JayKimDevolved/deepseek

c011401 verified 3 months ago

30.2 kB

	"""
	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.

	"""
	from __future__ import division, absolute_import, print_function

	import string
	import sys
	import warnings
	from number import Number

	from numpy import ModuleDeprecationWarning

	warnings.warn("The polytemplate module will be removed in Numpy 1.10.0.",
	ModuleDeprecationWarning)

	polytemplate = string.Template('''
	from __future__ import division, absolute_import, print_function
	import numpy as np
	import warnings
	from . 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
	# Not hashable
	__hash__ = None

	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 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, 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) :
	"""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) :
	# An instance of PolyBase is not considered a
	# Number.
	return NotImplemented

	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 type(domain) is list and len(domain) == 0:
	domain = $domain

	if type(window) is list and len(window) == 0:
	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.
	domain : {array_like, None}, optional
	Domain for the resulting instance of $name. If none the domain
	is the interval from the smallest root to the largest. The
	default is $domain.
	window : array_like, optional
	Window for the resulting instance of $name. The default value
	is $domain.

	Returns
	-------
	object : $name instance
	Series with the specified roots.

	See Also
	--------
	${nick}fromroots : equivalent function

	"""
	[roots] = pu.as_series([roots], trim=False)
	if domain is None :
	domain = pu.getdomain(roots)
	deg = len(roots)
	off, scl = pu.mapparms(domain, window)
	rnew = off + scl*roots
	coef = ${nick}fromroots(rnew) / scl**deg
	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)

	''')