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B �5�g[V � @ s� d Z ddlZddlZddlZddlZddddddd gZG d d� de�Zd d� Z d+d d�Z d,dd�Z dd� Z dd � Z dd� Zdd� Zdd� Zdd� Zdd� Zdd� Zd d � Zd d � Zd d!� Zd"d#� Zd-d%d&�Zd'd(� Zd)d*� ZdS ).a 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� RankWarning� as_series�trimseq�trimcoef� getdomain� mapdomain�mapparmsc @ s e Zd ZdZdS )r z;Issued by chebfit when the design matrix is rank deficient.N)�__name__� __module__� __qualname__�__doc__� r r �P/home/digitalm-up/venv/lib/python3.7/site-packages/numpy/polynomial/polyutils.pyr + s c C sP t | �dkr| S x*tt | �d dd�D ]}| | dkr&P q&W | d|d � S dS )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�ir r r r 2 s Tc s( dd� | D �}t dd� |D ��dkr,td��tdd� |D ��rFtd��|rXd d� |D �}td d� |D ��r�g }x�|D ]Z}|jt�t�kr�tjt|�t�t�d �}|d d � |d d �< |�|� qt|�|� � � qtW nPytj |� � W n. t k �r } ztd �|�W d d }~X Y nX � 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. ])] c S s g | ]}t j|d dd��qS )r F)Zndmin�copy)�np�array)�.0�ar r r � <listcomp>� s z as_series.<locals>.<listcomp>c S s g | ] }|j �qS r )�size)r r r r r r � s r zCoefficient array is emptyc s s | ]}|j d kV qdS )r N)�ndim)r r r r r � <genexpr>� s z as_series.<locals>.<genexpr>z Coefficient array is not 1-dc S s g | ] }t |��qS r )r )r r r r r r � s c s s | ]}|j t� t�kV qd S )N)�dtyper �object)r r r r r r � s )r Nz&Coefficient arrays have no common typec s g | ]}t j|d � d��qS )T)r r )r r )r r )r r r r � s ) �min� ValueError�anyr r r �emptyr �appendr Z common_type� Exception)�alist�trimZarrays�retr �tmp�er )r r r P s* 2 c C sj |dk rt d��t| g�\} t�t�| �|k�\}t|�dkrN| dd� d S | d|d d � �� S dS )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]) r ztol must be non-negativeNr r )r! r r Znonzero�absr r )�cZtol�indr r r r � s , c C s� t | gdd�\} | jjtjd krh| j�� | j�� }}| j�� | j�� }}t� t ||�t ||�f�S t� | �� | �� f�S dS )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)r' �ComplexN) r r �charr � typecodes�realr �max�imagr �complex)�xZrminZrmaxZiminZimaxr r r r � s &c C sT | 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)) r r r )�old�newZoldlenZnewlen�off�sclr r r r s )$c C s$ t �| �} t||�\}}||| S )a5 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 )r5 r6 r7 r8 r9 r r r r / s ? c C s t jg| }td �|| < t|�S )N)r Znewaxis�slice�tuple)r r �slr r r � _nth_slices s r> c s� t ����t ��kr,td�� dt ��� ����t � �krPtd�� dt � �� ����dkr`td��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 r z9Unable to guess a dtype or shape when no points are givenF)r g c 3 s2 | ]*}�| �| � | �d t |�� V qdS )).N)r> )r r )�degrees�n_dims�points� vander_fsr r r � s z _vander_nd.<locals>.<genexpr>) r r! r< r r r � functools�reduce�operator�mul)rB rA r? Z vander_arraysr )r? r@ rA rB r � _vander_ndy s - rG c C s* t | ||�}|�|jdt|� � d �S )z� Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis Used to implement the public ``<type>vander<n>d`` functions. N)r )rG Zreshape�shaper )rB rA r? �vr r r �_vander_nd_flat� s rJ c s� t |�dkrt�d�S t|gdd�\}|�� � fdd�|D ��t ��}xV|dkr�t|d�\�}���fdd�t��D �}|r��|d �d �|d< |��}qJW �d S d S ) a� Helper function used to implement the ``<type>fromroots`` functions. Parameters ---------- 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 r r F)r' c s g | ]}� | d ��qS )r r )r �r)�line_fr r r � s z _fromroots.<locals>.<listcomp>� c s" g | ]}��| �|� ��qS r r )r r )�m�mul_f�pr r r � s r N)r r Zonesr �sort�divmodr )rL rO �roots�nrK r) r )rL rN rO rP r � _fromroots� s rU c s� dd� |D �}|d j � t� fdd�|dd� D ��sjt|�dkrLtd ��n t|�d krbtd ��ntd ��t|�}t|�}| ||�}x|D ]}| ||d d�}q�W |S )a4 Helper function used to implement the ``<type>val<n>d`` functions. Parameters ---------- 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 c S s g | ]}t �|��qS r )r r: )r r r r r r � s z_valnd.<locals>.<listcomp>r c 3 s | ]}|j � kV qd S )N)rH )r r )�shape0r r r � s z_valnd.<locals>.<genexpr>r N� zx, y, z are incompatiblerM zx, y are incompatiblezordinates are incompatibleF)Ztensor)rH �allr r! �iter�next)�val_fr, �args�itZx0�xir )rV r �_valnd� s r_ c G s x|D ]}| ||�}qW |S )a6 Helper function used to implement the ``<type>grid<n>d`` functions. Parameters ---------- 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 )r[ r, r\ r^ r r r �_gridnd s r` c C s t ||g�\}}|d dkr"t� �t|�}t|�}||k rN|dd� d |fS |dkrr||d |dd� d fS tj|| d |jd�}|}xbt|| dd�D ]N}| dg| dg |�}|d |d } |dd� | |dd� }| ||< q�W |t|�fS dS )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 r r Nr )r )r �ZeroDivisionErrorr r r# r r r ) rO �c1�c2Zlc1Zlc2Zquo�remr rP �qr r r �_div s" rf c C s^ t | |g�\} }t| �t|�kr<| d|j� |7 < | }n|d| j� | 7 < |}t|�S )z@ Helper function used to implement the ``<type>add`` functions. N)r r r r )rb rc r( r r r �_add8 s rg c C sd t | |g�\} }t| �t|�kr<| d|j� |8 < | }n | }|d| j� | 7 < |}t|�S )z@ Helper function used to implement the ``<type>sub`` functions. N)r r r r )rb rc r( r r r �_subE s rh Fc C s� t �|�d }t �|�d }t �|�}|jdksF|jjdksF|jdkrNtd��|�� dk rbtd��|jdkrttd��|jdkr�td��|jdk s�|jd kr�td ��t |�t |�kr�td ��|jdkr�|}|d }| ||�} n0t � |�}|d }t |�}| ||�d d �|f } | j } |j } |d k �rnt �|�d }|jdk�rDtd��t |�t |�k�r^td��| | } | | } |d k�r�t |�t � |j�j }t| jjt j��r�t �t �| j�t �| j� �d��} nt �t �| ��d��} d| | dk< t j�| j | | j |�\} }}}| j | j } |jdk�rl| jd k�rLt j|d | jd f| jd�}nt j|d | jd�}| ||< |} ||k�r�|�s�d}tj|t d d� |�r�| ||||gfS | S d S )a 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 g r �iur z0deg must be an int or non-empty 1-D array of intzexpected deg >= 0zexpected 1D vector for xz expected non-empty vector for xrM z expected 1D or 2D array for yz$expected x and y to have same lengthr Nzexpected 1D vector for wz$expected x and w to have same length)r z!The fit may be poorly conditioned)� stacklevel) r Zasarrayr r �kindr � TypeErrorr r! r rQ �TZfinfoZeps� issubclass�typeZcomplexfloating�sqrtZsquarer1 r3 �sumZlinalgZlstsqZzerosrH �warnings�warnr )Zvander_fr5 �y�degZrcond�full�wZlmax�orderZvan�lhs�rhsr9 r, ZresidsZrank�s�cc�msgr r r �_fitS sj & r~ c C s� t |g�\}t|�}||ks$|dk r.td��nj|dk rH||krHtd��nP|dkrbtjdg|jd�S |dkrn|S |}x td|d �D ]}| ||�}q�W |S dS )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 See the ``<type>pow`` functions for more detail r z%Power must be a non-negative integer.NzPower is too larger )r rM )r �intr! r r r r )rO r, �powZmaxpower�powerZprdr r r r �_pow� s r� c C s� y t �| �S tk r� } zZy t| �}W n tk r< Y n&X || krbtjd|� d�tdd� |S t|� d��|�W dd}~X Y nX dS )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.rW )rj z must be an integerN)rE �indexrl r rr rs �DeprecationWarning)r5 �descr* Zixr r r �_deprecate_as_int� s r� )T)r )NFN)r rE rC rr �numpyr �__all__� UserWarningr r r r r r r r> rG rJ rU r_ r` rf rg rh r~ r� r� r r r r �<module> s2 L 6./DE $ X!
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