"""
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.
"""
from __future__ import division, absolute_import, print_function
from abc import ABCMeta, abstractmethod, abstractproperty
import numbers
import numpy as np
from . import polyutils as pu
__all__ = ['ABCPolyBase']
class ABCPolyBase(object):
"""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.
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.
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.
"""
__metaclass__ = ABCMeta
# Not hashable
__hash__ = None
# Opt out of numpy ufuncs and Python ops with ndarray subclasses.
__array_ufunc__ = None
# Limit runaway size. T_n^m has degree n*m
maxpower = 100
@abstractproperty
def domain(self):
pass
@abstractproperty
def window(self):
pass
@abstractproperty
def nickname(self):
pass
@abstractproperty
def basis_name(self):
pass
@abstractmethod
def _add(self):
pass
@abstractmethod
def _sub(self):
pass
@abstractmethod
def _mul(self):
pass
@abstractmethod
def _div(self):
pass
@abstractmethod
def _pow(self):
pass
@abstractmethod
def _val(self):
pass
@abstractmethod
def _int(self):
pass
@abstractmethod
def _der(self):
pass
@abstractmethod
def _fit(self):
pass
@abstractmethod
def _line(self):
pass
@abstractmethod
def _roots(self):
pass
@abstractmethod
def _fromroots(self):
pass
def has_samecoef(self, other):
"""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.
"""
if len(self.coef) != len(other.coef):
return False
elif not np.all(self.coef == other.coef):
return False
else:
return True
def has_samedomain(self, other):
"""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.
"""
return np.all(self.domain == other.domain)
def has_samewindow(self, other):
"""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.
"""
return np.all(self.window == other.window)
def has_sametype(self, other):
"""Check if types match.
.. versionadded:: 1.7.0
Parameters
----------
other : object
Class instance.
Returns
-------
bool : boolean
True if other is same class as self
"""
return isinstance(other, self.__class__)
def _get_coefficients(self, other):
"""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.
"""
if isinstance(other, ABCPolyBase):
if not isinstance(other, self.__class__):
raise TypeError("Polynomial types differ")
elif not np.all(self.domain == other.domain):
raise TypeError("Domains differ")
elif not np.all(self.window == other.window):
raise TypeError("Windows differ")
return other.coef
return other
def __init__(self, coef, domain=None, window=None):
[coef] = pu.as_series([coef], trim=False)
self.coef = coef
if domain is not None:
[domain] = pu.as_series([domain], trim=False)
if len(domain) != 2:
raise ValueError("Domain has wrong number of elements.")
self.domain = domain
if window is not None:
[window] = pu.as_series([window], trim=False)
if len(window) != 2:
raise ValueError("Window has wrong number of elements.")
self.window = window
def __repr__(self):
format = "%s(%s, domain=%s, window=%s)"
coef = repr(self.coef)[6:-1]
domain = repr(self.domain)[6:-1]
window = repr(self.window)[6:-1]
name = self.__class__.__name__
return format % (name, coef, domain, window)
def __str__(self):
format = "%s(%s)"
coef = str(self.coef)
name = self.nickname
return format % (name, coef)
@classmethod
def _repr_latex_term(cls, i, arg_str, needs_parens):
if cls.basis_name is None:
raise NotImplementedError(
"Subclasses must define either a basis name, or override "
"_repr_latex_term(i, arg_str, needs_parens)")
# since we always add parens, we don't care if the expression needs them
return "{{{basis}}}_{{{i}}}({arg_str})".format(
basis=cls.basis_name, i=i, arg_str=arg_str
)
@staticmethod
def _repr_latex_scalar(x):
# TODO: we're stuck with disabling math formatting until we handle
# exponents in this function
return r'\text{{{}}}'.format(x)
def _repr_latex_(self):
# get the scaled argument string to the basis functions
off, scale = self.mapparms()
if off == 0 and scale == 1:
term = 'x'
needs_parens = False
elif scale == 1:
term = '{} + x'.format(
self._repr_latex_scalar(off)
)
needs_parens = True
elif off == 0:
term = '{}x'.format(
self._repr_latex_scalar(scale)
)
needs_parens = True
else:
term = '{} + {}x'.format(
self._repr_latex_scalar(off),
self._repr_latex_scalar(scale)
)
needs_parens = True
mute = r"\color{{LightGray}}{{{}}}".format
parts = []
for i, c in enumerate(self.coef):
# prevent duplication of + and - signs
if i == 0:
coef_str = '{}'.format(self._repr_latex_scalar(c))
elif not isinstance(c, numbers.Real):
coef_str = ' + ({})'.format(self._repr_latex_scalar(c))
elif not np.signbit(c):
coef_str = ' + {}'.format(self._repr_latex_scalar(c))
else:
coef_str = ' - {}'.format(self._repr_latex_scalar(-c))
# produce the string for the term
term_str = self._repr_latex_term(i, term, needs_parens)
if term_str == '1':
part = coef_str
else:
part = r'{}\,{}'.format(coef_str, term_str)
if c == 0:
part = mute(part)
parts.append(part)
if parts:
body = ''.join(parts)
else:
# in case somehow there are no coefficients at all
body = '0'
return r'$x \mapsto {}$'.format(body)
# Pickle and copy
def __getstate__(self):
ret = self.__dict__.copy()
ret['coef'] = self.coef.copy()
ret['domain'] = self.domain.copy()
ret['window'] = self.window.copy()
return ret
def __setstate__(self, dict):
self.__dict__ = dict
# Call
def __call__(self, arg):
off, scl = pu.mapparms(self.domain, self.window)
arg = off + scl*arg
return self._val(arg, self.coef)
def __iter__(self):
return iter(self.coef)
def __len__(self):
return len(self.coef)
# Numeric properties.
def __neg__(self):
return self.__class__(-self.coef, self.domain, self.window)
def __pos__(self):
return self
def __add__(self, other):
othercoef = self._get_coefficients(other)
try:
coef = self._add(self.coef, othercoef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __sub__(self, other):
othercoef = self._get_coefficients(other)
try:
coef = self._sub(self.coef, othercoef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __mul__(self, other):
othercoef = self._get_coefficients(other)
try:
coef = self._mul(self.coef, othercoef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __div__(self, other):
# this can be removed when python 2 support is dropped.
return self.__floordiv__(other)
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.
# It is hard to see where n would come from, though.
if not isinstance(other, numbers.Number) or isinstance(other, bool):
form = "unsupported types for true division: '%s', '%s'"
raise TypeError(form % (type(self), type(other)))
return self.__floordiv__(other)
def __floordiv__(self, other):
res = self.__divmod__(other)
if res is NotImplemented:
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 as e:
raise e
except Exception:
return NotImplemented
quo = self.__class__(quo, self.domain, self.window)
rem = self.__class__(rem, self.domain, self.window)
return quo, rem
def __pow__(self, other):
coef = self._pow(self.coef, other, maxpower=self.maxpower)
res = self.__class__(coef, self.domain, self.window)
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)
def __rsub__(self, other):
try:
coef = self._sub(other, self.coef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __rmul__(self, other):
try:
coef = self._mul(other, self.coef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __rdiv__(self, other):
# set to __floordiv__ /.
return self.__rfloordiv__(other)
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 as e:
raise e
except Exception:
return NotImplemented
quo = self.__class__(quo, self.domain, self.window)
rem = self.__class__(rem, self.domain, self.window)
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))
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)
def degree(self):
"""The degree of the series.
.. versionadded:: 1.5.0
Returns
-------
degree : int
Degree of the series, one less than the number of coefficients.
"""
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.
.. 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.
"""
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
Contains the new set of coefficients.
"""
coef = pu.trimcoef(self.coef, tol)
return self.__class__(coef, self.domain, self.window)
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)
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.
Examples
--------
"""
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))
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)
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)
def roots(self):
"""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.
"""
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.
.. 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.
"""
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):
"""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,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
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 contribution of each point
``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
weights are chosen so that the errors of the products
``w[i]*y[i]`` all have the same variance. 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
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)
elif type(domain) is 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), status
else:
coef = res
return cls(coef, domain=domain, window=window)
@classmethod
def fromroots(cls, roots, domain=[], window=None):
"""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.
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 type(domain) is 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)
@classmethod
def identity(cls, domain=None, window=None):
"""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.
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)
@classmethod
def basis(cls, deg, domain=None, window=None):
"""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.
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)
@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.
.. 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
"""
if domain is None:
domain = cls.domain
if window is None:
window = cls.window
return series.convert(domain, cls, window)