Python core.atleast_1d函数代码示例

def polyadd(a1, a2):
    """
    Find the sum of two polynomials.

    Returns the polynomial resulting from the sum of two input polynomials.
    Each input must be either a poly1d object or a 1D sequence of polynomial
    coefficients, from highest to lowest degree.

    Parameters
    ----------
    a1, a2 : array_like or poly1d object
        Input polynomials.

    Returns
    -------
    out : ndarray or poly1d object
        The sum of the inputs. If either input is a poly1d object, then the
        output is also a poly1d object. Otherwise, it is a 1D array of
        polynomial coefficients from highest to lowest degree.

    See Also
    --------
    poly1d : A one-dimensional polynomial class.
    poly, polyadd, polyder, polydiv, polyfit, polyint, polysub, polyval

    Examples
    --------
    >>> np.polyadd([1, 2], [9, 5, 4])
    array([9, 6, 6])

    Using poly1d objects:

    >>> p1 = np.poly1d([1, 2])
    >>> p2 = np.poly1d([9, 5, 4])
    >>> print p1
    1 x + 2
    >>> print p2
       2
    9 x + 5 x + 4
    >>> print np.polyadd(p1, p2)
       2
    9 x + 6 x + 6

    """
    truepoly = (isinstance(a1, poly1d) or isinstance(a2, poly1d))
    a1 = atleast_1d(a1)
    a2 = atleast_1d(a2)
    diff = len(a2) - len(a1)
    if diff == 0:
        val = a1 + a2
    elif diff > 0:
        zr = NX.zeros(diff, a1.dtype)
        val = NX.concatenate((zr, a1)) + a2
    else:
        zr = NX.zeros(abs(diff), a2.dtype)
        val = a1 + NX.concatenate((zr, a2))
    if truepoly:
        val = poly1d(val)
    return val

 def test_r1array(self):
     """ Test to make sure equivalent Travis O's r1array function
     """
     assert_(atleast_1d(3).shape == (1,))
     assert_(atleast_1d(3j).shape == (1,))
     assert_(atleast_1d(long(3)).shape == (1,))
     assert_(atleast_1d(3.0).shape == (1,))
     assert_(atleast_1d([[2, 3], [4, 5]]).shape == (2, 2))

 def test_3D_array(self):
     a = array([[1, 2], [1, 2]])
     b = array([[2, 3], [2, 3]])
     a = array([a, a])
     b = array([b, b])
     res = [atleast_1d(a), atleast_1d(b)]
     desired = [a, b]
     assert_array_equal(res, desired)

def polydiv(u, v):
    """
    Returns the quotient and remainder of polynomial division.

    The input arrays specify the polynomial terms in turn with a length equal
    to the polynomial degree plus 1.

    Parameters
    ----------
    u : {array_like, poly1d}
        Dividend polynomial.
    v : {array_like, poly1d}
        Divisor polynomial.

    Returns
    -------
    q : ndarray
        Polynomial terms of quotient.
    r : ndarray
        Remainder of polynomial division.

    See Also
    --------
    poly, polyadd, polyder, polydiv, polyfit, polyint, polymul, polysub,
    polyval

    Examples
    --------
    .. math:: \\frac{3x^2 + 5x + 2}{2x + 1} = 1.5x + 1.75, remainder 0.25

    >>> x = np.array([3.0, 5.0, 2.0])
    >>> y = np.array([2.0, 1.0])
    >>> np.polydiv(x, y)
    >>> (array([ 1.5 ,  1.75]), array([ 0.25]))

    """
    truepoly = (isinstance(u, poly1d) or isinstance(u, poly1d))
    u = atleast_1d(u) + 0.0
    v = atleast_1d(v) + 0.0
    # w has the common type
    w = u[0] + v[0]
    m = len(u) - 1
    n = len(v) - 1
    scale = 1. / v[0]
    q = NX.zeros((max(m - n + 1, 1),), w.dtype)
    r = u.copy()
    for k in range(0, m-n+1):
        d = scale * r[k]
        q[k] = d
        r[k:k+n+1] -= d*v
    while NX.allclose(r[0], 0, rtol=1e-14) and (r.shape[-1] > 1):
        r = r[1:]
    if truepoly:
        return poly1d(q), poly1d(r)
    return q, r

    def __init__(self, c_or_r, r=False, variable=None):
        if isinstance(c_or_r, poly1d):
            self._variable = c_or_r._variable
            self._coeffs = c_or_r._coeffs

            if set(c_or_r.__dict__) - set(self.__dict__):
                msg = ("In the future extra properties will not be copied "
                       "across when constructing one poly1d from another")
                warnings.warn(msg, FutureWarning, stacklevel=2)
                self.__dict__.update(c_or_r.__dict__)

            if variable is not None:
                self._variable = variable
            return
        if r:
            c_or_r = poly(c_or_r)
        c_or_r = atleast_1d(c_or_r)
        if c_or_r.ndim > 1:
            raise ValueError("Polynomial must be 1d only.")
        c_or_r = trim_zeros(c_or_r, trim='f')
        if len(c_or_r) == 0:
            c_or_r = NX.array([0.])
        self._coeffs = c_or_r
        if variable is None:
            variable = 'x'
        self._variable = variable

def polysub(a1, a2):
    """
    Returns difference from subtraction of two polynomials input as sequences.

    Returns difference of polynomials; `a1` - `a2`.  Input polynomials are
    represented as an array_like sequence of terms or a poly1d object.

    Parameters
    ----------
    a1 : {array_like, poly1d}
        Minuend polynomial as sequence of terms.
    a2 : {array_like, poly1d}
        Subtrahend polynomial as sequence of terms.

    Returns
    -------
    out : {ndarray, poly1d}
        Array representing the polynomial terms.

    See Also
    --------
    polyval, polydiv, polymul, polyadd

    Examples
    --------
    .. math:: (2 x^2 + 10 x - 2) - (3 x^2 + 10 x -4) = (-x^2 + 2)

    >>> np.polysub([2, 10, -2], [3, 10, -4])
    array([-1,  0,  2])

    """
    truepoly = (isinstance(a1, poly1d) or isinstance(a2, poly1d))
    a1 = atleast_1d(a1)
    a2 = atleast_1d(a2)
    diff = len(a2) - len(a1)
    if diff == 0:
        val = a1 - a2
    elif diff > 0:
        zr = NX.zeros(diff, a1.dtype)
        val = NX.concatenate((zr, a1)) - a2
    else:
        zr = NX.zeros(abs(diff), a2.dtype)
        val = a1 - NX.concatenate((zr, a2))
    if truepoly:
        val = poly1d(val)
    return val

def polysub(a1, a2):
    """
    Difference (subtraction) of two polynomials.

    Given two polynomials `a1` and `a2`, returns ``a1 - a2``.
    `a1` and `a2` can be either array_like sequences of the polynomials'
    coefficients (including coefficients equal to zero), or `poly1d` objects.

    Parameters
    ----------
    a1, a2 : array_like or poly1d
        Minuend and subtrahend polynomials, respectively.

    Returns
    -------
    out : ndarray or poly1d
        Array or `poly1d` object of the difference polynomial's coefficients.

    See Also
    --------
    polyval, polydiv, polymul, polyadd

    Examples
    --------
    .. math:: (2 x^2 + 10 x - 2) - (3 x^2 + 10 x -4) = (-x^2 + 2)

    >>> np.polysub([2, 10, -2], [3, 10, -4])
    array([-1,  0,  2])

    """
    truepoly = (isinstance(a1, poly1d) or isinstance(a2, poly1d))
    a1 = atleast_1d(a1)
    a2 = atleast_1d(a2)
    diff = len(a2) - len(a1)
    if diff == 0:
        val = a1 - a2
    elif diff > 0:
        zr = NX.zeros(diff, a1.dtype)
        val = NX.concatenate((zr, a1)) - a2
    else:
        zr = NX.zeros(abs(diff), a2.dtype)
        val = a1 - NX.concatenate((zr, a2))
    if truepoly:
        val = poly1d(val)
    return val

def polyadd(a1, a2):
    """
    Returns sum of two polynomials.

    Returns sum of polynomials; `a1` + `a2`.  Input polynomials are
    represented as an array_like sequence of terms or a poly1d object.

    Parameters
    ----------
    a1 : {array_like, poly1d}
        Polynomial as sequence of terms.
    a2 : {array_like, poly1d}
        Polynomial as sequence of terms.

    Returns
    -------
    out : {ndarray, poly1d}
        Array representing the polynomial terms.

    See Also
    --------
    polyval, polydiv, polymul, polyadd

    """
    truepoly = (isinstance(a1, poly1d) or isinstance(a2, poly1d))
    a1 = atleast_1d(a1)
    a2 = atleast_1d(a2)
    diff = len(a2) - len(a1)
    if diff == 0:
        val = a1 + a2
    elif diff > 0:
        zr = NX.zeros(diff, a1.dtype)
        val = NX.concatenate((zr, a1)) + a2
    else:
        zr = NX.zeros(abs(diff), a2.dtype)
        val = a1 + NX.concatenate((zr, a2))
    if truepoly:
        val = poly1d(val)
    return val

 def __init__(self, c_or_r, r=0, variable=None):
     if isinstance(c_or_r, poly1d):
         for key in c_or_r.__dict__.keys():
             self.__dict__[key] = c_or_r.__dict__[key]
         if variable is not None:
             self.__dict__['variable'] = variable
         return
     if r:
         c_or_r = poly(c_or_r)
     c_or_r = atleast_1d(c_or_r)
     if len(c_or_r.shape) > 1:
         raise ValueError("Polynomial must be 1d only.")
     c_or_r = trim_zeros(c_or_r, trim='f')
     if len(c_or_r) == 0:
         c_or_r = NX.array([0.])
     self.__dict__['coeffs'] = c_or_r
     self.__dict__['order'] = len(c_or_r) - 1
     if variable is None:
         variable = 'x'
     self.__dict__['variable'] = variable

def poly(seq_of_zeros):
    """
    Find the coefficients of a polynomial with the given sequence of roots.

    Returns the coefficients of the polynomial whose leading coefficient
    is one for the given sequence of zeros (multiple roots must be included
    in the sequence as many times as their multiplicity; see Examples).
    A square matrix (or array, which will be treated as a matrix) can also
    be given, in which case the coefficients of the characteristic polynomial
    of the matrix are returned.

    Parameters
    ----------
    seq_of_zeros : array_like, shape (N,) or (N, N)
        A sequence of polynomial roots, or a square array or matrix object.

    Returns
    -------
    c : ndarray
        1D array of polynomial coefficients from highest to lowest degree:

        ``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]``
        where c[0] always equals 1.

    Raises
    ------
    ValueError
        If input is the wrong shape (the input must be a 1-D or square
        2-D array).

    See Also
    --------
    polyval : Evaluate a polynomial at a point.
    roots : Return the roots of a polynomial.
    polyfit : Least squares polynomial fit.
    poly1d : A one-dimensional polynomial class.

    Notes
    -----
    Specifying the roots of a polynomial still leaves one degree of
    freedom, typically represented by an undetermined leading
    coefficient. [1]_ In the case of this function, that coefficient -
    the first one in the returned array - is always taken as one. (If
    for some reason you have one other point, the only automatic way
    presently to leverage that information is to use ``polyfit``.)

    The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n`
    matrix **A** is given by

        :math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`,

    where **I** is the `n`-by-`n` identity matrix. [2]_

    References
    ----------
    .. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry,
       Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996.

    .. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition,"
       Academic Press, pg. 182, 1980.

    Examples
    --------
    Given a sequence of a polynomial's zeros:

    >>> np.poly((0, 0, 0)) # Multiple root example
    array([1, 0, 0, 0])

    The line above represents z**3 + 0*z**2 + 0*z + 0.

    >>> np.poly((-1./2, 0, 1./2))
    array([ 1.  ,  0.  , -0.25,  0.  ])

    The line above represents z**3 - z/4

    >>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0]))
    array([ 1.        , -0.77086955,  0.08618131,  0.        ]) #random

    Given a square array object:

    >>> P = np.array([[0, 1./3], [-1./2, 0]])
    >>> np.poly(P)
    array([ 1.        ,  0.        ,  0.16666667])

    Or a square matrix object:

    >>> np.poly(np.matrix(P))
    array([ 1.        ,  0.        ,  0.16666667])

    Note how in all cases the leading coefficient is always 1.

    """
    seq_of_zeros = atleast_1d(seq_of_zeros)
    sh = seq_of_zeros.shape
    if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0:
        seq_of_zeros = eigvals(seq_of_zeros)
    elif len(sh) == 1:
        pass
    else:
        raise ValueError("input must be 1d or square 2d array.")
#.........这里部分代码省略.........

def polyint(p, m=1, k=None):
    """
    Return an antiderivative (indefinite integral) of a polynomial.

    The returned order `m` antiderivative `P` of polynomial `p` satisfies
    :math:`\\frac{d^m}{dx^m}P(x) = p(x)` and is defined up to `m - 1`
    integration constants `k`. The constants determine the low-order
    polynomial part

    .. math:: \\frac{k_{m-1}}{0!} x^0 + \\ldots + \\frac{k_0}{(m-1)!}x^{m-1}

    of `P` so that :math:`P^{(j)}(0) = k_{m-j-1}`.

    Parameters
    ----------
    p : {array_like, poly1d}
        Polynomial to differentiate.
        A sequence is interpreted as polynomial coefficients, see `poly1d`.
    m : int, optional
        Order of the antiderivative. (Default: 1)
    k : {None, list of `m` scalars, scalar}, optional
        Integration constants. They are given in the order of integration:
        those corresponding to highest-order terms come first.

        If ``None`` (default), all constants are assumed to be zero.
        If `m = 1`, a single scalar can be given instead of a list.

    See Also
    --------
    polyder : derivative of a polynomial
    poly1d.integ : equivalent method

    Examples
    --------
    The defining property of the antiderivative:

    >>> p = np.poly1d([1,1,1])
    >>> P = np.polyint(p)
    >>> P
    poly1d([ 0.33333333,  0.5       ,  1.        ,  0.        ])
    >>> np.polyder(P) == p
    True

    The integration constants default to zero, but can be specified:

    >>> P = np.polyint(p, 3)
    >>> P(0)
    0.0
    >>> np.polyder(P)(0)
    0.0
    >>> np.polyder(P, 2)(0)
    0.0
    >>> P = np.polyint(p, 3, k=[6,5,3])
    >>> P
    poly1d([ 0.01666667,  0.04166667,  0.16666667,  3. ,  5. ,  3. ])

    Note that 3 = 6 / 2!, and that the constants are given in the order of
    integrations. Constant of the highest-order polynomial term comes first:

    >>> np.polyder(P, 2)(0)
    6.0
    >>> np.polyder(P, 1)(0)
    5.0
    >>> P(0)
    3.0

    """
    m = int(m)
    if m < 0:
        raise ValueError("Order of integral must be positive (see polyder)")
    if k is None:
        k = NX.zeros(m, float)
    k = atleast_1d(k)
    if len(k) == 1 and m > 1:
        k = k[0]*NX.ones(m, float)
    if len(k) < m:
        raise ValueError(
              "k must be a scalar or a rank-1 array of length 1 or >m.")

    truepoly = isinstance(p, poly1d)
    p = NX.asarray(p)
    if m == 0:
        if truepoly:
            return poly1d(p)
        return p
    else:
        # Note: this must work also with object and integer arrays
        y = NX.concatenate((p.__truediv__(NX.arange(len(p), 0, -1)), [k[0]]))
        val = polyint(y, m - 1, k=k[1:])
        if truepoly:
            return poly1d(val)
        return val

def roots(p):
    """
    Return the roots of a polynomial with coefficients given in p.

    The values in the rank-1 array `p` are coefficients of a polynomial.
    If the length of `p` is n+1 then the polynomial is described by::

      p[0] * x**n + p[1] * x**(n-1) + ... + p[n-1]*x + p[n]

    Parameters
    ----------
    p : array_like
        Rank-1 array of polynomial coefficients.

    Returns
    -------
    out : ndarray
        An array containing the complex roots of the polynomial.

    Raises
    ------
    ValueError :
        When `p` cannot be converted to a rank-1 array.

    See also
    --------
    poly : Find the coefficients of a polynomial with a given sequence
           of roots.
    polyval : Evaluate a polynomial at a point.
    polyfit : Least squares polynomial fit.
    poly1d : A one-dimensional polynomial class.

    Notes
    -----
    The algorithm relies on computing the eigenvalues of the
    companion matrix [1]_.

    References
    ----------
    .. [1] R. A. Horn & C. R. Johnson, *Matrix Analysis*.  Cambridge, UK:
        Cambridge University Press, 1999, pp. 146-7.

    Examples
    --------
    >>> coeff = [3.2, 2, 1]
    >>> np.roots(coeff)
    array([-0.3125+0.46351241j, -0.3125-0.46351241j])

    """
    # If input is scalar, this makes it an array
    p = atleast_1d(p)
    if len(p.shape) != 1:
        raise ValueError("Input must be a rank-1 array.")

    # find non-zero array entries
    non_zero = NX.nonzero(NX.ravel(p))[0]

    # Return an empty array if polynomial is all zeros
    if len(non_zero) == 0:
        return NX.array([])

    # find the number of trailing zeros -- this is the number of roots at 0.
    trailing_zeros = len(p) - non_zero[-1] - 1

    # strip leading and trailing zeros
    p = p[int(non_zero[0]):int(non_zero[-1])+1]

    # casting: if incoming array isn't floating point, make it floating point.
    if not issubclass(p.dtype.type, (NX.floating, NX.complexfloating)):
        p = p.astype(float)

    N = len(p)
    if N > 1:
        # build companion matrix and find its eigenvalues (the roots)
        A = diag(NX.ones((N-2,), p.dtype), -1)
        A[0, :] = -p[1:] / p[0]
        roots = eigvals(A)
    else:
        roots = NX.array([])

    # tack any zeros onto the back of the array
    roots = hstack((roots, NX.zeros(trailing_zeros, roots.dtype)))
    return roots

 def test_1D_array(self):
     a = array([1, 2])
     b = array([2, 3])
     res = [atleast_1d(a), atleast_1d(b)]
     desired = [array([1, 2]), array([2, 3])]
     assert_array_equal(res, desired)

 def test_0D_array(self):
     a = array(1)
     b = array(2)
     res = [atleast_1d(a), atleast_1d(b)]
     desired = [array([1]), array([2])]
     assert_array_equal(res, desired)

def roots(p):
    """
    Return the roots of a polynomial with coefficients given in p.

    The values in the rank-1 array `p` are coefficients of a polynomial.
    If the length of `p` is n+1 then the polynomial is described by
    p[0] * x**n + p[1] * x**(n-1) + ... + p[n-1]*x + p[n]

    Parameters
    ----------
    p : array_like of shape(M,)
        Rank-1 array of polynomial co-efficients.

    Returns
    -------
    out : ndarray
        An array containing the complex roots of the polynomial.

    Raises
    ------
    ValueError:
        When `p` cannot be converted to a rank-1 array.

    Examples
    --------

    >>> coeff = [3.2, 2, 1]
    >>> print np.roots(coeff)
    [-0.3125+0.46351241j -0.3125-0.46351241j]

    """
    # If input is scalar, this makes it an array
    p = atleast_1d(p)
    if len(p.shape) != 1:
        raise ValueError,"Input must be a rank-1 array."

    # find non-zero array entries
    non_zero = NX.nonzero(NX.ravel(p))[0]

    # Return an empty array if polynomial is all zeros
    if len(non_zero) == 0:
        return NX.array([])

    # find the number of trailing zeros -- this is the number of roots at 0.
    trailing_zeros = len(p) - non_zero[-1] - 1

    # strip leading and trailing zeros
    p = p[int(non_zero[0]):int(non_zero[-1])+1]

    # casting: if incoming array isn't floating point, make it floating point.
    if not issubclass(p.dtype.type, (NX.floating, NX.complexfloating)):
        p = p.astype(float)

    N = len(p)
    if N > 1:
        # build companion matrix and find its eigenvalues (the roots)
        A = diag(NX.ones((N-2,), p.dtype), -1)
        A[0, :] = -p[1:] / p[0]
        roots = eigvals(A)
    else:
        roots = NX.array([])

    # tack any zeros onto the back of the array
    roots = hstack((roots, NX.zeros(trailing_zeros, roots.dtype)))
    return roots