| Current File : //usr/lib64/python2.7/site-packages/numpy/polynomial/polytemplate.pyo |
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Template for the Chebyshev and Polynomial classes.
This module houses a Python string module Template object (see, e.g.,
http://docs.python.org/library/string.html#template-strings) used by
the `polynomial` and `chebyshev` modules to implement their respective
`Polynomial` and `Chebyshev` classes. It provides a mechanism for easily
creating additional specific polynomial classes (e.g., Legendre, Jacobi,
etc.) in the future, such that all these classes will have a common API.
i����Ni i s
from . importt imports�q
from __future__ import division
import numpy as np
import warnings
REL_IMPORT polyutils as pu
class $name(pu.PolyBase) :
"""A $name series class.
$name instances provide the standard Python numerical methods '+',
'-', '*', '//', '%', 'divmod', '**', and '()' as well as the listed
methods.
Parameters
----------
coef : array_like
$name coefficients, in increasing order. For example,
``(1, 2, 3)`` implies ``P_0 + 2P_1 + 3P_2`` where the
``P_i`` are a graded polynomial basis.
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 $domain.
window : (2,) array_like, optional
Window, see ``domain`` for its use. The default value is $domain.
.. versionadded:: 1.6.0
Attributes
----------
coef : (N,) ndarray
$name coefficients, from low to high.
domain : (2,) ndarray
Domain that is mapped to ``window``.
window : (2,) ndarray
Window that ``domain`` is mapped to.
Class Attributes
----------------
maxpower : int
Maximum power allowed, i.e., the largest number ``n`` such that
``p(x)**n`` is allowed. This is to limit runaway polynomial size.
domain : (2,) ndarray
Default domain of the class.
window : (2,) ndarray
Default window of the class.
Notes
-----
It is important to specify the domain in many cases, for instance in
fitting data, because many of the important properties of the
polynomial basis only hold in a specified interval and consequently
the data must be mapped into that interval in order to benefit.
Examples
--------
"""
# Limit runaway size. T_n^m has degree n*2^m
maxpower = 16
# Default domain
domain = np.array($domain)
# Default window
window = np.array($domain)
# Don't let participate in array operations. Value doesn't matter.
__array_priority__ = 1000
def has_samecoef(self, other):
"""Check if coefficients match.
Parameters
----------
other : class instance
The other class must have the ``coef`` attribute.
Returns
-------
bool : boolean
True if the coefficients are the same, False otherwise.
Notes
-----
.. versionadded:: 1.6.0
"""
if len(self.coef) != len(other.coef):
return False
elif not np.all(self.coef == other.coef):
return False
else:
return True
def has_samedomain(self, other):
"""Check if domains match.
Parameters
----------
other : class instance
The other class must have the ``domain`` attribute.
Returns
-------
bool : boolean
True if the domains are the same, False otherwise.
Notes
-----
.. versionadded:: 1.6.0
"""
return np.all(self.domain == other.domain)
def has_samewindow(self, other):
"""Check if windows match.
Parameters
----------
other : class instance
The other class must have the ``window`` attribute.
Returns
-------
bool : boolean
True if the windows are the same, False otherwise.
Notes
-----
.. versionadded:: 1.6.0
"""
return np.all(self.window == other.window)
def has_sametype(self, other):
"""Check if types match.
Parameters
----------
other : object
Class instance.
Returns
-------
bool : boolean
True if other is same class as self
Notes
-----
.. versionadded:: 1.7.0
"""
return isinstance(other, self.__class__)
def __init__(self, coef, domain=$domain, window=$domain) :
[coef, dom, win] = pu.as_series([coef, domain, window], trim=False)
if len(dom) != 2 :
raise ValueError("Domain has wrong number of elements.")
if len(win) != 2 :
raise ValueError("Window has wrong number of elements.")
self.coef = coef
self.domain = dom
self.window = win
def __repr__(self):
format = "%s(%s, %s, %s)"
coef = repr(self.coef)[6:-1]
domain = repr(self.domain)[6:-1]
window = repr(self.window)[6:-1]
return format % ('$name', coef, domain, window)
def __str__(self) :
format = "%s(%s)"
coef = str(self.coef)
return format % ('$nick', coef)
# 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 ${nick}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) :
"""Returns sum"""
if isinstance(other, pu.PolyBase):
if not self.has_sametype(other):
raise TypeError("Polynomial types differ")
elif not self.has_samedomain(other):
raise TypeError("Domains differ")
elif not self.has_samewindow(other):
raise TypeError("Windows differ")
else:
coef = ${nick}add(self.coef, other.coef)
else :
try :
coef = ${nick}add(self.coef, other)
except :
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __sub__(self, other) :
"""Returns difference"""
if isinstance(other, pu.PolyBase):
if not self.has_sametype(other):
raise TypeError("Polynomial types differ")
elif not self.has_samedomain(other):
raise TypeError("Domains differ")
elif not self.has_samewindow(other):
raise TypeError("Windows differ")
else:
coef = ${nick}sub(self.coef, other.coef)
else :
try :
coef = ${nick}sub(self.coef, other)
except :
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __mul__(self, other) :
"""Returns product"""
if isinstance(other, pu.PolyBase):
if not self.has_sametype(other):
raise TypeError("Polynomial types differ")
elif not self.has_samedomain(other):
raise TypeError("Domains differ")
elif not self.has_samewindow(other):
raise TypeError("Windows differ")
else:
coef = ${nick}mul(self.coef, other.coef)
else :
try :
coef = ${nick}mul(self.coef, other)
except :
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __div__(self, other):
# set to __floordiv__, /, for now.
return self.__floordiv__(other)
def __truediv__(self, other) :
# there is no true divide if the rhs is not a scalar, although it
# could return the first n elements of an infinite series.
# It is hard to see where n would come from, though.
if np.isscalar(other) :
# this might be overly restrictive
coef = self.coef/other
return self.__class__(coef, self.domain, self.window)
else :
return NotImplemented
def __floordiv__(self, other) :
"""Returns the quotient."""
if isinstance(other, pu.PolyBase):
if not self.has_sametype(other):
raise TypeError("Polynomial types differ")
elif not self.has_samedomain(other):
raise TypeError("Domains differ")
elif not self.has_samewindow(other):
raise TypeError("Windows differ")
else:
quo, rem = ${nick}div(self.coef, other.coef)
else :
try :
quo, rem = ${nick}div(self.coef, other)
except :
return NotImplemented
return self.__class__(quo, self.domain, self.window)
def __mod__(self, other) :
"""Returns the remainder."""
if isinstance(other, pu.PolyBase):
if not self.has_sametype(other):
raise TypeError("Polynomial types differ")
elif not self.has_samedomain(other):
raise TypeError("Domains differ")
elif not self.has_samewindow(other):
raise TypeError("Windows differ")
else:
quo, rem = ${nick}div(self.coef, other.coef)
else :
try :
quo, rem = ${nick}div(self.coef, other)
except :
return NotImplemented
return self.__class__(rem, self.domain, self.window)
def __divmod__(self, other) :
"""Returns quo, remainder"""
if isinstance(other, self.__class__) :
if not self.has_samedomain(other):
raise TypeError("Domains are not equal")
elif not self.has_samewindow(other):
raise TypeError("Windows are not equal")
else:
quo, rem = ${nick}div(self.coef, other.coef)
else :
try :
quo, rem = ${nick}div(self.coef, other)
except :
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) :
try :
coef = ${nick}pow(self.coef, other, maxpower = self.maxpower)
except :
raise
return self.__class__(coef, self.domain, self.window)
def __radd__(self, other) :
try :
coef = ${nick}add(other, self.coef)
except :
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __rsub__(self, other):
try :
coef = ${nick}sub(other, self.coef)
except :
return NotImplemented
return self.__class__(coef, self.domain, self.window)
def __rmul__(self, other) :
try :
coef = ${nick}mul(other, self.coef)
except :
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) :
# there is no true divide if the rhs is not a scalar, although it
# could return the first n elements of an infinite series.
# It is hard to see where n would come from, though.
if len(self.coef) == 1 :
try :
quo, rem = ${nick}div(other, self.coef[0])
except :
return NotImplemented
return self.__class__(quo, self.domain, self.window)
def __rfloordiv__(self, other) :
try :
quo, rem = ${nick}div(other, self.coef)
except :
return NotImplemented
return self.__class__(quo, self.domain, self.window)
def __rmod__(self, other) :
try :
quo, rem = ${nick}div(other, self.coef)
except :
return NotImplemented
return self.__class__(rem, self.domain, self.window)
def __rdivmod__(self, other) :
try :
quo, rem = ${nick}div(other, self.coef)
except :
return NotImplemented
quo = self.__class__(quo, self.domain, self.window)
rem = self.__class__(rem, self.domain, self.window)
return quo, rem
# Enhance me
# some augmented arithmetic operations could be added here
def __eq__(self, other) :
res = isinstance(other, self.__class__) and self.has_samecoef(other) and self.has_samedomain(other) and self.has_samewindow(other)
return res
def __ne__(self, other) :
return not self.__eq__(other)
#
# Extra methods.
#
def copy(self) :
"""Return a copy.
Return a copy of the current $name instance.
Returns
-------
new_instance : $name
Copy of current instance.
"""
return self.__class__(self.coef, self.domain, self.window)
def degree(self) :
"""The degree of the series.
Notes
-----
.. versionadded:: 1.5.0
"""
return len(self) - 1
def cutdeg(self, deg) :
"""Truncate series to the given degree.
Reduce the degree of the $name 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_instance : $name
New instance of $name with reduced degree.
Notes
-----
.. versionadded:: 1.5.0
"""
return self.truncate(deg + 1)
def trim(self, tol=0) :
"""Remove small leading coefficients
Remove leading 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 $name 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_instance : $name
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 $name 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_instance : $name
New instance of $name 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 to different class and/or domain.
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_instance : `kind`
The returned class can be of different type than the current
instance and/or have a different domain.
Notes
-----
Conversion between domains and class types can result in
numerically ill defined series.
Examples
--------
"""
if kind is None:
kind = $name
if domain is None:
domain = kind.domain
if window is None:
window = kind.window
return self(kind.identity(domain, window=window))
def mapparms(self) :
"""Return the mapping parameters.
The returned values define a linear map ``off + scl*x`` that is
applied to the input arguments before the series is evaluated. The
map depends on the ``domain`` and ``window``; if the current
``domain`` is equal to the ``window`` the resulting map is the
identity. If the coefficients of the ``$name`` 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 : floats or complex
The mapping function is defined by ``off + scl*x``.
Notes
-----
If the current domain is the interval ``[l_1, r_1]`` and the window
is ``[l_2, r_2]``, then the linear mapping function ``L`` is
defined by the equations::
L(l_1) = l_2
L(r_1) = r_2
"""
return pu.mapparms(self.domain, self.window)
def integ(self, m=1, k=[], lbnd=None) :
"""Integrate.
Return an instance of $name that is the definite integral of the
current series. Refer to `${nick}int` for full documentation.
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
-------
integral : $name
The integral of the series using the same domain.
See Also
--------
${nick}int : similar function.
${nick}der : similar function for derivative.
"""
off, scl = self.mapparms()
if lbnd is None :
lbnd = 0
else :
lbnd = off + scl*lbnd
coef = ${nick}int(self.coef, m, k, lbnd, 1./scl)
return self.__class__(coef, self.domain, self.window)
def deriv(self, m=1):
"""Differentiate.
Return an instance of $name that is the derivative of the current
series. Refer to `${nick}der` for full documentation.
Parameters
----------
m : non-negative int
The number of integrations to perform.
Returns
-------
derivative : $name
The derivative of the series using the same domain.
See Also
--------
${nick}der : similar function.
${nick}int : similar function for integration.
"""
off, scl = self.mapparms()
coef = ${nick}der(self.coef, m, scl)
return self.__class__(coef, self.domain, self.window)
def roots(self) :
"""Return list of roots.
Return ndarray of roots for this series. See `${nick}roots` for
full documentation. Note that the accuracy of the roots is likely to
decrease the further outside the domain they lie.
See Also
--------
${nick}roots : similar function
${nick}fromroots : function to go generate series from roots.
"""
roots = ${nick}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 x, y values at `n` linearly spaced points across domain.
Here y is the value of the polynomial at the points x. By default
the domain is the same as that of the $name instance. This method
is intended mostly as a plotting aid.
Parameters
----------
n : int, optional
Number of point pairs to return. The default value is 100.
domain : {None, array_like}
If not None, the specified domain is used instead of that of
the calling instance. It should be of the form ``[beg,end]``.
The default is None.
Returns
-------
x, y : ndarrays
``x`` is equal to linspace(self.domain[0], self.domain[1], n)
``y`` is the polynomial evaluated at ``x``.
.. versionadded:: 1.5.0
"""
if domain is None:
domain = self.domain
x = np.linspace(domain[0], domain[1], n)
y = self(x)
return x, y
@staticmethod
def fit(x, y, deg, domain=None, rcond=None, full=False, w=None,
window=$domain):
"""Least squares fit to data.
Return a `$name` instance that is the least squares fit to the data
`y` sampled at `x`. Unlike `${nick}fit`, the domain of the returned
instance can be specified and this will often result in a superior
fit with less chance of ill conditioning. Support for NA was added
in version 1.7.0. See `${nick}fit` for full documentation of the
implementation.
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
Degree of the fitting polynomial.
domain : {None, [beg, end], []}, optional
Domain to use for the returned $name instance. If ``None``,
then a minimal domain that covers the points `x` is chosen. If
``[]`` the default domain ``$domain`` is used. The default
value is $domain in numpy 1.4.x and ``None`` in later versions.
The ``'[]`` value 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 $name instance. The default
value is ``$domain``
.. versionadded:: 1.6.0
Returns
-------
least_squares_fit : instance of $name
The $name instance is the least squares fit to the data and
has the domain specified in the call.
[residuals, rank, singular_values, rcond] : only if `full` = True
Residuals of the least squares fit, the effective rank of the
scaled Vandermonde matrix and its singular values, and the
specified value of `rcond`. For more details, see
`linalg.lstsq`.
See Also
--------
${nick}fit : similar function
"""
if domain is None:
domain = pu.getdomain(x)
elif domain == []:
domain = $domain
if window == []:
window = $domain
xnew = pu.mapdomain(x, domain, window)
res = ${nick}fit(xnew, y, deg, w=w, rcond=rcond, full=full)
if full :
[coef, status] = res
return $name(coef, domain=domain, window=window), status
else :
coef = res
return $name(coef, domain=domain, window=window)
@staticmethod
def fromroots(roots, domain=$domain, window=$domain) :
"""Return $name instance with specified roots.
Returns an instance of $name representing the product
``(x - r[0])*(x - r[1])*...*(x - r[n-1])``, where ``r`` is the
list of roots.
Parameters
----------
roots : array_like
List of roots.
Returns
-------
object : $name instance
Series with the specified roots.
See Also
--------
${nick}fromroots : equivalent function
"""
if domain is None :
domain = pu.getdomain(roots)
rnew = pu.mapdomain(roots, domain, window)
coef = ${nick}fromroots(rnew)
return $name(coef, domain=domain, window=window)
@staticmethod
def identity(domain=$domain, window=$domain) :
"""Identity function.
If ``p`` is the returned $name object, then ``p(x) == x`` for all
values of x.
Parameters
----------
domain : array_like
The resulting array must be of the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the domain.
window : array_like
The resulting array must be if the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the window.
Returns
-------
identity : $name instance
"""
off, scl = pu.mapparms(window, domain)
coef = ${nick}line(off, scl)
return $name(coef, domain, window)
@staticmethod
def basis(deg, domain=$domain, window=$domain):
"""$name polynomial of degree `deg`.
Returns an instance of the $name polynomial of degree `d`.
Parameters
----------
deg : int
Degree of the $name polynomial. Must be >= 0.
domain : array_like
The resulting array must be of the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the domain.
window : array_like
The resulting array must be if the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the window.
Returns
p : $name instance
Notes
-----
.. versionadded:: 1.7.0
"""
ideg = int(deg)
if ideg != deg or ideg < 0:
raise ValueError("deg must be non-negative integer")
return $name([0]*ideg + [1], domain, window)
@staticmethod
def cast(series, domain=$domain, window=$domain):
"""Convert instance to equivalent $name series.
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 instance series to be converted.
domain : array_like
The resulting array must be of the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the domain.
window : array_like
The resulting array must be if the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the window.
Returns
p : $name instance
A $name series equal to the `poly` series.
See Also
--------
convert -- similar instance method
Notes
-----
.. versionadded:: 1.7.0
"""
return series.convert(domain, $name, window)
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