True,
              check_finite=True):
    """Common code for cholesky() and cho_factor()."""

    a1 = asarray_chkfinite(a) if check_finite else asarray(a)
    a1 = atleast_2d(a1)

    # Dimension check
    if a1.ndim != 2:
        raise ValueError('Input array needs to be 2 dimensional but received '
                         'a {}d-array.'.format(a1.ndim))
    # Squareness check
    if a1.shape[0] != a1.shape[1]:
        raise ValueError('Input array is expected to be square but has '
                         'the shape: {}.'.format(a1.shape))

    # Quick return for square empty array
    if a1.size == 0:
        return a1.copy(), lower

    overwrite_a = overwrite_a or _datacopied(a1, a)
    potrf, = get_lapack_funcs(('potrf',), (a1,))
    c, info = potrf(a1, lower=lower, overwrite_a=overwrite_a, clean=clean)
    if info > 0:
        raise LinAlgError("%d-th leading minor of the array is not positive "
                          "definite" % info)
    if info < 0:
        raise ValueError('LAPACK reported an illegal value in {}-th argument'
                         'on entry to "POTRF".'.format(-info))
    return c, lower 

def __init__(self, shape, **kwargs):
        if '_array' not in kwargs:
            raise ValueError('_array must be specified in kwargs')
        else:
            kwargs['_array'] = np.asarray_chkfinite(kwargs['_array'])
        super().__init__(shape, **kwargs) 

def pinvh(a, cond=None, rcond=None, lower=True):
    '''Compute the (Moore-Penrose) pseudo-inverse of a hermetian matrix.

    Calculate a generalized inverse of a symmetric matrix using its
    eigenvalue decomposition and including all 'large' eigenvalues.

    Parameters
    ----------
    a : array, shape (N, N)
        Real symmetric or complex hermetian matrix to be pseudo-inverted
    cond, rcond : float or None
        Cutoff for 'small' eigenvalues.
        Singular values smaller than rcond * largest_eigenvalue are considered
        zero.

        If None or -1, suitable machine precision is used.
    lower : boolean
        Whether the pertinent array data is taken from the lower or upper
        triangle of a. (Default: lower)

    Returns
    -------
    B : array, shape (N, N)

    Raises
    ------
    LinAlgError
        If eigenvalue does not converge

    Examples
    --------
    >>> import numpy as np
    >>> 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 = np.asarray_chkfinite(a)
    s, u = linalg.eigh(a, lower=lower)

    if rcond is not None:
        cond = rcond
    if cond in [None, -1]:
        t = u.dtype.char.lower()
        factor = {'f': 1E3, 'd': 1E6}
        cond = factor[t] * np.finfo(t).eps

    # unlike svd case, eigh can lead to negative eigenvalues
    above_cutoff = (abs(s) > cond * np.max(abs(s)))
    psigma_diag = np.zeros_like(s)
    psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]

    return np.dot(u * psigma_diag, np.conjugate(u).T) 

def pinvh(a, cond=None, rcond=None, lower=True):
    '''Compute the (Moore-Penrose) pseudo-inverse of a hermetian matrix.

    Calculate a generalized inverse of a symmetric matrix using its
    eigenvalue decomposition and including all 'large' eigenvalues.

    Parameters
    ----------
    a : array, shape (N, N)
        Real symmetric or complex hermetian matrix to be pseudo-inverted
    cond, rcond : float or None
        Cutoff for 'small' eigenvalues.
        Singular values smaller than rcond * largest_eigenvalue are considered
        zero.

        If None or -1, suitable machine precision is used.
    lower : boolean
        Whether the pertinent array data is taken from the lower or upper
        triangle of a. (Default: lower)

    Returns
    -------
    B : array, shape (N, N)

    Raises
    ------
    LinAlgError
        If eigenvalue does not converge

    Examples
    --------
    >>> import numpy as np
    >>> 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 = np.asarray_chkfinite(a)
    s, u = linalg.eigh(a, lower=lower)

    if rcond is not None:
        cond = rcond
    if cond in [None, -1]:
        t = u.dtype.char.lower()
        factor = {'f': 1E3, 'd': 1E6}
        cond = factor[t] * np.finfo(t).eps

    # unlike svd case, eigh can lead to negative eigenvalues
    above_cutoff = (abs(s) > cond * np.max(abs(s)))
    psigma_diag = np.zeros_like(s)
    psigma_diag[above_cutoff] = 1.0 / s[above_cutoff]

    return np.dot(u * psigma_diag, np.conjugate(u).T) 

def nnls(A, b, maxiter=None):
    """
    Solve ``argmin_x || Ax - b ||_2`` for ``x>=0``. This is a wrapper
    for a FORTRAN non-negative least squares solver.

    Parameters
    ----------
    A : ndarray
        Matrix ``A`` as shown above.
    b : ndarray
        Right-hand side vector.
    maxiter: int, optional
        Maximum number of iterations, optional.
        Default is ``3 * A.shape[1]``.

    Returns
    -------
    x : ndarray
        Solution vector.
    rnorm : float
        The residual, ``|| Ax-b ||_2``.

    Notes
    -----
    The FORTRAN code was published in the book below. The algorithm
    is an active set method. It solves the KKT (Karush-Kuhn-Tucker)
    conditions for the non-negative least squares problem.

    References
    ----------
    Lawson C., Hanson R.J., (1987) Solving Least Squares Problems, SIAM

    """

    A, b = map(asarray_chkfinite, (A, b))

    if len(A.shape) != 2:
        raise ValueError("expected matrix")
    if len(b.shape) != 1:
        raise ValueError("expected vector")

    m, n = A.shape

    if m != b.shape[0]:
        raise ValueError("incompatible dimensions")

    maxiter = -1 if maxiter is None else int(maxiter)

    w = zeros((n,), dtype=double)
    zz = zeros((m,), dtype=double)
    index = zeros((n,), dtype=int)

    x, rnorm, mode = _nnls.nnls(A, m, n, b, w, zz, index, maxiter)
    if mode != 1:
        raise RuntimeError("too many iterations")

    return x, rnorm 

def expm_cond(A, check_finite=True):
    """
    Relative condition number of the matrix exponential in the Frobenius norm.

    Parameters
    ----------
    A : 2d array_like
        Square input matrix with shape (N, N).
    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
    -------
    kappa : float
        The relative condition number of the matrix exponential
        in the Frobenius norm

    Notes
    -----
    A faster estimate for the condition number in the 1-norm
    has been published but is not yet implemented in scipy.

    .. versionadded:: 0.14.0

    See also
    --------
    expm : Compute the exponential of a matrix.
    expm_frechet : Compute the Frechet derivative of the matrix exponential.

    Examples
    --------
    >>> from scipy.linalg import expm_cond
    >>> A = np.array([[-0.3, 0.2, 0.6], [0.6, 0.3, -0.1], [-0.7, 1.2, 0.9]])
    >>> k = expm_cond(A)
    >>> k
    1.7787805864469866

    """
    if check_finite:
        A = np.asarray_chkfinite(A)
    else:
        A = np.asarray(A)
    if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
        raise ValueError('expected a square matrix')

    X = scipy.linalg.expm(A)
    K = expm_frechet_kronform(A, check_finite=False)

    # The following norm choices are deliberate.
    # The norms of A and X are Frobenius norms,
    # and the norm of K is the induced 2-norm.
    A_norm = scipy.linalg.norm(A, 'fro')
    X_norm = scipy.linalg.norm(X, 'fro')
    K_norm = scipy.linalg.norm(K, 2)

    kappa = (K_norm * A_norm) / X_norm
    return kappa 

def pinv2(a, cond=None, rcond=None):
    """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 : array, shape (M, N)
        Matrix to be pseudo-inverted
    cond, rcond : float or None
        Cutoff for 'small' singular values.
        Singular values smaller than rcond*largest_singular_value are
        considered zero.

        If None or -1, suitable machine precision is used.

    Returns
    -------
    B : array, shape (N, M)

    Raises LinAlgError if SVD computation does not converge

    Examples
    --------
    >>> from numpy import *
    >>> a = random.randn(9, 6)
    >>> B = linalg.pinv2(a)
    >>> allclose(a, dot(a, dot(B, a)))
    True
    >>> allclose(B, dot(B, dot(a, B)))
    True

    """
    a = asarray_chkfinite(a)
    u, s, vh = decomp_svd(a)
    t = u.dtype.char
    if rcond is not None:
        cond = rcond
    if cond in [None, -1]:
        cond = {0: feps*1e3, 1: eps*1e6}[_array_precision[t]]
    m, n = a.shape
    cutoff = cond*numpy.maximum.reduce(s)
    psigma = zeros((m, n), t)
    for i in range(len(s)):
        if s[i] > cutoff:
            psigma[i, i] = 1.0/conjugate(s[i])
    # XXX: use lapack/blas routines for dot
    return transpose(conjugate(dot(dot(u, psigma), vh))) 

def nnls(A, b):
    """
    Solve ``argmin_x || Ax - b ||_2`` for ``x>=0``. This is a wrapper
    for a FORTRAN non-negative least squares solver.

    Parameters
    ----------
    A : ndarray
        Matrix ``A`` as shown above.
    b : ndarray
        Right-hand side vector.

    Returns
    -------
    x : ndarray
        Solution vector.
    rnorm : float
        The residual, ``|| Ax-b ||_2``.

    Notes
    -----
    The FORTRAN code was published in the book below. The algorithm
    is an active set method. It solves the KKT (Karush-Kuhn-Tucker)
    conditions for the non-negative least squares problem.

    References
    ----------
    Lawson C., Hanson R.J., (1987) Solving Least Squares Problems, SIAM

    """

    A, b = map(asarray_chkfinite, (A, b))

    if len(A.shape) != 2:
        raise ValueError("expected matrix")
    if len(b.shape) != 1:
        raise ValueError("expected vector")

    m, n = A.shape

    if m != b.shape[0]:
        raise ValueError("incompatible dimensions")

    w = zeros((n,), dtype=double)
    zz = zeros((m,), dtype=double)
    index = zeros((n,), dtype=int)

    x, rnorm, mode = _nnls.nnls(A, m, n, b, w, zz, index)
    if mode != 1:
        raise RuntimeError("too many iterations")

    return x, rnorm 

def lu_factor(a, overwrite_a=False, check_finite=True):
    """
    Compute pivoted LU decomposition of a matrix.

    The decomposition is::

        A = P L U

    where P is a permutation matrix, L lower triangular with unit
    diagonal elements, and U upper triangular.

    Parameters
    ----------
    a : (M, M) array_like
        Matrix to decompose
    overwrite_a : bool, optional
        Whether to overwrite data in A (may increase 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
    -------
    lu : (N, N) ndarray
        Matrix containing U in its upper triangle, and L in its lower triangle.
        The unit diagonal elements of L are not stored.
    piv : (N,) ndarray
        Pivot indices representing the permutation matrix P:
        row i of matrix was interchanged with row piv[i].

    See also
    --------
    lu_solve : solve an equation system using the LU factorization of a matrix

    Notes
    -----
    This is a wrapper to the ``*GETRF`` routines from LAPACK.

    """
    if check_finite:
        a1 = asarray_chkfinite(a)
    else:
        a1 = asarray(a)
    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))
    getrf, = get_lapack_funcs(('getrf',), (a1,))
    lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
    if info < 0:
        raise ValueError('illegal value in %d-th argument of '
                                'internal getrf (lu_factor)' % -info)
    if info > 0:
        warn("Diagonal number %d is exactly zero. Singular matrix." % info,
                    RuntimeWarning)
    return lu, piv 

def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True):
    """Solve an equation system, a x = b, given the LU factorization of a

    Parameters
    ----------
    (lu, piv)
        Factorization of the coefficient matrix a, as given by lu_factor
    b : array
        Right-hand side
    trans : {0, 1, 2}, optional
        Type of system to solve:

        =====  =========
        trans  system
        =====  =========
        0      a x   = b
        1      a^T x = b
        2      a^H x = b
        =====  =========
    overwrite_b : bool, optional
        Whether to overwrite data in b (may increase performance)
    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 : array
        Solution to the system

    See also
    --------
    lu_factor : LU factorize a matrix

    """
    (lu, piv) = lu_and_piv
    if check_finite:
        b1 = asarray_chkfinite(b)
    else:
        b1 = asarray(b)
    overwrite_b = overwrite_b or _datacopied(b1, b)
    if lu.shape[0] != b1.shape[0]:
        raise ValueError("incompatible dimensions.")

    getrs, = get_lapack_funcs(('getrs',), (lu, b1))
    x,info = getrs(lu, piv, b1, trans=trans, overwrite_b=overwrite_b)
    if info == 0:
        return x
    raise ValueError('illegal value in %d-th argument of internal gesv|posv'
                                                                    % -info) 

def expm_cond(A, check_finite=True):
    """
    Relative condition number of the matrix exponential in the Frobenius norm.

    Parameters
    ----------
    A : 2d array_like
        Square input matrix with shape (N, N).
    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
    -------
    kappa : float
        The relative condition number of the matrix exponential
        in the Frobenius norm

    Notes
    -----
    A faster estimate for the condition number in the 1-norm
    has been published but is not yet implemented in scipy.

    .. versionadded:: 0.14.0

    See also
    --------
    expm : Compute the exponential of a matrix.
    expm_frechet : Compute the Frechet derivative of the matrix exponential.

    """
    if check_finite:
        A = np.asarray_chkfinite(A)
    else:
        A = np.asarray(A)
    if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
        raise ValueError('expected a square matrix')

    X = scipy.linalg.expm(A)
    K = expm_frechet_kronform(A, check_finite=False)

    # The following norm choices are deliberate.
    # The norms of A and X are Frobenius norms,
    # and the norm of K is the induced 2-norm.
    A_norm = scipy.linalg.norm(A, 'fro')
    X_norm = scipy.linalg.norm(X, 'fro')
    K_norm = scipy.linalg.norm(K, 2)

    kappa = (K_norm * A_norm) / X_norm
    return kappa 

def cho_solve(c_and_lower, b, overwrite_b=False, check_finite=True):
    """Solve the linear equations A x = b, given the Cholesky factorization of A.

    Parameters
    ----------
    (c, lower) : tuple, (array, bool)
        Cholesky factorization of a, as given by cho_factor
    b : array
        Right-hand side
    overwrite_b : bool, optional
        Whether to overwrite data in b (may improve performance)
    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 : array
        The solution to the system A x = b

    See also
    --------
    cho_factor : Cholesky factorization of a matrix

    """
    (c, lower) = c_and_lower
    if check_finite:
        b1 = asarray_chkfinite(b)
        c = asarray_chkfinite(c)
    else:
        b1 = asarray(b)
        c = asarray(c)
    if c.ndim != 2 or c.shape[0] != c.shape[1]:
        raise ValueError("The factored matrix c is not square.")
    if c.shape[1] != b1.shape[0]:
        raise ValueError("incompatible dimensions.")

    overwrite_b = overwrite_b or _datacopied(b1, b)

    potrs, = get_lapack_funcs(('potrs',), (c, b1))
    x, info = potrs(c, b1, lower=lower, overwrite_b=overwrite_b)
    if info != 0:
        raise ValueError('illegal value in %d-th argument of internal potrs'
                                                                    % -info)
    return x 

def cholesky_banded(ab, overwrite_ab=False, lower=False, check_finite=True):
    """
    Cholesky decompose a banded Hermitian positive-definite 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 *   *

    Parameters
    ----------
    ab : (u + 1, M) array_like
        Banded matrix
    overwrite_ab : bool, optional
        Discard data in ab (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 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
    -------
    c : (u + 1, M) ndarray
        Cholesky factorization of a, in the same banded format as ab

    """
    if check_finite:
        ab = asarray_chkfinite(ab)
    else:
        ab = asarray(ab)

    pbtrf, = get_lapack_funcs(('pbtrf',), (ab,))
    c, info = pbtrf(ab, lower=lower, overwrite_ab=overwrite_ab)
    if info > 0:
        raise LinAlgError("%d-th leading minor not positive definite" % info)
    if info < 0:
        raise ValueError('illegal value in %d-th argument of internal pbtrf'
                                                                    % -info)
    return c 

def cho_solve_banded(cb_and_lower, b, overwrite_b=False, check_finite=True):
    """Solve the linear equations A x = b, given the Cholesky factorization of A.

    Parameters
    ----------
    (cb, lower) : tuple, (array, bool)
        `cb` is the Cholesky factorization of A, as given by cholesky_banded.
        `lower` must be the same value that was given to cholesky_banded.
    b : array
        Right-hand side
    overwrite_b : bool, optional
        If True, the function will overwrite the values in `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
        (crashes, non-termination) if the inputs do contain infinities or NaNs.

    Returns
    -------
    x : array
        The solution to the system A x = b

    See also
    --------
    cholesky_banded : Cholesky factorization of a banded matrix

    Notes
    -----

    .. versionadded:: 0.8.0

    """
    (cb, lower) = cb_and_lower
    if check_finite:
        cb = asarray_chkfinite(cb)
        b = asarray_chkfinite(b)
    else:
        cb = asarray(cb)
        b = asarray(b)

    # Validate shapes.
    if cb.shape[-1] != b.shape[0]:
        raise ValueError("shapes of cb and b are not compatible.")

    pbtrs, = get_lapack_funcs(('pbtrs',), (cb, b))
    x, info = pbtrs(cb, b, lower=lower, overwrite_b=overwrite_b)
    if info > 0:
        raise LinAlgError("%d-th leading minor not positive definite" % info)
    if info < 0:
        raise ValueError('illegal value in %d-th argument of internal pbtrs'
                                                                    % -info)
    return x 

def nnls(A, b):
    """
    Solve ``argmin_x || Ax - b ||_2`` for ``x>=0``. This is a wrapper
    for a FORTAN non-negative least squares solver.

    Parameters
    ----------
    A : ndarray
        Matrix ``A`` as shown above.
    b : ndarray
        Right-hand side vector.

    Returns
    -------
    x : ndarray
        Solution vector.
    rnorm : float
        The residual, ``|| Ax-b ||_2``.

    Notes
    -----
    The FORTRAN code was published in the book below. The algorithm
    is an active set method. It solves the KKT (Karush-Kuhn-Tucker)
    conditions for the non-negative least squares problem.

    References
    ----------
    Lawson C., Hanson R.J., (1987) Solving Least Squares Problems, SIAM

    """

    A, b = map(asarray_chkfinite, (A, b))

    if len(A.shape) != 2:
        raise ValueError("expected matrix")
    if len(b.shape) != 1:
        raise ValueError("expected vector")

    m, n = A.shape

    if m != b.shape[0]:
        raise ValueError("incompatible dimensions")

    w = zeros((n,), dtype=double)
    zz = zeros((m,), dtype=double)
    index = zeros((n,), dtype=int)

    x, rnorm, mode = _nnls.nnls(A, m, n, b, w, zz, index)
    if mode != 1:
        raise RuntimeError("too many iterations")

    return x, rnorm 

def nnls(A, b):
    """
    Solve ``argmin_x || Ax - b ||_2`` for ``x>=0``. This is a wrapper
    for a FORTRAN non-negative least squares solver.

    Parameters
    ----------
    A : ndarray
        Matrix ``A`` as shown above.
    b : ndarray
        Right-hand side vector.

    Returns
    -------
    x : ndarray
        Solution vector.
    rnorm : float
        The residual, ``|| Ax-b ||_2``.

    Notes
    -----
    The FORTRAN code was published in the book below. The algorithm
    is an active set method. It solves the KKT (Karush-Kuhn-Tucker)
    conditions for the non-negative least squares problem.

    References
    ----------
    Lawson C., Hanson R.J., (1987) Solving Least Squares Problems, SIAM

    """

    A, b = map(asarray_chkfinite, (A, b))

    if len(A.shape) != 2:
        raise ValueError("expected matrix")
    if len(b.shape) != 1:
        raise ValueError("expected vector")

    m, n = A.shape

    if m != b.shape[0]:
        raise ValueError("incompatible dimensions")

    w = zeros((n,), dtype=double)
    zz = zeros((m,), dtype=double)
    index = zeros((n,), dtype=int)

    x, rnorm, mode = _nnls.nnls(A, m, n, b, w, zz, index)
    if mode != 1:
        raise RuntimeError("too many iterations")

    return x, rnorm 

def lu_factor(a, overwrite_a=False, check_finite=True):
    """
    Compute pivoted LU decomposition of a matrix.

    The decomposition is::

        A = P L U

    where P is a permutation matrix, L lower triangular with unit
    diagonal elements, and U upper triangular.

    Parameters
    ----------
    a : (M, M) array_like
        Matrix to decompose
    overwrite_a : bool, optional
        Whether to overwrite data in A (may increase 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
    -------
    lu : (N, N) ndarray
        Matrix containing U in its upper triangle, and L in its lower triangle.
        The unit diagonal elements of L are not stored.
    piv : (N,) ndarray
        Pivot indices representing the permutation matrix P:
        row i of matrix was interchanged with row piv[i].

    See also
    --------
    lu_solve : solve an equation system using the LU factorization of a matrix

    Notes
    -----
    This is a wrapper to the ``*GETRF`` routines from LAPACK.

    """
    if check_finite:
        a1 = asarray_chkfinite(a)
    else:
        a1 = asarray(a)
    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))
    getrf, = get_lapack_funcs(('getrf',), (a1,))
    lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
    if info < 0:
        raise ValueError('illegal value in %d-th argument of '
                                'internal getrf (lu_factor)' % -info)
    if info > 0:
        warn("Diagonal number %d is exactly zero. Singular matrix." % info,
                    RuntimeWarning)
    return lu, piv 

def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True):
    """Solve an equation system, a x = b, given the LU factorization of a

    Parameters
    ----------
    (lu, piv)
        Factorization of the coefficient matrix a, as given by lu_factor
    b : array
        Right-hand side
    trans : {0, 1, 2}, optional
        Type of system to solve:

        =====  =========
        trans  system
        =====  =========
        0      a x   = b
        1      a^T x = b
        2      a^H x = b
        =====  =========
    overwrite_b : bool, optional
        Whether to overwrite data in b (may increase performance)
    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 : array
        Solution to the system

    See also
    --------
    lu_factor : LU factorize a matrix

    """
    (lu, piv) = lu_and_piv
    if check_finite:
        b1 = asarray_chkfinite(b)
    else:
        b1 = asarray(b)
    overwrite_b = overwrite_b or _datacopied(b1, b)
    if lu.shape[0] != b1.shape[0]:
        raise ValueError("incompatible dimensions.")

    getrs, = get_lapack_funcs(('getrs',), (lu, b1))
    x,info = getrs(lu, piv, b1, trans=trans, overwrite_b=overwrite_b)
    if info == 0:
        return x
    raise ValueError('illegal value in %d-th argument of internal gesv|posv'
                                                                    % -info) 

def expm_frechet_kronform(A, method=None, check_finite=True):
    """
    Construct the Kronecker form of the Frechet derivative of expm.

    Parameters
    ----------
    A : array_like with shape (N, N)
        Matrix to be expm'd.
    method : str, optional
        Extra keyword to be passed to expm_frechet.
    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
    -------
    K : 2d ndarray with shape (N*N, N*N)
        Kronecker form of the Frechet derivative of the matrix exponential.

    Notes
    -----
    This function is used to help compute the condition number
    of the matrix exponential.

    See also
    --------
    expm : Compute a matrix exponential.
    expm_frechet : Compute the Frechet derivative of the matrix exponential.
    expm_cond : Compute the relative condition number of the matrix exponential
                in the Frobenius norm.

    """
    if check_finite:
        A = np.asarray_chkfinite(A)
    else:
        A = np.asarray(A)
    if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
        raise ValueError('expected a square matrix')

    n = A.shape[0]
    ident = np.identity(n)
    cols = []
    for i in range(n):
        for j in range(n):
            E = np.outer(ident[i], ident[j])
            F = expm_frechet(A, E,
                    method=method, compute_expm=False, check_finite=False)
            cols.append(vec(F))
    return np.vstack(cols).T