1192 lines
38 KiB
Python
1192 lines
38 KiB
Python
"""
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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
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for the various polynomial classes. It operates as a mixin, but uses the
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abc module from the stdlib, hence it is only available for Python >= 2.6.
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"""
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import abc
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import numbers
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import os
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from collections.abc import Callable
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import numpy as np
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from . import polyutils as pu
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__all__ = ['ABCPolyBase']
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class ABCPolyBase(abc.ABC):
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"""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
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Series coefficients in order of increasing degree, i.e.,
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``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``, where
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``P_i`` is the basis polynomials of degree ``i``.
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domain : (2,) array_like, optional
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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.
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The default value is the derived class domain.
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window : (2,) array_like, optional
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Window, see domain for its use. The default value is the
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derived class window.
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symbol : str, optional
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Symbol used to represent the independent variable in string
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representations of the polynomial expression, e.g. for printing.
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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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----------
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coef : (N,) ndarray
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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
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Window that domain is mapped to.
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symbol : str
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Symbol representing the independent variable.
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Class Attributes
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----------------
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maxpower : int
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Maximum power allowed, i.e., the largest number ``n`` such that
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``p(x)**n`` is allowed. This is to limit runaway polynomial size.
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domain : (2,) ndarray
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Default domain of the class.
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window : (2,) ndarray
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Default window of the class.
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"""
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# Not hashable
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__hash__ = None
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# Opt out of numpy ufuncs and Python ops with ndarray subclasses.
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__array_ufunc__ = None
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# Limit runaway size. T_n^m has degree n*m
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maxpower = 100
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# Unicode character mappings for improved __str__
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_superscript_mapping = str.maketrans({
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"0": "⁰",
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"1": "¹",
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"2": "²",
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"3": "³",
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"4": "⁴",
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"5": "⁵",
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"6": "⁶",
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"7": "⁷",
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"8": "⁸",
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"9": "⁹"
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})
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_subscript_mapping = str.maketrans({
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"0": "₀",
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"1": "₁",
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"2": "₂",
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"3": "₃",
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"4": "₄",
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"5": "₅",
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"6": "₆",
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"7": "₇",
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"8": "₈",
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"9": "₉"
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})
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# Some fonts don't support full unicode character ranges necessary for
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# the full set of superscripts and subscripts, including common/default
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# fonts in Windows shells/terminals. Therefore, default to ascii-only
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# printing on windows.
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_use_unicode = not os.name == 'nt'
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@property
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def symbol(self):
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return self._symbol
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@property
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@abc.abstractmethod
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def domain(self):
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pass
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@property
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@abc.abstractmethod
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def window(self):
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pass
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@property
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@abc.abstractmethod
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def basis_name(self):
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pass
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@staticmethod
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@abc.abstractmethod
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def _add(c1, c2):
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pass
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@staticmethod
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@abc.abstractmethod
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def _sub(c1, c2):
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pass
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@staticmethod
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@abc.abstractmethod
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def _mul(c1, c2):
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pass
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@staticmethod
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@abc.abstractmethod
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def _div(c1, c2):
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pass
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@staticmethod
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@abc.abstractmethod
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def _pow(c, pow, maxpower=None):
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pass
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@staticmethod
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@abc.abstractmethod
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def _val(x, c):
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pass
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@staticmethod
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@abc.abstractmethod
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def _int(c, m, k, lbnd, scl):
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pass
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@staticmethod
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@abc.abstractmethod
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def _der(c, m, scl):
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pass
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@staticmethod
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@abc.abstractmethod
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def _fit(x, y, deg, rcond, full):
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pass
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@staticmethod
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@abc.abstractmethod
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def _line(off, scl):
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pass
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@staticmethod
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@abc.abstractmethod
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def _roots(c):
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pass
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@staticmethod
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@abc.abstractmethod
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def _fromroots(r):
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pass
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def has_samecoef(self, other):
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"""Check if coefficients match.
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Parameters
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----------
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other : class instance
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The other class must have the ``coef`` attribute.
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Returns
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-------
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bool : boolean
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True if the coefficients are the same, False otherwise.
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"""
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return (
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len(self.coef) == len(other.coef)
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and np.all(self.coef == other.coef)
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)
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def has_samedomain(self, other):
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"""Check if domains match.
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Parameters
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----------
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other : class instance
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The other class must have the ``domain`` attribute.
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Returns
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-------
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bool : boolean
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True if the domains are the same, False otherwise.
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"""
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return np.all(self.domain == other.domain)
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def has_samewindow(self, other):
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"""Check if windows match.
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Parameters
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----------
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other : class instance
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The other class must have the ``window`` attribute.
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Returns
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-------
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bool : boolean
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True if the windows are the same, False otherwise.
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"""
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return np.all(self.window == other.window)
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def has_sametype(self, other):
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"""Check if types match.
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Parameters
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----------
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other : object
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Class instance.
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Returns
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-------
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bool : boolean
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True if other is same class as self
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"""
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return isinstance(other, self.__class__)
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def _get_coefficients(self, other):
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"""Interpret other as polynomial coefficients.
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The `other` argument is checked to see if it is of the same
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class as self with identical domain and window. If so,
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return its coefficients, otherwise return `other`.
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Parameters
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----------
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other : anything
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Object to be checked.
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Returns
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-------
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coef
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The coefficients of`other` if it is a compatible instance,
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of ABCPolyBase, otherwise `other`.
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Raises
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------
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TypeError
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When `other` is an incompatible instance of ABCPolyBase.
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"""
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if isinstance(other, ABCPolyBase):
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if not isinstance(other, self.__class__):
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raise TypeError("Polynomial types differ")
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elif not np.all(self.domain == other.domain):
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raise TypeError("Domains differ")
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elif not np.all(self.window == other.window):
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raise TypeError("Windows differ")
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elif self.symbol != other.symbol:
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raise ValueError("Polynomial symbols differ")
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return other.coef
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return other
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def __init__(self, coef, domain=None, window=None, symbol='x'):
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[coef] = pu.as_series([coef], trim=False)
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self.coef = coef
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if domain is not None:
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[domain] = pu.as_series([domain], trim=False)
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if len(domain) != 2:
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raise ValueError("Domain has wrong number of elements.")
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self.domain = domain
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if window is not None:
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[window] = pu.as_series([window], trim=False)
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if len(window) != 2:
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raise ValueError("Window has wrong number of elements.")
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self.window = window
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# Validation for symbol
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try:
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if not symbol.isidentifier():
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raise ValueError(
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"Symbol string must be a valid Python identifier"
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)
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# If a user passes in something other than a string, the above
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# results in an AttributeError. Catch this and raise a more
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# informative exception
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except AttributeError:
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raise TypeError("Symbol must be a non-empty string")
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self._symbol = symbol
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def __repr__(self):
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coef = repr(self.coef)[6:-1]
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domain = repr(self.domain)[6:-1]
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window = repr(self.window)[6:-1]
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name = self.__class__.__name__
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return (f"{name}({coef}, domain={domain}, window={window}, "
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f"symbol='{self.symbol}')")
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def __format__(self, fmt_str):
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if fmt_str == '':
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return self.__str__()
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if fmt_str not in ('ascii', 'unicode'):
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raise ValueError(
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f"Unsupported format string '{fmt_str}' passed to "
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f"{self.__class__}.__format__. Valid options are "
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f"'ascii' and 'unicode'"
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)
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if fmt_str == 'ascii':
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return self._generate_string(self._str_term_ascii)
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return self._generate_string(self._str_term_unicode)
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def __str__(self):
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if self._use_unicode:
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return self._generate_string(self._str_term_unicode)
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return self._generate_string(self._str_term_ascii)
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def _generate_string(self, term_method):
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"""
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Generate the full string representation of the polynomial, using
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``term_method`` to generate each polynomial term.
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"""
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# Get configuration for line breaks
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linewidth = np.get_printoptions().get('linewidth', 75)
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if linewidth < 1:
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linewidth = 1
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out = pu.format_float(self.coef[0])
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off, scale = self.mapparms()
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scaled_symbol, needs_parens = self._format_term(pu.format_float,
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off, scale)
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if needs_parens:
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scaled_symbol = '(' + scaled_symbol + ')'
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for i, coef in enumerate(self.coef[1:]):
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out += " "
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power = str(i + 1)
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# Polynomial coefficient
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# The coefficient array can be an object array with elements that
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# will raise a TypeError with >= 0 (e.g. strings or Python
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# complex). In this case, represent the coefficient as-is.
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try:
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if coef >= 0:
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next_term = "+ " + pu.format_float(coef, parens=True)
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else:
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next_term = "- " + pu.format_float(-coef, parens=True)
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except TypeError:
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next_term = f"+ {coef}"
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# Polynomial term
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next_term += term_method(power, scaled_symbol)
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# Length of the current line with next term added
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line_len = len(out.split('\n')[-1]) + len(next_term)
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# If not the last term in the polynomial, it will be two
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# characters longer due to the +/- with the next term
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if i < len(self.coef[1:]) - 1:
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line_len += 2
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# Handle linebreaking
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if line_len >= linewidth:
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next_term = next_term.replace(" ", "\n", 1)
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out += next_term
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return out
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@classmethod
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def _str_term_unicode(cls, i, arg_str):
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"""
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String representation of single polynomial term using unicode
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characters for superscripts and subscripts.
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"""
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if cls.basis_name is None:
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raise NotImplementedError(
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"Subclasses must define either a basis_name, or override "
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"_str_term_unicode(cls, i, arg_str)"
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)
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return (f"·{cls.basis_name}{i.translate(cls._subscript_mapping)}"
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f"({arg_str})")
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@classmethod
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def _str_term_ascii(cls, i, arg_str):
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"""
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String representation of a single polynomial term using ** and _ to
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represent superscripts and subscripts, respectively.
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"""
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if cls.basis_name is None:
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raise NotImplementedError(
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"Subclasses must define either a basis_name, or override "
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"_str_term_ascii(cls, i, arg_str)"
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)
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return f" {cls.basis_name}_{i}({arg_str})"
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@classmethod
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def _repr_latex_term(cls, i, arg_str, needs_parens):
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if cls.basis_name is None:
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raise NotImplementedError(
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"Subclasses must define either a basis name, or override "
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"_repr_latex_term(i, arg_str, needs_parens)")
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# since we always add parens, we don't care if the expression needs them
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return f"{{{cls.basis_name}}}_{{{i}}}({arg_str})"
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@staticmethod
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def _repr_latex_scalar(x, parens=False):
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# TODO: we're stuck with disabling math formatting until we handle
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# exponents in this function
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return fr'\text{{{pu.format_float(x, parens=parens)}}}'
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def _format_term(self, scalar_format: Callable, off: float, scale: float):
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""" Format a single term in the expansion """
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if off == 0 and scale == 1:
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term = self.symbol
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needs_parens = False
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elif scale == 1:
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term = f"{scalar_format(off)} + {self.symbol}"
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needs_parens = True
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elif off == 0:
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term = f"{scalar_format(scale)}{self.symbol}"
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needs_parens = True
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else:
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term = (
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f"{scalar_format(off)} + "
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f"{scalar_format(scale)}{self.symbol}"
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)
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needs_parens = True
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return term, needs_parens
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def _repr_latex_(self):
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# get the scaled argument string to the basis functions
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off, scale = self.mapparms()
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term, needs_parens = self._format_term(self._repr_latex_scalar,
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off, scale)
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mute = r"\color{{LightGray}}{{{}}}".format
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parts = []
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for i, c in enumerate(self.coef):
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# prevent duplication of + and - signs
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if i == 0:
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coef_str = f"{self._repr_latex_scalar(c)}"
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elif not isinstance(c, numbers.Real):
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coef_str = f" + ({self._repr_latex_scalar(c)})"
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elif c >= 0:
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coef_str = f" + {self._repr_latex_scalar(c, parens=True)}"
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else:
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coef_str = f" - {self._repr_latex_scalar(-c, parens=True)}"
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# produce the string for the term
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term_str = self._repr_latex_term(i, term, needs_parens)
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if term_str == '1':
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part = coef_str
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else:
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part = rf"{coef_str}\,{term_str}"
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if c == 0:
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part = mute(part)
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parts.append(part)
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if parts:
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body = ''.join(parts)
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else:
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# in case somehow there are no coefficients at all
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body = '0'
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return rf"${self.symbol} \mapsto {body}$"
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|
|
# Pickle and copy
|
|
|
|
def __getstate__(self):
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ret = self.__dict__.copy()
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ret['coef'] = self.coef.copy()
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ret['domain'] = self.domain.copy()
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ret['window'] = self.window.copy()
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ret['symbol'] = self.symbol
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return ret
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|
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def __setstate__(self, dict):
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self.__dict__ = dict
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|
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# Call
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|
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def __call__(self, arg):
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arg = pu.mapdomain(arg, self.domain, self.window)
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return self._val(arg, self.coef)
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|
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def __iter__(self):
|
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return iter(self.coef)
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|
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|
def __len__(self):
|
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return len(self.coef)
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|
|
# Numeric properties.
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|
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def __neg__(self):
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|
return self.__class__(
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-self.coef, self.domain, self.window, self.symbol
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)
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def __pos__(self):
|
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return self
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|
|
|
def __add__(self, other):
|
|
othercoef = self._get_coefficients(other)
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|
try:
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coef = self._add(self.coef, othercoef)
|
|
except Exception:
|
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return NotImplemented
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|
return self.__class__(coef, self.domain, self.window, self.symbol)
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|
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def __sub__(self, other):
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|
othercoef = self._get_coefficients(other)
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try:
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coef = self._sub(self.coef, othercoef)
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|
except Exception:
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return NotImplemented
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|
return self.__class__(coef, self.domain, self.window, self.symbol)
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|
|
|
def __mul__(self, other):
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|
othercoef = self._get_coefficients(other)
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try:
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coef = self._mul(self.coef, othercoef)
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|
except Exception:
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return NotImplemented
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|
return self.__class__(coef, self.domain, self.window, self.symbol)
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|
|
|
def __truediv__(self, other):
|
|
# there is no true divide if the rhs is not a Number, although it
|
|
# could return the first n elements of an infinite series.
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|
# It is hard to see where n would come from, though.
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|
if not isinstance(other, numbers.Number) or isinstance(other, bool):
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|
raise TypeError(
|
|
f"unsupported types for true division: "
|
|
f"'{type(self)}', '{type(other)}'"
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|
)
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|
return self.__floordiv__(other)
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|
|
def __floordiv__(self, other):
|
|
res = self.__divmod__(other)
|
|
if res is NotImplemented:
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|
return res
|
|
return res[0]
|
|
|
|
def __mod__(self, other):
|
|
res = self.__divmod__(other)
|
|
if res is NotImplemented:
|
|
return res
|
|
return res[1]
|
|
|
|
def __divmod__(self, other):
|
|
othercoef = self._get_coefficients(other)
|
|
try:
|
|
quo, rem = self._div(self.coef, othercoef)
|
|
except ZeroDivisionError:
|
|
raise
|
|
except Exception:
|
|
return NotImplemented
|
|
quo = self.__class__(quo, self.domain, self.window, self.symbol)
|
|
rem = self.__class__(rem, self.domain, self.window, self.symbol)
|
|
return quo, rem
|
|
|
|
def __pow__(self, other):
|
|
coef = self._pow(self.coef, other, maxpower=self.maxpower)
|
|
res = self.__class__(coef, self.domain, self.window, self.symbol)
|
|
return res
|
|
|
|
def __radd__(self, other):
|
|
try:
|
|
coef = self._add(other, self.coef)
|
|
except Exception:
|
|
return NotImplemented
|
|
return self.__class__(coef, self.domain, self.window, self.symbol)
|
|
|
|
def __rsub__(self, other):
|
|
try:
|
|
coef = self._sub(other, self.coef)
|
|
except Exception:
|
|
return NotImplemented
|
|
return self.__class__(coef, self.domain, self.window, self.symbol)
|
|
|
|
def __rmul__(self, other):
|
|
try:
|
|
coef = self._mul(other, self.coef)
|
|
except Exception:
|
|
return NotImplemented
|
|
return self.__class__(coef, self.domain, self.window, self.symbol)
|
|
|
|
def __rtruediv__(self, other):
|
|
# An instance of ABCPolyBase is not considered a
|
|
# Number.
|
|
return NotImplemented
|
|
|
|
def __rfloordiv__(self, other):
|
|
res = self.__rdivmod__(other)
|
|
if res is NotImplemented:
|
|
return res
|
|
return res[0]
|
|
|
|
def __rmod__(self, other):
|
|
res = self.__rdivmod__(other)
|
|
if res is NotImplemented:
|
|
return res
|
|
return res[1]
|
|
|
|
def __rdivmod__(self, other):
|
|
try:
|
|
quo, rem = self._div(other, self.coef)
|
|
except ZeroDivisionError:
|
|
raise
|
|
except Exception:
|
|
return NotImplemented
|
|
quo = self.__class__(quo, self.domain, self.window, self.symbol)
|
|
rem = self.__class__(rem, self.domain, self.window, self.symbol)
|
|
return quo, rem
|
|
|
|
def __eq__(self, other):
|
|
res = (isinstance(other, self.__class__) and
|
|
np.all(self.domain == other.domain) and
|
|
np.all(self.window == other.window) and
|
|
(self.coef.shape == other.coef.shape) and
|
|
np.all(self.coef == other.coef) and
|
|
(self.symbol == other.symbol))
|
|
return res
|
|
|
|
def __ne__(self, other):
|
|
return not self.__eq__(other)
|
|
|
|
#
|
|
# Extra methods.
|
|
#
|
|
|
|
def copy(self):
|
|
"""Return a copy.
|
|
|
|
Returns
|
|
-------
|
|
new_series : series
|
|
Copy of self.
|
|
|
|
"""
|
|
return self.__class__(self.coef, self.domain, self.window, self.symbol)
|
|
|
|
def degree(self):
|
|
"""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
|
|
|
|
"""
|
|
return len(self) - 1
|
|
|
|
def cutdeg(self, deg):
|
|
"""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.
|
|
|
|
"""
|
|
return self.truncate(deg + 1)
|
|
|
|
def trim(self, tol=0):
|
|
"""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.
|
|
|
|
"""
|
|
coef = pu.trimcoef(self.coef, tol)
|
|
return self.__class__(coef, self.domain, self.window, self.symbol)
|
|
|
|
def truncate(self, size):
|
|
"""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.
|
|
|
|
"""
|
|
isize = int(size)
|
|
if isize != size or isize < 1:
|
|
raise ValueError("size must be a positive integer")
|
|
if isize >= len(self.coef):
|
|
coef = self.coef
|
|
else:
|
|
coef = self.coef[:isize]
|
|
return self.__class__(coef, self.domain, self.window, self.symbol)
|
|
|
|
def convert(self, domain=None, kind=None, window=None):
|
|
"""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.
|
|
|
|
"""
|
|
if kind is None:
|
|
kind = self.__class__
|
|
if domain is None:
|
|
domain = kind.domain
|
|
if window is None:
|
|
window = kind.window
|
|
return self(kind.identity(domain, window=window, symbol=self.symbol))
|
|
|
|
def mapparms(self):
|
|
"""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
|
|
|
|
"""
|
|
return pu.mapparms(self.domain, self.window)
|
|
|
|
def integ(self, m=1, k=[], lbnd=None):
|
|
"""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.
|
|
|
|
"""
|
|
off, scl = self.mapparms()
|
|
if lbnd is None:
|
|
lbnd = 0
|
|
else:
|
|
lbnd = off + scl * lbnd
|
|
coef = self._int(self.coef, m, k, lbnd, 1. / scl)
|
|
return self.__class__(coef, self.domain, self.window, self.symbol)
|
|
|
|
def deriv(self, m=1):
|
|
"""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.
|
|
|
|
"""
|
|
off, scl = self.mapparms()
|
|
coef = self._der(self.coef, m, scl)
|
|
return self.__class__(coef, self.domain, self.window, self.symbol)
|
|
|
|
def roots(self):
|
|
"""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.
|
|
|
|
"""
|
|
roots = self._roots(self.coef)
|
|
return pu.mapdomain(roots, self.window, self.domain)
|
|
|
|
def linspace(self, n=100, domain=None):
|
|
"""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.
|
|
|
|
"""
|
|
if domain is None:
|
|
domain = self.domain
|
|
x = np.linspace(domain[0], domain[1], n)
|
|
y = self(x)
|
|
return x, y
|
|
|
|
@classmethod
|
|
def fit(cls, x, y, deg, domain=None, rcond=None, full=False, w=None,
|
|
window=None, symbol='x'):
|
|
"""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`.
|
|
|
|
"""
|
|
if domain is None:
|
|
domain = pu.getdomain(x)
|
|
if domain[0] == domain[1]:
|
|
domain[0] -= 1
|
|
domain[1] += 1
|
|
elif isinstance(domain, list) and len(domain) == 0:
|
|
domain = cls.domain
|
|
|
|
if window is None:
|
|
window = cls.window
|
|
|
|
xnew = pu.mapdomain(x, domain, window)
|
|
res = cls._fit(xnew, y, deg, w=w, rcond=rcond, full=full)
|
|
if full:
|
|
[coef, status] = res
|
|
return (
|
|
cls(coef, domain=domain, window=window, symbol=symbol), status
|
|
)
|
|
else:
|
|
coef = res
|
|
return cls(coef, domain=domain, window=window, symbol=symbol)
|
|
|
|
@classmethod
|
|
def fromroots(cls, roots, domain=[], window=None, symbol='x'):
|
|
"""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.
|
|
|
|
"""
|
|
[roots] = pu.as_series([roots], trim=False)
|
|
if domain is None:
|
|
domain = pu.getdomain(roots)
|
|
elif isinstance(domain, list) and len(domain) == 0:
|
|
domain = cls.domain
|
|
|
|
if window is None:
|
|
window = cls.window
|
|
|
|
deg = len(roots)
|
|
off, scl = pu.mapparms(domain, window)
|
|
rnew = off + scl * roots
|
|
coef = cls._fromroots(rnew) / scl**deg
|
|
return cls(coef, domain=domain, window=window, symbol=symbol)
|
|
|
|
@classmethod
|
|
def identity(cls, domain=None, window=None, symbol='x'):
|
|
"""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.
|
|
|
|
"""
|
|
if domain is None:
|
|
domain = cls.domain
|
|
if window is None:
|
|
window = cls.window
|
|
off, scl = pu.mapparms(window, domain)
|
|
coef = cls._line(off, scl)
|
|
return cls(coef, domain, window, symbol)
|
|
|
|
@classmethod
|
|
def basis(cls, deg, domain=None, window=None, symbol='x'):
|
|
"""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.
|
|
|
|
"""
|
|
if domain is None:
|
|
domain = cls.domain
|
|
if window is None:
|
|
window = cls.window
|
|
ideg = int(deg)
|
|
|
|
if ideg != deg or ideg < 0:
|
|
raise ValueError("deg must be non-negative integer")
|
|
return cls([0] * ideg + [1], domain, window, symbol)
|
|
|
|
@classmethod
|
|
def cast(cls, series, domain=None, window=None):
|
|
"""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
|
|
|
|
"""
|
|
if domain is None:
|
|
domain = cls.domain
|
|
if window is None:
|
|
window = cls.window
|
|
return series.convert(domain, cls, window)
|