Ë
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Kñúhe ã óp d Z ddlZddlZddlZddlmZ ddlZddlm Z
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dgZ G d dej « Z y)a
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Abstract base class for the various polynomial Classes.
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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.
|
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é N)ÚCallableé )Ú polyutilsÚ ABCPolyBasec óP e Zd ZdZdZdZdZej dddddd d
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d d d d
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« Z ej ddddddddddd
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« Z
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e j dk( Z ed « Zeej" d « « Zeej" d « « Zeej" d « « Zeej" d « « Zeej" d « « Zeej" d « « Zeej" d! « « Zeej" dgd"« « Zeej" d# « « Zeej" d$ « « Z eej" d% « « Z eej" d& « « Z eej" d' « « Z eej" d( « « Z eej" d) « « Z!d* Z"d+ Z#d, Z$d- Z%d. Z&dhd0Z'd1 Z(d2 Z)d3 Z*d4 Z+e,d5 « Z-e,d6 « Z.e,d7 « Z/edid8« Z0d9e1d:e2d;e2fd<Z3d= Z4d> Z5d? Z6d@ Z7dA Z8dB Z9dC Z:dD Z;dE Z<dF Z=dG Z>dH Z?dI Z@dJ ZAdK ZBdL ZCdM ZDdN ZEdO ZFdP ZGdQ ZHdR ZIdS ZJdT ZKdU ZLdV ZMdW ZNdX ZOdjdYZPdZ ZQdkd[ZRd\ ZSd]g dfd^ZTdld_ZUd` ZVdmdaZWe, dndb« ZXe,g dd/fdc« ZYe,dhdd« ZZe,dhde« Z[e,dodf« Z\y)pr aê An abstract base class for immutable series classes.
|
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ABCPolyBase provides the standard Python numerical methods
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'+', '-', '*', '//', '%', 'divmod', '**', and '()' along with the
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methods listed below.
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Parameters
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----------
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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
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to the interval ``[window[0], window[1]]`` by shifting and scaling.
|
The default value is the derived class domain.
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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'.
|
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.. versionadded:: 1.24
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Attributes
|
----------
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coef : (N,) ndarray
|
Series coefficients in order of increasing degree.
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domain : (2,) ndarray
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Domain that is mapped to window.
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window : (2,) ndarray
|
Window that domain is mapped to.
|
symbol : str
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Symbol representing the independent variable.
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Class Attributes
|
----------------
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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.
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domain : (2,) ndarray
|
Default domain of the class.
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window : (2,) ndarray
|
Default window of the class.
|
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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 ó | j S ©N)Ú_symbol©Úselfs úMH:\Change_password\venv_build\Lib\site-packages\numpy/polynomial/_polybase.pyÚsymbolzABCPolyBase.symboln s à||Ðó c ó y r © r s r ÚdomainzABCPolyBase.domainr ó ð r c ó y r r r s r ÚwindowzABCPolyBase.windoww r" r c ó y r r r s r Ú
|
basis_namezABCPolyBase.basis_name| r" r c ó y r r ©Úc1Úc2s r Ú_addzABCPolyBase._add r" r c ó y r r r( s r Ú_subzABCPolyBase._sub r" r c ó y r r r( s r Ú_mulzABCPolyBase._mul r" r c ó y r r r( s r Ú_divzABCPolyBase._div r" r c ó y r r )ÚcÚpowÚmaxpowers r Ú_powzABCPolyBase._pow r" r c ó y r r )Úxr3 s r Ú_valzABCPolyBase._val r" r c ó y r r )r3 ÚmÚkÚlbndÚscls r Ú_intzABCPolyBase._int r" r c ó y r r )r3 r; r> s r Ú_derzABCPolyBase._der¤ r" r c ó y r r )r8 ÚyÚdegÚrcondÚfulls r Ú_fitzABCPolyBase._fit© r" r c ó y r r )Úoffr> s r Ú_linezABCPolyBase._line® r" r c ó y r r )r3 s r Ú_rootszABCPolyBase._roots³ r" r c ó y r r )Úrs r Ú
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_fromrootszABCPolyBase._fromroots¸ r" r c ó´ t | j « t |j « k( xr, t j | j |j k( « S )a& Check if coefficients match.
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|
Parameters
|
----------
|
other : class instance
|
The other class must have the ``coef`` attribute.
|
|
Returns
|
-------
|
bool : boolean
|
True if the coefficients are the same, False otherwise.
|
|
)ÚlenÚcoefÚnpÚall©r Úothers r Ú has_samecoefzABCPolyBase.has_samecoef½ s? ô Nc %§*¡*oÑ -ò 0Ütyy E§J¡JÑ.Ó/ð
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r c óZ t j | j |j k( « S )a Check if domains match.
|
|
Parameters
|
----------
|
other : class instance
|
The other class must have the ``domain`` attribute.
|
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Returns
|
-------
|
bool : boolean
|
True if the domains are the same, False otherwise.
|
|
)rS rT r! rU s r Úhas_samedomainzABCPolyBase.has_samedomainÐ ó ô vvdkk U§\¡\Ñ1Ó2Ð2r c óZ t j | j |j k( « S )a Check if windows match.
|
|
Parameters
|
----------
|
other : class instance
|
The other class must have the ``window`` attribute.
|
|
Returns
|
-------
|
bool : boolean
|
True if the windows are the same, False otherwise.
|
|
)rS rT r$ rU s r Úhas_samewindowzABCPolyBase.has_samewindowà rZ r c ó. t || j « S )záCheck if types match.
|
|
Parameters
|
----------
|
other : object
|
Class instance.
|
|
Returns
|
-------
|
bool : boolean
|
True if other is same class as self
|
|
)Ú
|
isinstanceÚ __class__rU s r Ú has_sametypezABCPolyBase.has_sametypeð s ô % § ¡ Ó0Ð0r c ó¤ t |t « r¿t || j « s t d« t j
|
| j |j k( « s t d« t j
|
| j |j k( « s t d« | j |j k7 r t d« |j S |S )ap 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`.
|
|
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) r^ r r_ Ú TypeErrorrS rT r! r$ r Ú
|
ValueErrorrR rU s r Ú_get_coefficientsz ABCPolyBase._get_coefficients s ô0 e[Ô )Ü e T§^¡^Ô4Ü Ð 9Ó:Ð:ÜVVDKK¨5¯<©<Ñ 7Ô8Ü Ð 0Ó1Ð1ÜVVDKK¨5¯<©<Ñ 7Ô8Ü Ð 0Ó1Ð1Ø § ¡ Ò,Ü Ð!<Ó=Ð=Ø::Ð Ø r r8 c ó® t j |gd¬« \ }|| _ |:t j |gd¬« \ }t |« dk7 r t d« || _ |:t j |gd¬« \ }t |« dk7 r t d« || _ |j « s t d« || _
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y # t $ r t d« w xY w)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_seriesrR rQ rc r! r$ Ú isidentifierÚAttributeErrorrb r )r rR r! r$ r s r Ú__init__zABCPolyBase.__init__$ sÞ Ü tf¨5Ô1Ø à Ð Ü|| V H°5Ô9HVÜ6{aÒ Ü Ð!GÓHÐHØ DKà Ð Ü|| V H°5Ô9HVÜ6{aÒ Ü Ð!GÓHÐHØ DKð AØ×&Ñ&Ô(Ü ØEóð ð )ð øô ò AÜÐ ?Ó@Ð @ð Aús ÂB? Â?Cc
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óô t | j « dd }t | j « dd }t | j « dd }| j j
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}| d| d| d| d| j d
|
S )Né éÿÿÿÿú(z , domain=z , window=z
|
, symbol='z'))ÚreprrR r! r$ r_ Ú__name__r )r rR r! r$ Únames r Ú__repr__zABCPolyBase.__repr__B s ÜDIIq Ð$ÜdkkÓ" 1 RÐ(ÜdkkÓ" 1 RÐ(Ø~~×&Ñ&Ø&$y¨¨° ¸&¸ð BØ ;;- rð+ð ,r c óâ |dk( r| j « S |dvr t d| d| j d« |dk( r| j | j « S | j | j
|
« S )NÚ )ÚasciiÚunicodezUnsupported format string 'z ' passed to z4.__format__. Valid options are 'ascii' and 'unicode'rx )Ú__str__rc r_ Ú_generate_stringÚ_str_term_asciiÚ_str_term_unicode)r Úfmt_strs r Ú
|
__format__zABCPolyBase.__format__J s Ø b=Ø<<>Ð !Ø Ð.Ñ .Ü Ø-¨g¨Y°lØ>>Ð"ð #(ð)óð ð
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gÒ Ø×(Ñ(¨×)=Ñ)=Ó>Ð >Ø×$Ñ$ T×%;Ñ%;Ó<Ð<r c ó | j r| j | j « S | j | j « S r )Ú _use_unicoder{ r} r| r s r rz zABCPolyBase.__str__W s9 Ø × Ò Ø×(Ñ(¨×)?Ñ)?Ó@Ð @Ø×$Ñ$ T×%9Ñ%9Ó:Ð:r c ó t j « j dd« }|dk rd}t j | j
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d « }| j « \ }}| j t j ||« \ }}|rd|z dz }t | j
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dd « D ]É \ }} |dz }t |dz « }
|
| dk\ rd t j | d
|
¬ « z } nd t j | d
|
¬ « z } | ||
|
|« z } t |j d « d « t | « z } |t | j
|
dd « dz
|
k r| dz } | |k\ r| j dd d« } || z }Ë |S # t $ r d | } Y w xY w)z
|
Generate the full string representation of the polynomial, using
|
``term_method`` to generate each polynomial term.
|
Ú linewidthéK r r rq ú)Nú z+ T©Úparensz- ú
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rp rh )rS Úget_printoptionsÚgetri Ú format_floatrR ÚmapparmsÚ _format_termÚ enumerateÚstrrb rQ ÚsplitÚreplace) r Ú term_methodr ÚoutrI ÚscaleÚ scaled_symbolÚ needs_parensÚirR ÚpowerÚ next_termÚline_lens r r{ z ABCPolyBase._generate_string\ s ô ×'Ñ'Ó)×-Ñ-¨k¸2Ó> Ø q=ØIÜoodii¨lÓ+à]]_
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Uà&*×&7Ñ&7¼¿¹Ø8;¸Uó'DÑ# |á Ø -Ñ /°#Ñ 5Mä §¡¨1¨2 Ó/ò GAtØ 3JCÜAJEð
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(Ø19Ø $¤r§¡°tÀDÔ'IÑ IIà $¤r§¡¸°uÀTÔ'JÑ JIð U¨MÓ:Ñ :Iä399 T?¨2Ñ.Ó/´#°i³.Ñ@Hð 3tyy }Ó%¨Ñ)Ò)ØA à9Ò$Ø %× -Ñ -¨c°4¸Ó ; Ø 9Ñ Cð3 ð4
|
øô ò (Ø K ð (ús Ã;E5Å5FÆFc ó | j t d« d| j |j | j « d| dS )z
|
String representation of single polynomial term using unicode
|
characters for superscripts and subscripts.
|
zZSubclasses must define either a basis_name, or override _str_term_unicode(cls, i, arg_str)õ ·rq r
)r& ÚNotImplementedErrorÚ translateÚ_subscript_mapping©Úclsr Úarg_strs r r} z ABCPolyBase._str_term_unicode sU ð >>Ð !Ü%ð5óð ð S^^Ð$ Q§[¡[°×1GÑ1GÓ%HÐ$IØG9Að ð r c ó\ | j t d« d| j d| d| dS )z
|
String representation of a single polynomial term using ** and _ to
|
represent superscripts and subscripts, respectively.
|
zXSubclasses must define either a basis_name, or override _str_term_ascii(cls, i, arg_str)r Ú_rq r
©r& r r¡ s r r| zABCPolyBase._str_term_ascii sB ð >>Ð !Ü%ð3óð ð 3>>Ð" ! A 3 a¨ y°Ð2Ð2r c ó\ | j t d« d| j d| d| dS )NzbSubclasses must define either a basis name, or override _repr_latex_term(i, arg_str, needs_parens)ú{z}_{z}(r
r¦ )r¢ r r£ r s r Ú_repr_latex_termz ABCPolyBase._repr_latex_term¥ s@ à >>Ð !Ü%ð=ó>ð >ð CNNÐ# 5¨¨¨3¨w¨i°qÐ9Ð9r c ó8 dt j | |¬« dS )Nz\text{r ú})ri r )r8 r s r Ú_repr_latex_scalarz ABCPolyBase._repr_latex_scalar® s ð "//¨!°FÔ;Ð<¸BÐ?Ð?r Ú scalar_formatrI r c ó
|
|dk( r|dk( r| j }d}||fS |dk( r ||« d| j }d}||fS |dk( r ||« | j }d}||fS ||« d ||« | j }d}||fS )z' Format a single term in the expansion r r Fú + T)r )r r rI r Útermr s r r zABCPolyBase._format_term´ sÐ à !8
|
Ø;;DØ Lð \Ð!Ð!ð aZÙ# CÓ(Ð)¨¨T¯[©[¨MÐ:DØ Lð \Ð!Ð!ð AXÙ# EÓ*Ð+¨D¯K©K¨=Ð9DØ Lð \Ð!Ð!ñ ! Ó%Ð& cÙ Ó'Ð(¨¯©¨ ð7ð ð LØ\Ð!Ð!r c ó | j « \ }}| j | j ||« \ }}dj }g }t | j
|
« D ]À \ }}|dk( r| j |« } nct |t j « sd| j |« d} n3|dk\ rd| j |d¬« } nd| j | d¬« } | j |||« }
|
|
|
d k( r| } n| d
|
|
|
} |dk( r || « } |j | « Â |rd j |« } nd } d | j d| d S )Nz\color{{LightGray}}{{{}}}r z + (r
r¯ Tr z - r z\,rw r ú$z \mapsto ) r r r¬ Úformatr rR r^ ÚnumbersÚRealr© ÚappendÚjoinr ) r rI r r° r ÚmuteÚpartsr r3 Úcoef_strÚterm_strÚpartÚbodys r Ú _repr_latex_zABCPolyBase._repr_latex_Ç sf à]]_
|
UØ !× .Ñ .¨t×/FÑ/FØ/2°Eó ;Ñlð ,×2Ñ2àÜ diiÓ(ò DAqàAvØ "× 5Ñ 5°aÓ 8Ð 9Ü ¤7§<¡<Ô0Ø ! $×"9Ñ"9¸!Ó"<Ð!=¸QÐ?ØaØ ×!8Ñ!8¸À4Ð!8Ó!HÐ IÐJà ×!8Ñ!8¸!¸ÀDÐ!8Ó!IÐ JÐKð ×,Ñ,¨Q°°lÓCHØ3Ø à#* B x jÐ1àAvÙDzà LLÕ ð+ ñ. Ø775>Dð DàDKK= ¨$¨¨qÐ1Ð1r c ó | j j « }| j j « |d< | j j « |d< | j j « |d< | j
|
|d< |S )NrR r! r$ r )Ú__dict__ÚcopyrR r! r$ r )r Úrets r Ú __getstate__zABCPolyBase.__getstate__ñ sf Ømm× Ñ Ó"ØiinnÓ&F Ø ×(Ñ(Ó*H Ø ×(Ñ(Ó*H Ø H Ø
|
r c ó || _ y r )rÀ )r Údicts r Ú __setstate__zABCPolyBase.__setstate__ù s Ø r c ó t j || j | j « }| j || j
|
« S r )ri Ú mapdomainr! r$ r9 rR )r Úargs r Ú__call__zABCPolyBase.__call__þ s1 Üll3 § ¡ ¨T¯[©[Ó9ØyydiiÓ(Ð(r c ó, t | j « S r )ÚiterrR r s r Ú__iter__zABCPolyBase.__iter__ s ÜDIIÐ r c ó, t | j « S r )rQ rR r s r Ú__len__zABCPolyBase.__len__ s Ü499~Ð r c ó| | j | j | j | j | j « S r ©r_ rR r! r$ r r s r Ú__neg__zABCPolyBase.__neg__
|
s. Ø~~Ø YYJ T§[¡[°$·+±+ó
|
ð
|
r c ó | S r r r s r Ú__pos__zABCPolyBase.__pos__ s Ø r c óì | j |« } | j | j |« }| j || j | j | j « S # t $ r t cY S w xY wr ) rd r+ rR Ú ExceptionÚNotImplementedr_ r! r$ r ©r rV Ú othercoefrR s r Ú__add__zABCPolyBase.__add__ óe Ø×*Ñ*¨5Ó1 ð "Ø99TYY¨ Ó2Dð ~~d D§K¡K°·±¸d¿k¹kÓJÐJøô ò "Ü!Ò !ð "úó A! Á!A3Á2A3c óì | j |« } | j | j |« }| j || j | j | j « S # t $ r t cY S w xY wr ) rd r- rR rÖ r× r_ r! r$ r rØ s r Ú__sub__zABCPolyBase.__sub__ rÛ rÜ c óì | j |« } | j | j |« }| j || j | j | j « S # t $ r t cY S w xY wr ) rd r/ rR rÖ r× r_ r! r$ r rØ s r Ú__mul__zABCPolyBase.__mul__" rÛ rÜ c óÀ t |t j « rt |t « r$t dt | « dt |« d« | j |« S )Nz&unsupported types for true division: 'z', 'ú')r^ r´ ÚNumberÚboolrb ÚtypeÚ __floordiv__rU s r Ú __truediv__zABCPolyBase.__truediv__* sZ ô %¤§ ¡ Ô0´J¸uÄdÔ4KÜðÜJ<t¤D¨£K =°ð3óð ð × Ñ Ó'Ð'r c óB | j |« }|t u r|S |d S ©Nr ©Ú
|
__divmod__r× ©r rV Úress r ræ zABCPolyBase.__floordiv__5 ó& ØooeÓ$Ø .Ñ ØJØ1v r c óB | j |« }|t u r|S |d S ©Nr rê rì s r Ú__mod__zABCPolyBase.__mod__; rî r c óp | j |« } | j | j |« \ }}| j || j | j | j « }| j || j | j | j « }||fS # t $ r t $ r t
|
cY S w xY wr )
|
rd r1 rR ÚZeroDivisionErrorrÖ r× r_ r! r$ r )r rV rÙ ÚquoÚrems r rë zABCPolyBase.__divmod__A s Ø×*Ñ*¨5Ó1 ð "Øyy §¡¨IÓ6HCð
|
nnS $§+¡+¨t¯{©{¸D¿K¹KÓHØnnS $§+¡+¨t¯{©{¸D¿K¹KÓHØCxøô !ò Ø Üò "Ü!Ò !ð "ús B ÂB5Â4B5c óº | j | j || j ¬« }| j || j | j
|
| j « }|S )N)r5 )r6 rR r5 r_ r! r$ r )r rV rR rí s r Ú__pow__zABCPolyBase.__pow__M sC Øyy E°D·M±MyÓBØnnT 4§;¡;°· ± ¸T¿[¹[ÓIØ
|
r c óÊ | j || j « }| j || j
|
| j | j « S # t $ r t cY S w xY wr )r+ rR rÖ r× r_ r! r$ r ©r rV rR s r Ú__radd__zABCPolyBase.__radd__R óU ð "Ø99U D§I¡IÓ.Dð ~~d D§K¡K°·±¸d¿k¹kÓJÐJøô ò "Ü!Ò !ð "úó A ÁA"Á!A"c óÊ | j || j « }| j || j
|
| j | j « S # t $ r t cY S w xY wr )r- rR rÖ r× r_ r! r$ r rù s r Ú__rsub__zABCPolyBase.__rsub__Y rû rü c óÊ | j || j « }| j || j
|
| j | j « S # t $ r t cY S w xY wr )r/ rR rÖ r× r_ r! r$ r rù s r Ú__rmul__zABCPolyBase.__rmul__` rû rü c ó t S r )r× rU s r Ú __rtruediv__zABCPolyBase.__rtruediv__g s
|
ô Ð r c óB | j |« }|t u r|S |d S ré ©Ú __rdivmod__r× rì s r Ú __rfloordiv__zABCPolyBase.__rfloordiv__l ó( Ø× Ñ uÓ%Ø .Ñ ØJØ1v r c óB | j |« }|t u r|S |d S rð r rì s r Ú__rmod__zABCPolyBase.__rmod__r r r c óN | j || j « \ }}| j || j | j | j « }| j || j | j | j « }||fS # t $ r t $ r t cY S w xY wr ) r1 rR ró rÖ r× r_ r! r$ r )r rV rô rõ s r r zABCPolyBase.__rdivmod__x s ð "Øyy ¨¯ © Ó2HCð
|
nnS $§+¡+¨t¯{©{¸D¿K¹KÓHØnnS $§+¡+¨t¯{©{¸D¿K¹KÓHØCxøô !ò Ø Üò "Ü!Ò !ð "ús B Â B$Â#B$c óÚ t || j « xrÒ t j | j |j k( « xr¤ t j | j
|
|j
|
k( « xrv | j j |j j k( xrG t j | j |j k( « xr | j |j k( }|S r ) r^ r_ rS rT r! r$ rR Úshaper rì s r Ú__eq__zABCPolyBase.__eq__ s° Ü% § ¡ Ó0ò -Üvvdkk U§\¡\Ñ1Ó2ò-ävvdkk U§\¡\Ñ1Ó2ò-ð 5§:¡:×#3Ñ#3Ñ3ò-ô vvdii 5§:¡:Ñ-Ó.ò -ð
|
u||Ñ+ð ð
|
r c ó& | j |« S r )r rU s r Ú__ne__zABCPolyBase.__ne__ s Ø;;uÓ%Ð%Ð%r c óz | j | j | j | j | j « S )zoReturn a copy.
|
|
Returns
|
-------
|
new_series : series
|
Copy of self.
|
|
rÑ r s r rÁ zABCPolyBase.copy s) ð ~~dii¨¯©°d·k±kÀ4Ç;Á;ÓOÐOr c ó t | « dz
|
S )u" The degree of the series.
|
|
Returns
|
-------
|
degree : int
|
Degree of the series, one less than the number of coefficients.
|
|
Examples
|
--------
|
|
Create a polynomial object for ``1 + 7*x + 4*x**2``:
|
|
>>> np.polynomial.set_default_printstyle("unicode")
|
>>> poly = np.polynomial.Polynomial([1, 7, 4])
|
>>> print(poly)
|
1.0 + 7.0·x + 4.0·x²
|
>>> poly.degree()
|
2
|
|
Note that this method does not check for non-zero coefficients.
|
You must trim the polynomial to remove any trailing zeroes:
|
|
>>> poly = np.polynomial.Polynomial([1, 7, 0])
|
>>> print(poly)
|
1.0 + 7.0·x + 0.0·x²
|
>>> poly.degree()
|
2
|
>>> poly.trim().degree()
|
1
|
|
r )rQ r s r ÚdegreezABCPolyBase.degree s ô@ 4y1}Ð r c ó* | j |dz « S )a± 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.
|
|
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 rD s r ÚcutdegzABCPolyBase.cutdegÀ s ð* }}S 1WÓ%Ð%r c ó¦ t j | j |« }| 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.
|
|
)ri ÚtrimcoefrR r_ r! r$ r )r ÚtolrR s r rg zABCPolyBase.trim× s9 ô( {{499 cÓ*Ø~~d D§K¡K°·±¸d¿k¹kÓJÐJr c ó t |« }||k7 s|dk r t d« |t | j « k\ r | j }n| j d| }| j || 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)Úintrc rQ rR r_ r! r$ r )r ÚsizeÚisizerR s r r zABCPolyBase.truncateî sp ô( D Ø D=E AIÜ Ð >Ó?Ð ?Ø C NÒ "Ø99Dà99VeÐ$DØ~~d D§K¡K°·±¸d¿k¹kÓJÐJr c ó | | j }| |j }| |j } | |j ||| j ¬« « S )aé Convert series to a different kind and/or domain and/or window.
|
|
Parameters
|
----------
|
domain : array_like, optional
|
The domain of the converted series. If the value is None,
|
the default domain of `kind` is used.
|
kind : class, optional
|
The polynomial series type class to which the current instance
|
should be converted. If kind is None, then the class of the
|
current instance is used.
|
window : array_like, optional
|
The window of the converted series. If the value is None,
|
the default window of `kind` is used.
|
|
Returns
|
-------
|
new_series : series
|
The returned class can be of different type than the current
|
instance and/or have a different domain and/or different
|
window.
|
|
Notes
|
-----
|
Conversion between domains and class types can result in
|
numerically ill defined series.
|
|
)r$ r )r_ r! r$ Úidentityr )r r! Úkindr$ s r ÚconvertzABCPolyBase.convert sL ð: <Ø>>DØ >Ø[[FØ >Ø[[FÙDMM &°ÀÇ Á MÓLÓMÐMr c óV t j | 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
|
|
)ri r r! r$ r s r r zABCPolyBase.mapparms0 s ô6 {{4;;¨¯ © Ó4Ð4r r c óê | j « \ }}|d}n|||z z }| j | j |||d|z « }| j || 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.
|
|
r g ð?)r r? rR r_ r! r$ r )r r; r< r= rI r> rR s r ÚintegzABCPolyBase.integM sj ð2 ==?SØ <ØDàtÑ#DØyy A q¨$°°S±Ó9Ø~~d D§K¡K°·±¸d¿k¹kÓJÐJr c óÆ | j « \ }}| j | j ||« }| j || j | j
|
| j « S )a§ Differentiate.
|
|
Return a series instance of that is the derivative of the current
|
series.
|
|
Parameters
|
----------
|
m : non-negative int
|
Find the derivative of order `m`.
|
|
Returns
|
-------
|
new_series : series
|
A new series representing the derivative. The domain is the same
|
as the domain of the differentiated series.
|
|
)r rA rR r_ r! r$ r )r r; rI r> rR s r ÚderivzABCPolyBase.derivn sJ ð$ ==?SØyy A sÓ+Ø~~d D§K¡K°·±¸d¿k¹kÓJÐJr c ó | j | j « }t j || j | j
|
« S )a/ Return the roots of the series polynomial.
|
|
Compute the roots for the series. Note that the accuracy of the
|
roots decreases the further outside the `domain` they lie.
|
|
Returns
|
-------
|
roots : ndarray
|
Array containing the roots of the series.
|
|
)rL rR ri rÈ r$ r! )r Úrootss r r' zABCPolyBase.roots s1 ð DIIÓ&Ü||E 4§;¡;°· ± Ó<Ð<r c óp | | j }t j |d |d |« } | |« }||fS )a 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.
|
|
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.
|
|
r r )r! rS Úlinspace)r Únr! r8 rC s r r) zABCPolyBase.linspace s> ð0 >Ø[[FÜ KKq 6¨!¡9¨aÓ 0Ù GØ!t r c
|
ó |;t j |« }|d |d k( rE|dxx dz cc< |dxx dz cc< n*t |t « rt |« dk( r | j
|
}| | j }t j |||« }
|
| j |
|
|||||¬« } |r| \ } } | | ||| ¬« | fS | } | | ||| ¬« S )aê 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.
|
window : {[beg, end]}, optional
|
Window to use for the returned series. The default
|
value is the default class domain
|
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`.
|
|
r r )ÚwrE rF ©r! r$ r ) ri Ú getdomainr^ ÚlistrQ r! r$ rÈ rG )r¢ r8 rC rD r! rE rF r, r$ r Úxnewrí rR Ústatuss r ÚfitzABCPolyBase.fit± s× ðJ >Ü\\ !_FØayF 1IÒ%Øq Q Øq Q Ü ¤Ô %¬#¨f«+¸Ò*:ØZZFà >ØZZFä||Av vÓ.ØhhtQ q°¸DhÓAÙ Ø NT6áD °¸vÔFÈðð ð DÙt F°6À&ÔIÐ Ir c ót t j |gd¬« \ }|t j |« }n*t |t « rt |« dk( r | j }| | j }t |« }t j ||« \ }}|||z z }| j |« ||z z } | | |||¬« S )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.
|
|
Frf r r- )
|
ri rj r. r^ r/ rQ r! r$ r rO )
|
r¢ r' r! r$ r rD rI r> ÚrnewrR s
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r Ú fromrootszABCPolyBase.fromroots sª ô8 ,, w¨UÔ3Ø >Ü\\ %Ó(FÜ ¤Ô %¬#¨f«+¸Ò*:ØZZFà >ØZZFä%jÜ;;v vÓ.SØS5[Ñ Ø~~dÓ# c¨3¡hÑ.Ù4 ¨v¸fÔEÐEr c ó¦ | | j }| | j }t j ||« \ }}| j ||« } | ||||« 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.
|
|
)r! r$ ri r rJ )r¢ r! r$ r rI r> rR s r r zABCPolyBase.identity7 sU ð6 >ØZZFØ >ØZZFÜ;;v vÓ.SØyycÓ"Ù4 ¨Ó0Ð0r c ó | | j }| | j }t |« }||k7 s|dk r t d« | dg|z dgz |||« S )a; Series basis polynomial of degree `deg`.
|
|
Returns the series representing the basis polynomial of degree `deg`.
|
|
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.
|
|
r z deg must be non-negative integerr )r! r$ r rc )r¢ rD r! r$ r Úidegs r ÚbasiszABCPolyBase.basisZ s` ð: >ØZZFØ >ØZZFÜ3xà 3;$ (Ü Ð ?Ó@Ð @ÙA3: Ñ# V¨V°VÓ<Ð<r c ó` | | j }| | j }|j || |« S )a© 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.
|
|
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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