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File: //usr/local/lib/python3.10/dist-packages/numpy/polynomial/__pycache__/polyutils.cpython-310.pyc
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���g�Z�@s�dZddlZddlZddlZddlZddlmZmZddl	m
Z
gd�ZGdd�de�Z
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d�Zdd�Zdd�Zdd�Zdd�Zdd�Zdd�Zdd�Zdd�Zdd �Zd!d"�Zd#d$�Zd%d&�Zd2d(d)�Zd*d+�Zd,d-�Zd3d.d/�Z dS)4a
Utility classes and functions for the polynomial modules.

This module provides: error and warning objects; a polynomial base class;
and some routines used in both the `polynomial` and `chebyshev` modules.

Warning objects
---------------

.. autosummary::
   :toctree: generated/

   RankWarning  raised in least-squares fit for rank-deficient matrix.

Functions
---------

.. autosummary::
   :toctree: generated/

   as_series    convert list of array_likes into 1-D arrays of common type.
   trimseq      remove trailing zeros.
   trimcoef     remove small trailing coefficients.
   getdomain    return the domain appropriate for a given set of abscissae.
   mapdomain    maps points between domains.
   mapparms     parameters of the linear map between domains.

�N)�dragon4_positional�dragon4_scientific)�absolute)�RankWarning�	as_series�trimseq�trimcoef�	getdomain�	mapdomain�mapparms�format_floatc@seZdZdZdS)rz;Issued by chebfit when the design matrix is rank deficient.N)�__name__�
__module__�__qualname__�__doc__�rr�E/usr/local/lib/python3.10/dist-packages/numpy/polynomial/polyutils.pyr/srcCsJt|�dkr|Stt|�ddd�D]
}||dkrnq|d|d�S)a�Remove small Poly series coefficients.

    Parameters
    ----------
    seq : sequence
        Sequence of Poly series coefficients. This routine fails for
        empty sequences.

    Returns
    -------
    series : sequence
        Subsequence with trailing zeros removed. If the resulting sequence
        would be empty, return the first element. The returned sequence may
        or may not be a view.

    Notes
    -----
    Do not lose the type info if the sequence contains unknown objects.

    r����N)�len�range)�seq�irrrr6s�rTc
sdd�|D�}tdd�|D��dkrtd��tdd�|D��r#td��|r,d	d�|D�}td
d�|D��rig}|D]-}|jt�t�kr_tjt|�t�t�d�}|dd�|dd�<|�|�q9|�|�	��q9|Sztj
|��Wnty�}ztd
�|�d}~ww�fdd�|D�}|S)a
    Return argument as a list of 1-d arrays.

    The returned list contains array(s) of dtype double, complex double, or
    object.  A 1-d argument of shape ``(N,)`` is parsed into ``N`` arrays of
    size one; a 2-d argument of shape ``(M,N)`` is parsed into ``M`` arrays
    of size ``N`` (i.e., is "parsed by row"); and a higher dimensional array
    raises a Value Error if it is not first reshaped into either a 1-d or 2-d
    array.

    Parameters
    ----------
    alist : array_like
        A 1- or 2-d array_like
    trim : boolean, optional
        When True, trailing zeros are removed from the inputs.
        When False, the inputs are passed through intact.

    Returns
    -------
    [a1, a2,...] : list of 1-D arrays
        A copy of the input data as a list of 1-d arrays.

    Raises
    ------
    ValueError
        Raised when `as_series` cannot convert its input to 1-d arrays, or at
        least one of the resulting arrays is empty.

    Examples
    --------
    >>> from numpy.polynomial import polyutils as pu
    >>> a = np.arange(4)
    >>> pu.as_series(a)
    [array([0.]), array([1.]), array([2.]), array([3.])]
    >>> b = np.arange(6).reshape((2,3))
    >>> pu.as_series(b)
    [array([0., 1., 2.]), array([3., 4., 5.])]

    >>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16)))
    [array([1.]), array([0., 1., 2.]), array([0., 1.])]

    >>> pu.as_series([2, [1.1, 0.]])
    [array([2.]), array([1.1])]

    >>> pu.as_series([2, [1.1, 0.]], trim=False)
    [array([2.]), array([1.1, 0. ])]

    cSsg|]
}tj|ddd��qS)rF)�ndmin�copy��np�array��.0�arrr�
<listcomp>��zas_series.<locals>.<listcomp>cSsg|]}|j�qSr)�sizerrrrr!�srzCoefficient array is emptycss�|]}|jdkVqdS)rN)�ndimrrrr�	<genexpr>���zas_series.<locals>.<genexpr>zCoefficient array is not 1-dcSsg|]}t|��qSr)rrrrrr!�scss �|]}|jt�t�kVqdS�N)�dtyper�objectrrrrr%�s��r(Nz&Coefficient arrays have no common typecsg|]
}tj|d�d��qS)T)rr(rrr*rrr!�r")�min�
ValueError�anyr(rr)�emptyr�appendr�common_type�	Exception)�alist�trim�arrays�retr �tmp�err*rrTs02�
��rcCsf|dkrtd��t|g�\}t�t�|�|k�\}t|�dkr'|dd�dS|d|dd���S)a0
    Remove "small" "trailing" coefficients from a polynomial.

    "Small" means "small in absolute value" and is controlled by the
    parameter `tol`; "trailing" means highest order coefficient(s), e.g., in
    ``[0, 1, 1, 0, 0]`` (which represents ``0 + x + x**2 + 0*x**3 + 0*x**4``)
    both the 3-rd and 4-th order coefficients would be "trimmed."

    Parameters
    ----------
    c : array_like
        1-d array of coefficients, ordered from lowest order to highest.
    tol : number, optional
        Trailing (i.e., highest order) elements with absolute value less
        than or equal to `tol` (default value is zero) are removed.

    Returns
    -------
    trimmed : ndarray
        1-d array with trailing zeros removed.  If the resulting series
        would be empty, a series containing a single zero is returned.

    Raises
    ------
    ValueError
        If `tol` < 0

    See Also
    --------
    trimseq

    Examples
    --------
    >>> from numpy.polynomial import polyutils as pu
    >>> pu.trimcoef((0,0,3,0,5,0,0))
    array([0.,  0.,  3.,  0.,  5.])
    >>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed
    array([0.])
    >>> i = complex(0,1) # works for complex
    >>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3)
    array([0.0003+0.j   , 0.001 -0.001j])

    rztol must be non-negativeNrr)r,rr�nonzero�absrr)�c�tol�indrrrr�s,rcCs~t|gdd�\}|jjtjdvr4|j��|j��}}|j��|j��}}t�	t
||�t
||�f�St�	|��|��f�S)a;
    Return a domain suitable for given abscissae.

    Find a domain suitable for a polynomial or Chebyshev series
    defined at the values supplied.

    Parameters
    ----------
    x : array_like
        1-d array of abscissae whose domain will be determined.

    Returns
    -------
    domain : ndarray
        1-d array containing two values.  If the inputs are complex, then
        the two returned points are the lower left and upper right corners
        of the smallest rectangle (aligned with the axes) in the complex
        plane containing the points `x`. If the inputs are real, then the
        two points are the ends of the smallest interval containing the
        points `x`.

    See Also
    --------
    mapparms, mapdomain

    Examples
    --------
    >>> from numpy.polynomial import polyutils as pu
    >>> points = np.arange(4)**2 - 5; points
    array([-5, -4, -1,  4])
    >>> pu.getdomain(points)
    array([-5.,  4.])
    >>> c = np.exp(complex(0,1)*np.pi*np.arange(12)/6) # unit circle
    >>> pu.getdomain(c)
    array([-1.-1.j,  1.+1.j])

    F�r3�Complex)rr(�charr�	typecodes�realr+�max�imagr�complex)�x�rmin�rmax�imin�imaxrrrr	�s&r	cCsT|d|d}|d|d}|d|d|d|d|}||}||fS)a�
    Linear map parameters between domains.

    Return the parameters of the linear map ``offset + scale*x`` that maps
    `old` to `new` such that ``old[i] -> new[i]``, ``i = 0, 1``.

    Parameters
    ----------
    old, new : array_like
        Domains. Each domain must (successfully) convert to a 1-d array
        containing precisely two values.

    Returns
    -------
    offset, scale : scalars
        The map ``L(x) = offset + scale*x`` maps the first domain to the
        second.

    See Also
    --------
    getdomain, mapdomain

    Notes
    -----
    Also works for complex numbers, and thus can be used to calculate the
    parameters required to map any line in the complex plane to any other
    line therein.

    Examples
    --------
    >>> from numpy.polynomial import polyutils as pu
    >>> pu.mapparms((-1,1),(-1,1))
    (0.0, 1.0)
    >>> pu.mapparms((1,-1),(-1,1))
    (-0.0, -1.0)
    >>> i = complex(0,1)
    >>> pu.mapparms((-i,-1),(1,i))
    ((1+1j), (1-0j))

    rrr)�old�new�oldlen�newlen�off�sclrrrrs
)$rcCs$t�|�}t||�\}}|||S)a3
    Apply linear map to input points.

    The linear map ``offset + scale*x`` that maps the domain `old` to
    the domain `new` is applied to the points `x`.

    Parameters
    ----------
    x : array_like
        Points to be mapped. If `x` is a subtype of ndarray the subtype
        will be preserved.
    old, new : array_like
        The two domains that determine the map.  Each must (successfully)
        convert to 1-d arrays containing precisely two values.

    Returns
    -------
    x_out : ndarray
        Array of points of the same shape as `x`, after application of the
        linear map between the two domains.

    See Also
    --------
    getdomain, mapparms

    Notes
    -----
    Effectively, this implements:

    .. math::
        x\_out = new[0] + m(x - old[0])

    where

    .. math::
        m = \frac{new[1]-new[0]}{old[1]-old[0]}

    Examples
    --------
    >>> from numpy.polynomial import polyutils as pu
    >>> old_domain = (-1,1)
    >>> new_domain = (0,2*np.pi)
    >>> x = np.linspace(-1,1,6); x
    array([-1. , -0.6, -0.2,  0.2,  0.6,  1. ])
    >>> x_out = pu.mapdomain(x, old_domain, new_domain); x_out
    array([ 0.        ,  1.25663706,  2.51327412,  3.76991118,  5.02654825, # may vary
            6.28318531])
    >>> x - pu.mapdomain(x_out, new_domain, old_domain)
    array([0., 0., 0., 0., 0., 0.])

    Also works for complex numbers (and thus can be used to map any line in
    the complex plane to any other line therein).

    >>> i = complex(0,1)
    >>> old = (-1 - i, 1 + i)
    >>> new = (-1 + i, 1 - i)
    >>> z = np.linspace(old[0], old[1], 6); z
    array([-1. -1.j , -0.6-0.6j, -0.2-0.2j,  0.2+0.2j,  0.6+0.6j,  1. +1.j ])
    >>> new_z = pu.mapdomain(z, old, new); new_z
    array([-1.0+1.j , -0.6+0.6j, -0.2+0.2j,  0.2-0.2j,  0.6-0.6j,  1.0-1.j ]) # may vary

    )r�
asanyarrayr)rErJrKrNrOrrrr
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?r
cCs tjg|}td�||<t|�Sr')r�newaxis�slice�tuple)rr$�slrrr�
_nth_slicewsrUcs�t����t��krtd��dt�������t��kr(td��dt�������dkr0td��ttjt��dd�d������fd	d
�t��D�}t�tj	|�S)am
    A generalization of the Vandermonde matrix for N dimensions

    The result is built by combining the results of 1d Vandermonde matrices,

    .. math::
        W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{V_k(x_k)[i_0, \ldots, i_M, j_k]}

    where

    .. math::
        N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\
        M &= \texttt{points[k].ndim} \\
        V_k &= \texttt{vander\_fs[k]} \\
        x_k &= \texttt{points[k]} \\
        0 \le j_k &\le \texttt{degrees[k]}

    Expanding the one-dimensional :math:`V_k` functions gives:

    .. math::
        W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{B_{k, j_k}(x_k[i_0, \ldots, i_M])}

    where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along
    dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`.

    Parameters
    ----------
    vander_fs : Sequence[function(array_like, int) -> ndarray]
        The 1d vander function to use for each axis, such as ``polyvander``
    points : Sequence[array_like]
        Arrays of point coordinates, all of the same shape. The dtypes
        will be converted to either float64 or complex128 depending on
        whether any of the elements are complex. Scalars are converted to
        1-D arrays.
        This must be the same length as `vander_fs`.
    degrees : Sequence[int]
        The maximum degree (inclusive) to use for each axis.
        This must be the same length as `vander_fs`.

    Returns
    -------
    vander_nd : ndarray
        An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``.
    z	Expected z" dimensions of sample points, got z dimensions of degrees, got rz9Unable to guess a dtype or shape when no points are givenF)r�c3s4�|]}�|�|�|�dt|��VqdS)).N)rU�rr��degrees�n_dims�points�	vander_fsrrr%�s
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    Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis

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    N)r)rb�reshape�shaper)r\r[rY�vrrr�_vander_nd_flat�srfcs�t|�dkrt�d�St|gdd�\}|���fdd�|D��t��}|dkrQt|d�\�}���fdd�t��D�}|rI�|d�d	�|d<|��}|dks(�dS)
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    Helper function used to implement the ``<type>fromroots`` functions.

    Parameters
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    line_f : function(float, float) -> ndarray
        The ``<type>line`` function, such as ``polyline``
    mul_f : function(array_like, array_like) -> ndarray
        The ``<type>mul`` function, such as ``polymul``
    roots
        See the ``<type>fromroots`` functions for more detail
    rrFr=csg|]}�|d��qS)rr)r�r)�line_frrr!�sz_fromroots.<locals>.<listcomp>�cs"g|]
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    Helper function used to implement the ``<type>val<n>d`` functions.

    Parameters
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    val_f : function(array_like, array_like, tensor: bool) -> array_like
        The ``<type>val`` function, such as ``polyval``
    c, args
        See the ``<type>val<n>d`` functions for more detail
    cSsg|]}t�|��qSr)rrPrrrrr!�sz_valnd.<locals>.<listcomp>rc3s�|]}|j�kVqdSr')rdr��shape0rrr%�r&z_valnd.<locals>.<genexpr>rN�zx, y, z are incompatiblerizx, y are incompatiblezordinates are incompatibleF)�tensor)rd�allrr,�iter�next)�val_fr:�args�it�x0�xirrsr�_valnd�s

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    Helper function used to implement the ``<type>grid<n>d`` functions.

    Parameters
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    val_f : function(array_like, array_like, tensor: bool) -> array_like
        The ``<type>val`` function, such as ``polyval``
    c, args
        See the ``<type>grid<n>d`` functions for more detail
    r)rzr:r{r~rrr�_gridndsr�c
Cs�t||g�\}}|ddkrt��t|�}t|�}||kr'|dd�d|fS|dkr9||d|dd�dfStj||d|jd�}|}t||dd�D]'}|dg|dg|�}|d|d}	|dd�|	|dd�}|	||<qO|t|�fS)a�
    Helper function used to implement the ``<type>div`` functions.

    Implementation uses repeated subtraction of c2 multiplied by the nth basis.
    For some polynomial types, a more efficient approach may be possible.

    Parameters
    ----------
    mul_f : function(array_like, array_like) -> array_like
        The ``<type>mul`` function, such as ``polymul``
    c1, c2
        See the ``<type>div`` functions for more detail
    rrNrr*)r�ZeroDivisionErrorrrr.r(rr)
rk�c1�c2�lc1�lc2�quo�remrrl�qrrr�_divs"
r�cCsdt||g�\}}t|�t|�kr!|d|j�|7<|}t|�S|d|j�|7<|}t|�S)z@ Helper function used to implement the ``<type>add`` functions. N�rrr#r�r�r�r5rrr�_add<s�r�cCsjt||g�\}}t|�t|�kr!|d|j�|8<|}t|�S|}|d|j�|7<|}t|�S)z@ Helper function used to implement the ``<type>sub`` functions. Nr�r�rrr�_subIs�r�FcCs�t�|�d}t�|�d}t�|�}|jdks#|jjdvs#|jdkr'td��|��dkr1td��|jdkr:td��|jdkrCtd��|jdksM|jd	krQtd
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|�}|d}t	|�}|||�d
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    Helper function used to implement the ``<type>fit`` functions.

    Parameters
    ----------
    vander_f : function(array_like, int) -> ndarray
        The 1d vander function, such as ``polyvander``
    c1, c2
        See the ``<type>fit`` functions for more detail
    rVr�iurz0deg must be an int or non-empty 1-D array of intzexpected deg >= 0zexpected 1D vector for xzexpected non-empty vector for xrizexpected 1D or 2D array for yz$expected x and y to have same lengthrNzexpected 1D vector for wz$expected x and w to have same lengthr*z!The fit may be poorly conditioned��
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&  r�cCs�t|g�\}t|�}||ks|dkrtd��|dur"||kr"td��|dkr/tjdg|jd�S|dkr5|S|}td|d�D]}|||�}q>|S)af
    Helper function used to implement the ``<type>pow`` functions.

    Parameters
    ----------
    mul_f : function(array_like, array_like) -> ndarray
        The ``<type>mul`` function, such as ``polymul``
    c : array_like
        1-D array of array of series coefficients
    pow, maxpower
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    rz%Power must be a non-negative integer.NzPower is too largerr*ri)r�intr,rrr(r)rkr:�pow�maxpower�power�prdrrrr�_pow�sr�cCs�zt�|�WStyB}z0zt|�}Wn	tyYnw||kr6tjd|�d�tdd�|WYd}~St|�d��|�d}~ww)a�
    Like `operator.index`, but emits a deprecation warning when passed a float

    Parameters
    ----------
    x : int-like, or float with integral value
        Value to interpret as an integer
    desc : str
        description to include in any error message

    Raises
    ------
    TypeError : if x is a non-integral float or non-numeric
    DeprecationWarning : if x is an integral float
    z)In future, this will raise TypeError, as z7 will need to be an integer not just an integral float.rur�Nz must be an integer)r_�indexr�r�r�r��DeprecationWarning)rE�descr7�ixrrr�_deprecate_as_int�s$�
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