U
|
P ±dà ã @ sL d Z ddlZddlZddlZddlZddlmZ dgZ G dd dej
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Z dS )a
|
Abstract base class for the various polynomial Classes.
|
|
The ABCPolyBase class provides the methods needed to implement the common API
|
for the various polynomial classes. It operates as a mixin, but uses the
|
abc module from the stdlib, hence it is only available for Python >= 2.6.
|
|
é Né )Ú polyutilsÚ ABCPolyBasec @ s e Zd ZdZdZdZdZe dddddd d
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d d d d
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¡Z e ddddddddddd
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¡Z
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e j dk Z edd Zeejd d Zeejd d Zeejd d! Zeejd"d# Zeejd$d% Zeejd&d' Zeejd(d) Zeejd®d*d+Zeejd,d- Zeejd.d/ Z eejd0d1 Z eejd2d3 Z eejd4d5 Z eejd6d7 Z eejd8d9 Z!d:d; Z"d<d= Z#d>d? Z$d@dA Z%dBdC Z&d¯dEdFZ'dGdH Z(dIdJ Z)dKdL Z*dMdN Z+e,dOdP Z-e,dQdR Z.e,dSdT Z/ed°dVdWZ0dXdY Z1dZd[ Z2d\d] Z3d^d_ Z4d`da Z5dbdc Z6ddde Z7dfdg Z8dhdi Z9djdk Z:dldm Z;dndo Z<dpdq Z=drds Z>dtdu Z?dvdw Z@dxdy ZAdzd{ ZBd|d} ZCd~d ZDdd ZEdd ZFdd
ZGdd ZHdd ZIdd ZJdd ZKdd ZLdd ZMd±ddZNdd ZOd²ddZPdd ZQdg dfddZRd³ddZSd d¡ ZTd´d¢d£ZUe,dµd¤d¥ZVe,g ddDfd¦d§ZWe,d¶d¨d©ZXe,d·dªd«ZYe,d¸d¬dZZdS )¹r a An abstract base class for immutable series classes.
|
|
ABCPolyBase provides the standard Python numerical methods
|
'+', '-', '*', '//', '%', 'divmod', '**', and '()' along with the
|
methods listed below.
|
|
.. versionadded:: 1.9.0
|
|
Parameters
|
----------
|
coef : array_like
|
Series coefficients in order of increasing degree, i.e.,
|
``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``, where
|
``P_i`` is the basis polynomials of degree ``i``.
|
domain : (2,) array_like, optional
|
Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
|
to the interval ``[window[0], window[1]]`` by shifting and scaling.
|
The default value is the derived class domain.
|
window : (2,) array_like, optional
|
Window, see domain for its use. The default value is the
|
derived class window.
|
symbol : str, optional
|
Symbol used to represent the independent variable in string
|
representations of the polynomial expression, e.g. for printing.
|
The symbol must be a valid Python identifier. Default value is 'x'.
|
|
.. versionadded:: 1.24
|
|
Attributes
|
----------
|
coef : (N,) ndarray
|
Series coefficients in order of increasing degree.
|
domain : (2,) ndarray
|
Domain that is mapped to window.
|
window : (2,) ndarray
|
Window that domain is mapped to.
|
symbol : str
|
Symbol representing the independent variable.
|
|
Class Attributes
|
----------------
|
maxpower : int
|
Maximum power allowed, i.e., the largest number ``n`` such that
|
``p(x)**n`` is allowed. This is to limit runaway polynomial size.
|
domain : (2,) ndarray
|
Default domain of the class.
|
window : (2,) ndarray
|
Default window of the class.
|
|
Néd u â°õ ¹õ ²õ ³u â´u âµu â¶u â·u â¸u â¹)
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Ú0Ú1Ú2Ú3Ú4Ú5Ú6Ú7Ú8Ú9u âu âu âu âu âu â
u âu âu âu âÚntc C s | j S ©N)Ú_symbol©Úself© r úQd:\z\workplace\vscode\pyvenv\venv\Lib\site-packages\numpy/polynomial/_polybase.pyÚsymboln s zABCPolyBase.symbolc C s d S r r r r r r Údomainr s zABCPolyBase.domainc C s d S r r r r r r Úwindoww s zABCPolyBase.windowc C s d S r r r r r r Ú
|
basis_name| s zABCPolyBase.basis_namec C s d S r r ©Zc1Úc2r r r Ú_add s zABCPolyBase._addc C s d S r r r r r r Ú_sub s zABCPolyBase._subc C s d S r r r r r r Ú_mul s zABCPolyBase._mulc C s d S r r r r r r Ú_div s zABCPolyBase._divc C s d S r r )ÚcÚpowÚmaxpowerr r r Ú_pow s zABCPolyBase._powc C s d S r r )Úxr$ r r r Ú_val s zABCPolyBase._valc C s d S r r )r$ ÚmÚkÚlbndÚsclr r r Ú_int s zABCPolyBase._intc C s d S r r )r$ r* r- r r r Ú_der¤ s zABCPolyBase._derc C s d S r r )r( ÚyÚdegÚrcondÚfullr r r Ú_fit© s zABCPolyBase._fitc C s d S r r )Úoffr- r r r Ú_line® s zABCPolyBase._linec C s d S r r )r$ r r r Ú_roots³ s zABCPolyBase._rootsc C s d S r r )Úrr r r Ú
|
_fromroots¸ s zABCPolyBase._fromrootsc C s6 t | jt |jkrdS t | j|jk¡s.dS dS dS )aG Check if coefficients match.
|
|
.. versionadded:: 1.6.0
|
|
Parameters
|
----------
|
other : class instance
|
The other class must have the ``coef`` attribute.
|
|
Returns
|
-------
|
bool : boolean
|
True if the coefficients are the same, False otherwise.
|
|
FTN)ÚlenÚcoefÚnpÚall©r Úotherr r r Ú has_samecoef½ s
|
zABCPolyBase.has_samecoefc C s t | j|jk¡S )a? Check if domains match.
|
|
.. versionadded:: 1.6.0
|
|
Parameters
|
----------
|
other : class instance
|
The other class must have the ``domain`` attribute.
|
|
Returns
|
-------
|
bool : boolean
|
True if the domains are the same, False otherwise.
|
|
)r< r= r r> r r r Úhas_samedomainÔ s zABCPolyBase.has_samedomainc C s t | j|jk¡S )a? Check if windows match.
|
|
.. versionadded:: 1.6.0
|
|
Parameters
|
----------
|
other : class instance
|
The other class must have the ``window`` attribute.
|
|
Returns
|
-------
|
bool : boolean
|
True if the windows are the same, False otherwise.
|
|
)r< r= r r> r r r Úhas_samewindowæ s zABCPolyBase.has_samewindowc C s t || jS )a Check if types match.
|
|
.. versionadded:: 1.7.0
|
|
Parameters
|
----------
|
other : object
|
Class instance.
|
|
Returns
|
-------
|
bool : boolean
|
True if other is same class as self
|
|
)Ú
|
isinstanceÚ __class__r> r r r Ú has_sametypeø s zABCPolyBase.has_sametypec C sv t |trrt || js tdnLt | j|jk¡s<tdn0t | j|jk¡sXtdn| j|jkrlt d|j
|
S |S )a Interpret other as polynomial coefficients.
|
|
The `other` argument is checked to see if it is of the same
|
class as self with identical domain and window. If so,
|
return its coefficients, otherwise return `other`.
|
|
.. versionadded:: 1.9.0
|
|
Parameters
|
----------
|
other : anything
|
Object to be checked.
|
|
Returns
|
-------
|
coef
|
The coefficients of`other` if it is a compatible instance,
|
of ABCPolyBase, otherwise `other`.
|
|
Raises
|
------
|
TypeError
|
When `other` is an incompatible instance of ABCPolyBase.
|
|
zPolynomial types differzDomains differzWindows differzPolynomial symbols differ) rC r rD Ú TypeErrorr< r= r r r Ú
|
ValueErrorr; r> r r r Ú_get_coefficients
|
s
|
|
|
|
z ABCPolyBase._get_coefficientsr( c C s¼ t j|gdd\}|| _|d k rLt j|gdd\}t|dkrFtd|| _|d k rt j|gdd\}t|dkrztd|| _z| ¡ stdW n tk
|
r° t dY nX || _
|
d S )NF©Útrimé z$Domain has wrong number of elements.z$Window has wrong number of elements.z/Symbol string must be a valid Python identifierz!Symbol must be a non-empty string) ÚpuÚ as_seriesr; r: rG r r Ú isidentifierÚAttributeErrorrF r )r r; r r r r r r Ú__init__0 s( ÿzABCPolyBase.__init__c
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C sb t | jdd
}t | jdd
}t | jdd
}| jj}| d| d| d| d| j d
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S )Né éÿÿÿÿú(z , domain=z , window=z
|
, symbol='z'))Úreprr; r r rD Ú__name__r )r r; r r Únamer r r Ú__repr__N s
|
zABCPolyBase.__repr__c C sP |dkr| ¡ S |dkr0td| d| j d|dkrD| | j¡S | | j¡S )NÚ )ÚasciiÚunicodezUnsupported format string 'z ' passed to z4.__format__. Valid options are 'ascii' and 'unicode'rY )Ú__str__rG rD Ú_generate_stringÚ_str_term_asciiÚ_str_term_unicode)r Zfmt_strr r r Ú
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__format__V s ÿ zABCPolyBase.__format__c C s | j r| | j¡S | | j¡S r )Ú _use_unicoder\ r^ r] r r r r r[ c s zABCPolyBase.__str__c C s t ¡ dd¡}|dk r d}t | jd ¡}t| jdd
D ]Ü\}}|d7 }t|d }z4|dkrxdtj|dd }nd
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tj| dd }W n tk
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r¬ d| }Y nX |||| j 7 }t
|
| d ¡d t
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| }|t
|
| jdd
d k rú|d 7 }||kr| dd d¡}||7 }q>|S )z
|
Generate the full string representation of the polynomial, using
|
``term_method`` to generate each polynomial term.
|
Ú linewidthéK r r Nú z+ T©Úparensz- Ú
|
rR rK ) r< Zget_printoptionsÚgetrL Ú format_floatr; Ú enumerateÚstrrF r r: ÚsplitÚreplace) r Z term_methodra ÚoutÚir; ÚpowerZ next_termZline_lenr r r r\ h s*
|
|
z ABCPolyBase._generate_stringc C s2 | j dkrtdd| j | | j¡ d| dS )z
|
String representation of single polynomial term using unicode
|
characters for superscripts and subscripts.
|
NzZSubclasses must define either a basis_name, or override _str_term_unicode(cls, i, arg_str)õ ·rS ú))r ÚNotImplementedErrorÚ translateÚ_subscript_mapping©Úclsrn Úarg_strr r r r^ s
|
|
ÿz ABCPolyBase._str_term_unicodec C s, | j dkrtdd| j d| d| dS )z
|
String representation of a single polynomial term using ** and _ to
|
represent superscripts and subscripts, respectively.
|
NzXSubclasses must define either a basis_name, or override _str_term_ascii(cls, i, arg_str)rc Ú_rS rq ©r rr ru r r r r] s
|
|
ÿzABCPolyBase._str_term_asciic C s, | j d krtdd| j d| d| dS )NzbSubclasses must define either a basis name, or override _repr_latex_term(i, arg_str, needs_parens)Ú{z}_{z}(rq ry )rv rn rw Ú needs_parensr r r Ú_repr_latex_term© s
|
|
ÿz ABCPolyBase._repr_latex_termFc C s d tj| |d¡S )Nz \text{{{}}}rd )ÚformatrL rh )r( re r r r Ú_repr_latex_scalar² s z ABCPolyBase._repr_latex_scalarc C s | ¡ \}}|dkr(|dkr(| j}d}nj|dkrL| |¡ d| j }d}nF|dkrn| |¡ | j }d}n$| |¡ d| |¡ | j }d}dj}g }t| jD ]¾\}}|dkrÄ| |¡ } nXt|tjsäd| |¡ d} n8t
|
|¡sd| j|dd } nd
|
| j| dd } | |||¡}
|
|
|
d kr:| } n| d |
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} |dkrZ|| } | | ¡ q¦|rxd |¡} nd} d| j d| dS )Nr r Fz + Tz\color{{LightGray}}{{{}}}z + (rq rd z - r
|
z\,rX r ú$z \mapsto )Úmapparmsr r~ r} ri r; rC ÚnumbersÚRealr< Zsignbitr| ÚappendÚjoin) r r5 ZscaleZtermr{ ZmuteÚpartsrn r$ Zcoef_strZterm_strÚpartÚbodyr r r Ú _repr_latex_¸ sD ÿ
|
|
zABCPolyBase._repr_latex_c C sB | j ¡ }| j ¡ |d< | j ¡ |d< | j ¡ |d< | j|d< |S )Nr; r r r )Ú__dict__Úcopyr; r r r )r Úretr r r Ú __getstate__ñ s
|
|
zABCPolyBase.__getstate__c C s
|
|| _ d S r )r )r Údictr r r Ú __setstate__ù s zABCPolyBase.__setstate__c C s. t | j| j¡\}}||| }| || j¡S r )rL r r r r) r; )r Úargr5 r- r r r Ú__call__þ s zABCPolyBase.__call__c C s
|
t | jS r )Úiterr; r r r r Ú__iter__ s zABCPolyBase.__iter__c C s
|
t | jS r )r: r; r r r r Ú__len__ s zABCPolyBase.__len__c C s | | j | j| j| j¡S r ©rD r; r r r r r r r Ú__neg__ s ÿzABCPolyBase.__neg__c C s | S r r r r r r Ú__pos__ s zABCPolyBase.__pos__c C sL | |¡}z| | j|¡}W n tk
|
r4 t Y S X | || j| j| j¡S r ) rH r r; Ú ExceptionÚNotImplementedrD r r r ©r r? Ú othercoefr; r r r Ú__add__ s
|
|
zABCPolyBase.__add__c C sL | |¡}z| | j|¡}W n tk
|
r4 t Y S X | || j| j| j¡S r ) rH r! r; r r rD r r r r r r r Ú__sub__ s
|
|
zABCPolyBase.__sub__c C sL | |¡}z| | j|¡}W n tk
|
r4 t Y S X | || j| j| j¡S r ) rH r" r; r r rD r r r r r r r Ú__mul__# s
|
|
zABCPolyBase.__mul__c C s> t |tjrt |tr4tdt| dt| d| |¡S )Nz&unsupported types for true division: 'z', 'ú')rC r ÚNumberÚboolrF ÚtypeÚ __floordiv__r> r r r Ú __truediv__+ s
|
ÿzABCPolyBase.__truediv__c C s | |¡}|tkr|S |d S ©Nr ©Ú
|
__divmod__r ©r r? Úresr r r r¢ 6 s
|
zABCPolyBase.__floordiv__c C s | |¡}|tkr|S |d S ©Nr r¥ r§ r r r Ú__mod__< s
|
zABCPolyBase.__mod__c C s | |¡}z| | j|¡\}}W n, tk
|
r6 Y n tk
|
rL t Y S X | || j| j| j ¡}| || j| j| j ¡}||fS r )
|
rH r# r; ÚZeroDivisionErrorr r rD r r r )r r? r ÚquoÚremr r r r¦ B s
|
|
zABCPolyBase.__divmod__c C s. | j | j|| jd}| || j| j| j¡}|S )N)r& )r' r; r& rD r r r )r r? r; r¨ r r r Ú__pow__N s zABCPolyBase.__pow__c C sB z| || j¡}W n tk
|
r* t Y S X | || j| j| j¡S r )r r; r r rD r r r ©r r? r; r r r Ú__radd__S s
|
|
zABCPolyBase.__radd__c C sB z| || j¡}W n tk
|
r* t Y S X | || j| j| j¡S r )r! r; r r rD r r r r¯ r r r Ú__rsub__Z s
|
|
zABCPolyBase.__rsub__c C sB z| || j¡}W n tk
|
r* t Y S X | || j| j| j¡S r )r" r; r r rD r r r r¯ r r r Ú__rmul__a s
|
|
zABCPolyBase.__rmul__c C s
|
| |¡S r )Ú __rfloordiv__r> r r r Ú__rdiv__h s zABCPolyBase.__rdiv__c C s t S r )r r> r r r Ú __rtruediv__l s zABCPolyBase.__rtruediv__c C s | |¡}|tkr|S |d S r¤ ©Ú __rdivmod__r r§ r r r r³ q s
|
zABCPolyBase.__rfloordiv__c C s | |¡}|tkr|S |d S r© r¶ r§ r r r Ú__rmod__w s
|
zABCPolyBase.__rmod__c C sx z| || j¡\}}W n, tk
|
r, Y n tk
|
rB t Y S X | || j| j| j¡}| || j| j| j¡}||fS r ) r# r; r« r r rD r r r )r r? r¬ r r r r r· } s
|
zABCPolyBase.__rdivmod__c C sb t || jo\t | j|jk¡o\t | j|jk¡o\| jj|jjko\t | j|jk¡o\| j|jk}|S r ) rC rD r< r= r r r; Úshaper r§ r r r Ú__eq__ s ÿþýü
|
ûzABCPolyBase.__eq__c C s | |¡ S r )rº r> r r r Ú__ne__ s zABCPolyBase.__ne__c C s | | j| j| j| j¡S )zoReturn a copy.
|
|
Returns
|
-------
|
new_series : series
|
Copy of self.
|
|
r r r r r r s zABCPolyBase.copyc C s t | d S )zÆThe degree of the series.
|
|
.. versionadded:: 1.5.0
|
|
Returns
|
-------
|
degree : int
|
Degree of the series, one less than the number of coefficients.
|
|
r )r: r r r r Údegree£ s zABCPolyBase.degreec C s | |d ¡S )aÒ Truncate series to the given degree.
|
|
Reduce the degree of the series to `deg` by discarding the
|
high order terms. If `deg` is greater than the current degree a
|
copy of the current series is returned. This can be useful in least
|
squares where the coefficients of the high degree terms may be very
|
small.
|
|
.. versionadded:: 1.5.0
|
|
Parameters
|
----------
|
deg : non-negative int
|
The series is reduced to degree `deg` by discarding the high
|
order terms. The value of `deg` must be a non-negative integer.
|
|
Returns
|
-------
|
new_series : series
|
New instance of series with reduced degree.
|
|
r )Útruncate)r r1 r r r Úcutdeg° s zABCPolyBase.cutdegr c C s$ t | j|¡}| || j| j| j¡S )a Remove trailing coefficients
|
|
Remove trailing coefficients until a coefficient is reached whose
|
absolute value greater than `tol` or the beginning of the series is
|
reached. If all the coefficients would be removed the series is set
|
to ``[0]``. A new series instance is returned with the new
|
coefficients. The current instance remains unchanged.
|
|
Parameters
|
----------
|
tol : non-negative number.
|
All trailing coefficients less than `tol` will be removed.
|
|
Returns
|
-------
|
new_series : series
|
New instance of series with trimmed coefficients.
|
|
)rL Ztrimcoefr; rD r r r )r Ztolr; r r r rJ É s zABCPolyBase.trimc C sZ t |}||ks|dk r td|t| jkr6| j}n| jd|
}| || j| j| j¡S )a Truncate series to length `size`.
|
|
Reduce the series to length `size` by discarding the high
|
degree terms. The value of `size` must be a positive integer. This
|
can be useful in least squares where the coefficients of the
|
high degree terms may be very small.
|
|
Parameters
|
----------
|
size : positive int
|
The series is reduced to length `size` by discarding the high
|
degree terms. The value of `size` must be a positive integer.
|
|
Returns
|
-------
|
new_series : series
|
New instance of series with truncated coefficients.
|
|
r z size must be a positive integerN)ÚintrG r: r; rD r r r )r ÚsizeÚisizer; r r r r½ à s zABCPolyBase.truncatec C s@ |dkr| j }|dkr |j}|dkr*|j}| |j||| jdS )aé Convert series to a different kind and/or domain and/or window.
|
|
Parameters
|
----------
|
domain : array_like, optional
|
The domain of the converted series. If the value is None,
|
the default domain of `kind` is used.
|
kind : class, optional
|
The polynomial series type class to which the current instance
|
should be converted. If kind is None, then the class of the
|
current instance is used.
|
window : array_like, optional
|
The window of the converted series. If the value is None,
|
the default window of `kind` is used.
|
|
Returns
|
-------
|
new_series : series
|
The returned class can be of different type than the current
|
instance and/or have a different domain and/or different
|
window.
|
|
Notes
|
-----
|
Conversion between domains and class types can result in
|
numerically ill defined series.
|
|
N)r r )rD r r Úidentityr )r r Úkindr r r r Úconvertý s zABCPolyBase.convertc C s t | j| j¡S )aà Return the mapping parameters.
|
|
The returned values define a linear map ``off + scl*x`` that is
|
applied to the input arguments before the series is evaluated. The
|
map depends on the ``domain`` and ``window``; if the current
|
``domain`` is equal to the ``window`` the resulting map is the
|
identity. If the coefficients of the series instance are to be
|
used by themselves outside this class, then the linear function
|
must be substituted for the ``x`` in the standard representation of
|
the base polynomials.
|
|
Returns
|
-------
|
off, scl : float or complex
|
The mapping function is defined by ``off + scl*x``.
|
|
Notes
|
-----
|
If the current domain is the interval ``[l1, r1]`` and the window
|
is ``[l2, r2]``, then the linear mapping function ``L`` is
|
defined by the equations::
|
|
L(l1) = l2
|
L(r1) = r2
|
|
)rL r r r r r r r r " s zABCPolyBase.mapparmsr c C sT | ¡ \}}|dkrd}n ||| }| | j|||d| ¡}| || j| j| j¡S )a Integrate.
|
|
Return a series instance that is the definite integral of the
|
current series.
|
|
Parameters
|
----------
|
m : non-negative int
|
The number of integrations to perform.
|
k : array_like
|
Integration constants. The first constant is applied to the
|
first integration, the second to the second, and so on. The
|
list of values must less than or equal to `m` in length and any
|
missing values are set to zero.
|
lbnd : Scalar
|
The lower bound of the definite integral.
|
|
Returns
|
-------
|
new_series : series
|
A new series representing the integral. The domain is the same
|
as the domain of the integrated series.
|
|
Nr g ð?)r r. r; rD r r r )r r* r+ r, r5 r- r; r r r Úinteg? s zABCPolyBase.integc C s2 | ¡ \}}| | j||¡}| || j| j| j¡S )a§ Differentiate.
|
|
Return a series instance of that is the derivative of the current
|
series.
|
|
Parameters
|
----------
|
m : non-negative int
|
Find the derivative of order `m`.
|
|
Returns
|
-------
|
new_series : series
|
A new series representing the derivative. The domain is the same
|
as the domain of the differentiated series.
|
|
)r r/ r; rD r r r )r r* r5 r- r; r r r Úderiv` s zABCPolyBase.derivc C s | | j¡}t || j| j¡S )a, Return the roots of the series polynomial.
|
|
Compute the roots for the series. Note that the accuracy of the
|
roots decrease the further outside the domain they lie.
|
|
Returns
|
-------
|
roots : ndarray
|
Array containing the roots of the series.
|
|
)r7 r; rL Ú mapdomainr r )r Úrootsr r r rÈ v s zABCPolyBase.rootsc C s4 |dkr| j }t |d |d |¡}| |}||fS )a¾ Return x, y values at equally spaced points in domain.
|
|
Returns the x, y values at `n` linearly spaced points across the
|
domain. Here y is the value of the polynomial at the points x. By
|
default the domain is the same as that of the series instance.
|
This method is intended mostly as a plotting aid.
|
|
.. versionadded:: 1.5.0
|
|
Parameters
|
----------
|
n : int, optional
|
Number of point pairs to return. The default value is 100.
|
domain : {None, array_like}, optional
|
If not None, the specified domain is used instead of that of
|
the calling instance. It should be of the form ``[beg,end]``.
|
The default is None which case the class domain is used.
|
|
Returns
|
-------
|
x, y : ndarray
|
x is equal to linspace(self.domain[0], self.domain[1], n) and
|
y is the series evaluated at element of x.
|
|
Nr r )r r< Úlinspace)r Únr r( r0 r r r rÉ
s
|
zABCPolyBase.linspacec
|
C s |dkrt |¡}n t|tkr2t|dkr2| j}|dkr@| j}t |||¡}
|
| j|
|
|||||d} |r| \} } | | ||| d| fS | } | | ||| dS dS )a0 Least squares fit to data.
|
|
Return a series instance that is the least squares fit to the data
|
`y` sampled at `x`. The domain of the returned instance can be
|
specified and this will often result in a superior fit with less
|
chance of ill conditioning.
|
|
Parameters
|
----------
|
x : array_like, shape (M,)
|
x-coordinates of the M sample points ``(x[i], y[i])``.
|
y : array_like, shape (M,)
|
y-coordinates of the M sample points ``(x[i], y[i])``.
|
deg : int or 1-D array_like
|
Degree(s) of the fitting polynomials. If `deg` is a single integer
|
all terms up to and including the `deg`'th term are included in the
|
fit. For NumPy versions >= 1.11.0 a list of integers specifying the
|
degrees of the terms to include may be used instead.
|
domain : {None, [beg, end], []}, optional
|
Domain to use for the returned series. If ``None``,
|
then a minimal domain that covers the points `x` is chosen. If
|
``[]`` the class domain is used. The default value was the
|
class domain in NumPy 1.4 and ``None`` in later versions.
|
The ``[]`` option was added in numpy 1.5.0.
|
rcond : float, optional
|
Relative condition number of the fit. Singular values smaller
|
than this relative to the largest singular value will be
|
ignored. The default value is len(x)*eps, where eps is the
|
relative precision of the float type, about 2e-16 in most
|
cases.
|
full : bool, optional
|
Switch determining nature of return value. When it is False
|
(the default) just the coefficients are returned, when True
|
diagnostic information from the singular value decomposition is
|
also returned.
|
w : array_like, shape (M,), optional
|
Weights. If not None, the weight ``w[i]`` applies to the unsquared
|
residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
|
chosen so that the errors of the products ``w[i]*y[i]`` all have
|
the same variance. When using inverse-variance weighting, use
|
``w[i] = 1/sigma(y[i])``. The default value is None.
|
|
.. versionadded:: 1.5.0
|
window : {[beg, end]}, optional
|
Window to use for the returned series. The default
|
value is the default class domain
|
|
.. versionadded:: 1.6.0
|
symbol : str, optional
|
Symbol representing the independent variable. Default is 'x'.
|
|
Returns
|
-------
|
new_series : series
|
A series that represents the least squares fit to the data and
|
has the domain and window specified in the call. If the
|
coefficients for the unscaled and unshifted basis polynomials are
|
of interest, do ``new_series.convert().coef``.
|
|
[resid, rank, sv, rcond] : list
|
These values are only returned if ``full == True``
|
|
- resid -- sum of squared residuals of the least squares fit
|
- rank -- the numerical rank of the scaled Vandermonde matrix
|
- sv -- singular values of the scaled Vandermonde matrix
|
- rcond -- value of `rcond`.
|
|
For more details, see `linalg.lstsq`.
|
|
Nr )Úwr2 r3 ©r r r ) rL Ú getdomainr¡ Úlistr: r r rÇ r4 )rv r( r0 r1 r r2 r3 rË r r Zxnewr¨ r; Ústatusr r r Úfit¥ s I ÿzABCPolyBase.fitc
|
C s t j|gdd\}|dkr&t |¡}n t|tkrDt|dkrD| j}|dkrR| j}t|}t ||¡\}}||| }| |¡|| } | | |||dS )a Return series instance that has the specified roots.
|
|
Returns a series representing the product
|
``(x - r[0])*(x - r[1])*...*(x - r[n-1])``, where ``r`` is a
|
list of roots.
|
|
Parameters
|
----------
|
roots : array_like
|
List of roots.
|
domain : {[], None, array_like}, optional
|
Domain for the resulting series. If None the domain is the
|
interval from the smallest root to the largest. If [] the
|
domain is the class domain. The default is [].
|
window : {None, array_like}, optional
|
Window for the returned series. If None the class window is
|
used. The default is None.
|
symbol : str, optional
|
Symbol representing the independent variable. Default is 'x'.
|
|
Returns
|
-------
|
new_series : series
|
Series with the specified roots.
|
|
FrI Nr rÌ )
|
rL rM rÍ r¡ rÎ r: r r r r9 )
|
rv rÈ r r r r1 r5 r- Zrnewr; r r r Ú fromroots s zABCPolyBase.fromrootsc C sF |dkr| j }|dkr | j}t ||¡\}}| ||¡}| ||||S )a Identity function.
|
|
If ``p`` is the returned series, then ``p(x) == x`` for all
|
values of x.
|
|
Parameters
|
----------
|
domain : {None, array_like}, optional
|
If given, the array must be of the form ``[beg, end]``, where
|
``beg`` and ``end`` are the endpoints of the domain. If None is
|
given then the class domain is used. The default is None.
|
window : {None, array_like}, optional
|
If given, the resulting array must be if the form
|
``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
|
the window. If None is given then the class window is used. The
|
default is None.
|
symbol : str, optional
|
Symbol representing the independent variable. Default is 'x'.
|
|
Returns
|
-------
|
new_series : series
|
Series of representing the identity.
|
|
N)r r rL r r6 )rv r r r r5 r- r; r r r r , s zABCPolyBase.identityc C sV |dkr| j }|dkr | j}t|}||ks4|dk r<td| dg| dg |||S )a\ Series basis polynomial of degree `deg`.
|
|
Returns the series representing the basis polynomial of degree `deg`.
|
|
.. versionadded:: 1.7.0
|
|
Parameters
|
----------
|
deg : int
|
Degree of the basis polynomial for the series. Must be >= 0.
|
domain : {None, array_like}, optional
|
If given, the array must be of the form ``[beg, end]``, where
|
``beg`` and ``end`` are the endpoints of the domain. If None is
|
given then the class domain is used. The default is None.
|
window : {None, array_like}, optional
|
If given, the resulting array must be if the form
|
``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
|
the window. If None is given then the class window is used. The
|
default is None.
|
symbol : str, optional
|
Symbol representing the independent variable. Default is 'x'.
|
|
Returns
|
-------
|
new_series : series
|
A series with the coefficient of the `deg` term set to one and
|
all others zero.
|
|
Nr z deg must be non-negative integerr )r r r¿ rG )rv r1 r r r Zidegr r r ÚbasisO s zABCPolyBase.basisc C s* |dkr| j }|dkr | j}| || |¡S )aÊ Convert series to series of this class.
|
|
The `series` is expected to be an instance of some polynomial
|
series of one of the types supported by by the numpy.polynomial
|
module, but could be some other class that supports the convert
|
method.
|
|
.. versionadded:: 1.7.0
|
|
Parameters
|
----------
|
series : series
|
The series instance to be converted.
|
domain : {None, array_like}, optional
|
If given, the array must be of the form ``[beg, end]``, where
|
``beg`` and ``end`` are the endpoints of the domain. If None is
|
given then the class domain is used. The default is None.
|
window : {None, array_like}, optional
|
If given, the resulting array must be if the form
|
``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
|
the window. If None is given then the class window is used. The
|
default is None.
|
|
Returns
|
-------
|
new_series : series
|
A series of the same kind as the calling class and equal to
|
`series` when evaluated.
|
|
See Also
|
--------
|
convert : similar instance method
|
|
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