Chris@87: """
Chris@87: Template for the Chebyshev and Polynomial classes.
Chris@87: 
Chris@87: This module houses a Python string module Template object (see, e.g.,
Chris@87: http://docs.python.org/library/string.html#template-strings) used by
Chris@87: the `polynomial` and `chebyshev` modules to implement their respective
Chris@87: `Polynomial` and `Chebyshev` classes.  It provides a mechanism for easily
Chris@87: creating additional specific polynomial classes (e.g., Legendre, Jacobi,
Chris@87: etc.) in the future, such that all these classes will have a common API.
Chris@87: 
Chris@87: """
Chris@87: from __future__ import division, absolute_import, print_function
Chris@87: 
Chris@87: import string
Chris@87: import sys
Chris@87: import warnings
Chris@87: from number import Number
Chris@87: 
Chris@87: from numpy import ModuleDeprecationWarning
Chris@87: 
Chris@87: warnings.warn("The polytemplate module will be removed in Numpy 1.10.0.",
Chris@87:               ModuleDeprecationWarning)
Chris@87: 
Chris@87: polytemplate = string.Template('''
Chris@87: from __future__ import division, absolute_import, print_function
Chris@87: import numpy as np
Chris@87: import warnings
Chris@87: from . import polyutils as pu
Chris@87: 
Chris@87: class $name(pu.PolyBase) :
Chris@87:     """A $name series class.
Chris@87: 
Chris@87:     $name instances provide the standard Python numerical methods '+',
Chris@87:     '-', '*', '//', '%', 'divmod', '**', and '()' as well as the listed
Chris@87:     methods.
Chris@87: 
Chris@87:     Parameters
Chris@87:     ----------
Chris@87:     coef : array_like
Chris@87:         $name coefficients, in increasing order.  For example,
Chris@87:         ``(1, 2, 3)`` implies ``P_0 + 2P_1 + 3P_2`` where the
Chris@87:         ``P_i`` are a graded polynomial basis.
Chris@87:     domain : (2,) array_like, optional
Chris@87:         Domain to use. The interval ``[domain[0], domain[1]]`` is mapped to
Chris@87:         the interval ``[window[0], window[1]]`` by shifting and scaling.
Chris@87:         The default value is $domain.
Chris@87:     window : (2,) array_like, optional
Chris@87:         Window, see ``domain`` for its use. The default value is $domain.
Chris@87:         .. versionadded:: 1.6.0
Chris@87: 
Chris@87:     Attributes
Chris@87:     ----------
Chris@87:     coef : (N,) ndarray
Chris@87:         $name coefficients, from low to high.
Chris@87:     domain : (2,) ndarray
Chris@87:         Domain that is mapped to ``window``.
Chris@87:     window : (2,) ndarray
Chris@87:         Window that ``domain`` is mapped to.
Chris@87: 
Chris@87:     Class Attributes
Chris@87:     ----------------
Chris@87:     maxpower : int
Chris@87:         Maximum power allowed, i.e., the largest number ``n`` such that
Chris@87:         ``p(x)**n`` is allowed. This is to limit runaway polynomial size.
Chris@87:     domain : (2,) ndarray
Chris@87:         Default domain of the class.
Chris@87:     window : (2,) ndarray
Chris@87:         Default window of the class.
Chris@87: 
Chris@87:     Notes
Chris@87:     -----
Chris@87:     It is important to specify the domain in many cases, for instance in
Chris@87:     fitting data, because many of the important properties of the
Chris@87:     polynomial basis only hold in a specified interval and consequently
Chris@87:     the data must be mapped into that interval in order to benefit.
Chris@87: 
Chris@87:     Examples
Chris@87:     --------
Chris@87: 
Chris@87:     """
Chris@87:     # Limit runaway size. T_n^m has degree n*2^m
Chris@87:     maxpower = 16
Chris@87:     # Default domain
Chris@87:     domain = np.array($domain)
Chris@87:     # Default window
Chris@87:     window = np.array($domain)
Chris@87:     # Don't let participate in array operations. Value doesn't matter.
Chris@87:     __array_priority__ = 1000
Chris@87:     # Not hashable
Chris@87:     __hash__ = None
Chris@87: 
Chris@87:     def has_samecoef(self, other):
Chris@87:         """Check if coefficients match.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         other : class instance
Chris@87:             The other class must have the ``coef`` attribute.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         bool : boolean
Chris@87:             True if the coefficients are the same, False otherwise.
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         .. versionadded:: 1.6.0
Chris@87: 
Chris@87:         """
Chris@87:         if len(self.coef) != len(other.coef):
Chris@87:             return False
Chris@87:         elif not np.all(self.coef == other.coef):
Chris@87:             return False
Chris@87:         else:
Chris@87:             return True
Chris@87: 
Chris@87:     def has_samedomain(self, other):
Chris@87:         """Check if domains match.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         other : class instance
Chris@87:             The other class must have the ``domain`` attribute.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         bool : boolean
Chris@87:             True if the domains are the same, False otherwise.
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         .. versionadded:: 1.6.0
Chris@87: 
Chris@87:         """
Chris@87:         return np.all(self.domain == other.domain)
Chris@87: 
Chris@87:     def has_samewindow(self, other):
Chris@87:         """Check if windows match.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         other : class instance
Chris@87:             The other class must have the ``window`` attribute.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         bool : boolean
Chris@87:             True if the windows are the same, False otherwise.
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         .. versionadded:: 1.6.0
Chris@87: 
Chris@87:         """
Chris@87:         return np.all(self.window == other.window)
Chris@87: 
Chris@87:     def has_sametype(self, other):
Chris@87:         """Check if types match.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         other : object
Chris@87:             Class instance.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         bool : boolean
Chris@87:             True if other is same class as self
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         .. versionadded:: 1.7.0
Chris@87: 
Chris@87:         """
Chris@87:         return isinstance(other, self.__class__)
Chris@87: 
Chris@87:     def __init__(self, coef, domain=$domain, window=$domain) :
Chris@87:         [coef, dom, win] = pu.as_series([coef, domain, window], trim=False)
Chris@87:         if len(dom) != 2 :
Chris@87:             raise ValueError("Domain has wrong number of elements.")
Chris@87:         if len(win) != 2 :
Chris@87:             raise ValueError("Window has wrong number of elements.")
Chris@87:         self.coef = coef
Chris@87:         self.domain = dom
Chris@87:         self.window = win
Chris@87: 
Chris@87:     def __repr__(self):
Chris@87:         format = "%s(%s, %s, %s)"
Chris@87:         coef = repr(self.coef)[6:-1]
Chris@87:         domain = repr(self.domain)[6:-1]
Chris@87:         window = repr(self.window)[6:-1]
Chris@87:         return format % ('$name', coef, domain, window)
Chris@87: 
Chris@87:     def __str__(self) :
Chris@87:         format = "%s(%s)"
Chris@87:         coef = str(self.coef)
Chris@87:         return format % ('$nick', coef)
Chris@87: 
Chris@87:     # Pickle and copy
Chris@87: 
Chris@87:     def __getstate__(self) :
Chris@87:         ret = self.__dict__.copy()
Chris@87:         ret['coef'] = self.coef.copy()
Chris@87:         ret['domain'] = self.domain.copy()
Chris@87:         ret['window'] = self.window.copy()
Chris@87:         return ret
Chris@87: 
Chris@87:     def __setstate__(self, dict) :
Chris@87:         self.__dict__ = dict
Chris@87: 
Chris@87:     # Call
Chris@87: 
Chris@87:     def __call__(self, arg) :
Chris@87:         off, scl = pu.mapparms(self.domain, self.window)
Chris@87:         arg = off + scl*arg
Chris@87:         return ${nick}val(arg, self.coef)
Chris@87: 
Chris@87:     def __iter__(self) :
Chris@87:         return iter(self.coef)
Chris@87: 
Chris@87:     def __len__(self) :
Chris@87:         return len(self.coef)
Chris@87: 
Chris@87:     # Numeric properties.
Chris@87: 
Chris@87:     def __neg__(self) :
Chris@87:         return self.__class__(-self.coef, self.domain, self.window)
Chris@87: 
Chris@87:     def __pos__(self) :
Chris@87:         return self
Chris@87: 
Chris@87:     def __add__(self, other) :
Chris@87:         """Returns sum"""
Chris@87:         if isinstance(other, pu.PolyBase):
Chris@87:             if not self.has_sametype(other):
Chris@87:                 raise TypeError("Polynomial types differ")
Chris@87:             elif not self.has_samedomain(other):
Chris@87:                 raise TypeError("Domains differ")
Chris@87:             elif not self.has_samewindow(other):
Chris@87:                 raise TypeError("Windows differ")
Chris@87:             else:
Chris@87:                 coef = ${nick}add(self.coef, other.coef)
Chris@87:         else :
Chris@87:             try :
Chris@87:                 coef = ${nick}add(self.coef, other)
Chris@87:             except :
Chris@87:                 return NotImplemented
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def __sub__(self, other) :
Chris@87:         """Returns difference"""
Chris@87:         if isinstance(other, pu.PolyBase):
Chris@87:             if not self.has_sametype(other):
Chris@87:                 raise TypeError("Polynomial types differ")
Chris@87:             elif not self.has_samedomain(other):
Chris@87:                 raise TypeError("Domains differ")
Chris@87:             elif not self.has_samewindow(other):
Chris@87:                 raise TypeError("Windows differ")
Chris@87:             else:
Chris@87:                 coef = ${nick}sub(self.coef, other.coef)
Chris@87:         else :
Chris@87:             try :
Chris@87:                 coef = ${nick}sub(self.coef, other)
Chris@87:             except :
Chris@87:                 return NotImplemented
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def __mul__(self, other) :
Chris@87:         """Returns product"""
Chris@87:         if isinstance(other, pu.PolyBase):
Chris@87:             if not self.has_sametype(other):
Chris@87:                 raise TypeError("Polynomial types differ")
Chris@87:             elif not self.has_samedomain(other):
Chris@87:                 raise TypeError("Domains differ")
Chris@87:             elif not self.has_samewindow(other):
Chris@87:                 raise TypeError("Windows differ")
Chris@87:             else:
Chris@87:                 coef = ${nick}mul(self.coef, other.coef)
Chris@87:         else :
Chris@87:             try :
Chris@87:                 coef = ${nick}mul(self.coef, other)
Chris@87:             except :
Chris@87:                 return NotImplemented
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def __div__(self, other):
Chris@87:         # set to __floordiv__,  /, for now.
Chris@87:         return self.__floordiv__(other)
Chris@87: 
Chris@87:     def __truediv__(self, other) :
Chris@87:         # there is no true divide if the rhs is not a Number, although it
Chris@87:         # could return the first n elements of an infinite series.
Chris@87:         # It is hard to see where n would come from, though.
Chris@87:         if not isinstance(other, Number) or isinstance(other, bool):
Chris@87:             form = "unsupported types for true division: '%s', '%s'"
Chris@87:             raise TypeError(form % (type(self), type(other)))
Chris@87:         return self.__floordiv__(other)
Chris@87: 
Chris@87:     def __floordiv__(self, other) :
Chris@87:         """Returns the quotient."""
Chris@87:         if isinstance(other, pu.PolyBase):
Chris@87:             if not self.has_sametype(other):
Chris@87:                 raise TypeError("Polynomial types differ")
Chris@87:             elif not self.has_samedomain(other):
Chris@87:                 raise TypeError("Domains differ")
Chris@87:             elif not self.has_samewindow(other):
Chris@87:                 raise TypeError("Windows differ")
Chris@87:             else:
Chris@87:                 quo, rem = ${nick}div(self.coef, other.coef)
Chris@87:         else :
Chris@87:             try :
Chris@87:                 quo, rem = ${nick}div(self.coef, other)
Chris@87:             except :
Chris@87:                 return NotImplemented
Chris@87:         return self.__class__(quo, self.domain, self.window)
Chris@87: 
Chris@87:     def __mod__(self, other) :
Chris@87:         """Returns the remainder."""
Chris@87:         if isinstance(other, pu.PolyBase):
Chris@87:             if not self.has_sametype(other):
Chris@87:                 raise TypeError("Polynomial types differ")
Chris@87:             elif not self.has_samedomain(other):
Chris@87:                 raise TypeError("Domains differ")
Chris@87:             elif not self.has_samewindow(other):
Chris@87:                 raise TypeError("Windows differ")
Chris@87:             else:
Chris@87:                 quo, rem = ${nick}div(self.coef, other.coef)
Chris@87:         else :
Chris@87:             try :
Chris@87:                 quo, rem = ${nick}div(self.coef, other)
Chris@87:             except :
Chris@87:                 return NotImplemented
Chris@87:         return self.__class__(rem, self.domain, self.window)
Chris@87: 
Chris@87:     def __divmod__(self, other) :
Chris@87:         """Returns quo, remainder"""
Chris@87:         if isinstance(other, self.__class__) :
Chris@87:             if not self.has_samedomain(other):
Chris@87:                 raise TypeError("Domains are not equal")
Chris@87:             elif not self.has_samewindow(other):
Chris@87:                 raise TypeError("Windows are not equal")
Chris@87:             else:
Chris@87:                 quo, rem = ${nick}div(self.coef, other.coef)
Chris@87:         else :
Chris@87:             try :
Chris@87:                 quo, rem = ${nick}div(self.coef, other)
Chris@87:             except :
Chris@87:                 return NotImplemented
Chris@87:         quo = self.__class__(quo, self.domain, self.window)
Chris@87:         rem = self.__class__(rem, self.domain, self.window)
Chris@87:         return quo, rem
Chris@87: 
Chris@87:     def __pow__(self, other) :
Chris@87:         try :
Chris@87:             coef = ${nick}pow(self.coef, other, maxpower = self.maxpower)
Chris@87:         except :
Chris@87:             raise
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def __radd__(self, other) :
Chris@87:         try :
Chris@87:             coef = ${nick}add(other, self.coef)
Chris@87:         except :
Chris@87:             return NotImplemented
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def __rsub__(self, other):
Chris@87:         try :
Chris@87:             coef = ${nick}sub(other, self.coef)
Chris@87:         except :
Chris@87:             return NotImplemented
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def __rmul__(self, other) :
Chris@87:         try :
Chris@87:             coef = ${nick}mul(other, self.coef)
Chris@87:         except :
Chris@87:             return NotImplemented
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def __rdiv__(self, other):
Chris@87:         # set to __floordiv__ /.
Chris@87:         return self.__rfloordiv__(other)
Chris@87: 
Chris@87:     def __rtruediv__(self, other) :
Chris@87:         # An instance of PolyBase is not considered a
Chris@87:         # Number.
Chris@87:         return NotImplemented
Chris@87: 
Chris@87:     def __rfloordiv__(self, other) :
Chris@87:         try :
Chris@87:             quo, rem = ${nick}div(other, self.coef)
Chris@87:         except:
Chris@87:             return NotImplemented
Chris@87:         return self.__class__(quo, self.domain, self.window)
Chris@87: 
Chris@87:     def __rmod__(self, other) :
Chris@87:         try :
Chris@87:             quo, rem = ${nick}div(other, self.coef)
Chris@87:         except :
Chris@87:             return NotImplemented
Chris@87:         return self.__class__(rem, self.domain, self.window)
Chris@87: 
Chris@87:     def __rdivmod__(self, other) :
Chris@87:         try :
Chris@87:             quo, rem = ${nick}div(other, self.coef)
Chris@87:         except :
Chris@87:             return NotImplemented
Chris@87:         quo = self.__class__(quo, self.domain, self.window)
Chris@87:         rem = self.__class__(rem, self.domain, self.window)
Chris@87:         return quo, rem
Chris@87: 
Chris@87:     # Enhance me
Chris@87:     # some augmented arithmetic operations could be added here
Chris@87: 
Chris@87:     def __eq__(self, other) :
Chris@87:         res = isinstance(other, self.__class__) \
Chris@87:                 and self.has_samecoef(other) \
Chris@87:                 and self.has_samedomain(other) \
Chris@87:                 and self.has_samewindow(other)
Chris@87:         return res
Chris@87: 
Chris@87:     def __ne__(self, other) :
Chris@87:         return not self.__eq__(other)
Chris@87: 
Chris@87:     #
Chris@87:     # Extra methods.
Chris@87:     #
Chris@87: 
Chris@87:     def copy(self) :
Chris@87:         """Return a copy.
Chris@87: 
Chris@87:         Return a copy of the current $name instance.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         new_instance : $name
Chris@87:             Copy of current instance.
Chris@87: 
Chris@87:         """
Chris@87:         return self.__class__(self.coef, self.domain, self.window)
Chris@87: 
Chris@87:     def degree(self) :
Chris@87:         """The degree of the series.
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         .. versionadded:: 1.5.0
Chris@87: 
Chris@87:         """
Chris@87:         return len(self) - 1
Chris@87: 
Chris@87:     def cutdeg(self, deg) :
Chris@87:         """Truncate series to the given degree.
Chris@87: 
Chris@87:         Reduce the degree of the $name series to `deg` by discarding the
Chris@87:         high order terms. If `deg` is greater than the current degree a
Chris@87:         copy of the current series is returned. This can be useful in least
Chris@87:         squares where the coefficients of the high degree terms may be very
Chris@87:         small.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         deg : non-negative int
Chris@87:             The series is reduced to degree `deg` by discarding the high
Chris@87:             order terms. The value of `deg` must be a non-negative integer.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         new_instance : $name
Chris@87:             New instance of $name with reduced degree.
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         .. versionadded:: 1.5.0
Chris@87: 
Chris@87:         """
Chris@87:         return self.truncate(deg + 1)
Chris@87: 
Chris@87:     def trim(self, tol=0) :
Chris@87:         """Remove small leading coefficients
Chris@87: 
Chris@87:         Remove leading coefficients until a coefficient is reached whose
Chris@87:         absolute value greater than `tol` or the beginning of the series is
Chris@87:         reached. If all the coefficients would be removed the series is set to
Chris@87:         ``[0]``. A new $name instance is returned with the new coefficients.
Chris@87:         The current instance remains unchanged.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         tol : non-negative number.
Chris@87:             All trailing coefficients less than `tol` will be removed.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         new_instance : $name
Chris@87:             Contains the new set of coefficients.
Chris@87: 
Chris@87:         """
Chris@87:         coef = pu.trimcoef(self.coef, tol)
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def truncate(self, size) :
Chris@87:         """Truncate series to length `size`.
Chris@87: 
Chris@87:         Reduce the $name series to length `size` by discarding the high
Chris@87:         degree terms. The value of `size` must be a positive integer. This
Chris@87:         can be useful in least squares where the coefficients of the
Chris@87:         high degree terms may be very small.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         size : positive int
Chris@87:             The series is reduced to length `size` by discarding the high
Chris@87:             degree terms. The value of `size` must be a positive integer.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         new_instance : $name
Chris@87:             New instance of $name with truncated coefficients.
Chris@87: 
Chris@87:         """
Chris@87:         isize = int(size)
Chris@87:         if isize != size or isize < 1 :
Chris@87:             raise ValueError("size must be a positive integer")
Chris@87:         if isize >= len(self.coef) :
Chris@87:             coef = self.coef
Chris@87:         else :
Chris@87:             coef = self.coef[:isize]
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def convert(self, domain=None, kind=None, window=None) :
Chris@87:         """Convert to different class and/or domain.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         domain : array_like, optional
Chris@87:             The domain of the converted series. If the value is None,
Chris@87:             the default domain of `kind` is used.
Chris@87:         kind : class, optional
Chris@87:             The polynomial series type class to which the current instance
Chris@87:             should be converted. If kind is None, then the class of the
Chris@87:             current instance is used.
Chris@87:         window : array_like, optional
Chris@87:             The window of the converted series. If the value is None,
Chris@87:             the default window of `kind` is used.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         new_series_instance : `kind`
Chris@87:             The returned class can be of different type than the current
Chris@87:             instance and/or have a different domain.
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         Conversion between domains and class types can result in
Chris@87:         numerically ill defined series.
Chris@87: 
Chris@87:         Examples
Chris@87:         --------
Chris@87: 
Chris@87:         """
Chris@87:         if kind is None:
Chris@87:             kind = $name
Chris@87:         if domain is None:
Chris@87:             domain = kind.domain
Chris@87:         if window is None:
Chris@87:             window = kind.window
Chris@87:         return self(kind.identity(domain, window=window))
Chris@87: 
Chris@87:     def mapparms(self) :
Chris@87:         """Return the mapping parameters.
Chris@87: 
Chris@87:         The returned values define a linear map ``off + scl*x`` that is
Chris@87:         applied to the input arguments before the series is evaluated. The
Chris@87:         map depends on the ``domain`` and ``window``; if the current
Chris@87:         ``domain`` is equal to the ``window`` the resulting map is the
Chris@87:         identity.  If the coefficients of the ``$name`` instance are to be
Chris@87:         used by themselves outside this class, then the linear function
Chris@87:         must be substituted for the ``x`` in the standard representation of
Chris@87:         the base polynomials.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         off, scl : floats or complex
Chris@87:             The mapping function is defined by ``off + scl*x``.
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         If the current domain is the interval ``[l_1, r_1]`` and the window
Chris@87:         is ``[l_2, r_2]``, then the linear mapping function ``L`` is
Chris@87:         defined by the equations::
Chris@87: 
Chris@87:             L(l_1) = l_2
Chris@87:             L(r_1) = r_2
Chris@87: 
Chris@87:         """
Chris@87:         return pu.mapparms(self.domain, self.window)
Chris@87: 
Chris@87:     def integ(self, m=1, k=[], lbnd=None) :
Chris@87:         """Integrate.
Chris@87: 
Chris@87:         Return an instance of $name that is the definite integral of the
Chris@87:         current series. Refer to `${nick}int` for full documentation.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         m : non-negative int
Chris@87:             The number of integrations to perform.
Chris@87:         k : array_like
Chris@87:             Integration constants. The first constant is applied to the
Chris@87:             first integration, the second to the second, and so on. The
Chris@87:             list of values must less than or equal to `m` in length and any
Chris@87:             missing values are set to zero.
Chris@87:         lbnd : Scalar
Chris@87:             The lower bound of the definite integral.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         integral : $name
Chris@87:             The integral of the series using the same domain.
Chris@87: 
Chris@87:         See Also
Chris@87:         --------
Chris@87:         ${nick}int : similar function.
Chris@87:         ${nick}der : similar function for derivative.
Chris@87: 
Chris@87:         """
Chris@87:         off, scl = self.mapparms()
Chris@87:         if lbnd is None :
Chris@87:             lbnd = 0
Chris@87:         else :
Chris@87:             lbnd = off + scl*lbnd
Chris@87:         coef = ${nick}int(self.coef, m, k, lbnd, 1./scl)
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def deriv(self, m=1):
Chris@87:         """Differentiate.
Chris@87: 
Chris@87:         Return an instance of $name that is the derivative of the current
Chris@87:         series.  Refer to `${nick}der` for full documentation.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         m : non-negative int
Chris@87:             The number of integrations to perform.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         derivative : $name
Chris@87:             The derivative of the series using the same domain.
Chris@87: 
Chris@87:         See Also
Chris@87:         --------
Chris@87:         ${nick}der : similar function.
Chris@87:         ${nick}int : similar function for integration.
Chris@87: 
Chris@87:         """
Chris@87:         off, scl = self.mapparms()
Chris@87:         coef = ${nick}der(self.coef, m, scl)
Chris@87:         return self.__class__(coef, self.domain, self.window)
Chris@87: 
Chris@87:     def roots(self) :
Chris@87:         """Return list of roots.
Chris@87: 
Chris@87:         Return ndarray of roots for this series. See `${nick}roots` for
Chris@87:         full documentation. Note that the accuracy of the roots is likely to
Chris@87:         decrease the further outside the domain they lie.
Chris@87: 
Chris@87:         See Also
Chris@87:         --------
Chris@87:         ${nick}roots : similar function
Chris@87:         ${nick}fromroots : function to go generate series from roots.
Chris@87: 
Chris@87:         """
Chris@87:         roots = ${nick}roots(self.coef)
Chris@87:         return pu.mapdomain(roots, self.window, self.domain)
Chris@87: 
Chris@87:     def linspace(self, n=100, domain=None):
Chris@87:         """Return x,y values at equally spaced points in domain.
Chris@87: 
Chris@87:         Returns x, y values at `n` linearly spaced points across domain.
Chris@87:         Here y is the value of the polynomial at the points x. By default
Chris@87:         the domain is the same as that of the $name instance.  This method
Chris@87:         is intended mostly as a plotting aid.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         n : int, optional
Chris@87:             Number of point pairs to return. The default value is 100.
Chris@87:         domain : {None, array_like}
Chris@87:             If not None, the specified domain is used instead of that of
Chris@87:             the calling instance. It should be of the form ``[beg,end]``.
Chris@87:             The default is None.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         x, y : ndarrays
Chris@87:             ``x`` is equal to linspace(self.domain[0], self.domain[1], n)
Chris@87:             ``y`` is the polynomial evaluated at ``x``.
Chris@87: 
Chris@87:         .. versionadded:: 1.5.0
Chris@87: 
Chris@87:         """
Chris@87:         if domain is None:
Chris@87:             domain = self.domain
Chris@87:         x = np.linspace(domain[0], domain[1], n)
Chris@87:         y = self(x)
Chris@87:         return x, y
Chris@87: 
Chris@87: 
Chris@87: 
Chris@87:     @staticmethod
Chris@87:     def fit(x, y, deg, domain=None, rcond=None, full=False, w=None,
Chris@87:         window=$domain):
Chris@87:         """Least squares fit to data.
Chris@87: 
Chris@87:         Return a `$name` instance that is the least squares fit to the data
Chris@87:         `y` sampled at `x`. Unlike `${nick}fit`, the domain of the returned
Chris@87:         instance can be specified and this will often result in a superior
Chris@87:         fit with less chance of ill conditioning. Support for NA was added
Chris@87:         in version 1.7.0. See `${nick}fit` for full documentation of the
Chris@87:         implementation.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         x : array_like, shape (M,)
Chris@87:             x-coordinates of the M sample points ``(x[i], y[i])``.
Chris@87:         y : array_like, shape (M,) or (M, K)
Chris@87:             y-coordinates of the sample points. Several data sets of sample
Chris@87:             points sharing the same x-coordinates can be fitted at once by
Chris@87:             passing in a 2D-array that contains one dataset per column.
Chris@87:         deg : int
Chris@87:             Degree of the fitting polynomial.
Chris@87:         domain : {None, [beg, end], []}, optional
Chris@87:             Domain to use for the returned $name instance. If ``None``,
Chris@87:             then a minimal domain that covers the points `x` is chosen.  If
Chris@87:             ``[]`` the default domain ``$domain`` is used. The default
Chris@87:             value is $domain in numpy 1.4.x and ``None`` in later versions.
Chris@87:             The ``[]`` value was added in numpy 1.5.0.
Chris@87:         rcond : float, optional
Chris@87:             Relative condition number of the fit. Singular values smaller
Chris@87:             than this relative to the largest singular value will be
Chris@87:             ignored. The default value is len(x)*eps, where eps is the
Chris@87:             relative precision of the float type, about 2e-16 in most
Chris@87:             cases.
Chris@87:         full : bool, optional
Chris@87:             Switch determining nature of return value. When it is False
Chris@87:             (the default) just the coefficients are returned, when True
Chris@87:             diagnostic information from the singular value decomposition is
Chris@87:             also returned.
Chris@87:         w : array_like, shape (M,), optional
Chris@87:             Weights. If not None the contribution of each point
Chris@87:             ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
Chris@87:             weights are chosen so that the errors of the products
Chris@87:             ``w[i]*y[i]`` all have the same variance.  The default value is
Chris@87:             None.
Chris@87:             .. versionadded:: 1.5.0
Chris@87:         window : {[beg, end]}, optional
Chris@87:             Window to use for the returned $name instance. The default
Chris@87:             value is ``$domain``
Chris@87:             .. versionadded:: 1.6.0
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         least_squares_fit : instance of $name
Chris@87:             The $name instance is the least squares fit to the data and
Chris@87:             has the domain specified in the call.
Chris@87: 
Chris@87:         [residuals, rank, singular_values, rcond] : only if `full` = True
Chris@87:             Residuals of the least squares fit, the effective rank of the
Chris@87:             scaled Vandermonde matrix and its singular values, and the
Chris@87:             specified value of `rcond`. For more details, see
Chris@87:             `linalg.lstsq`.
Chris@87: 
Chris@87:         See Also
Chris@87:         --------
Chris@87:         ${nick}fit : similar function
Chris@87: 
Chris@87:         """
Chris@87:         if domain is None:
Chris@87:             domain = pu.getdomain(x)
Chris@87:         elif type(domain) is list and len(domain) == 0:
Chris@87:             domain = $domain
Chris@87: 
Chris@87:         if type(window) is list and len(window) == 0:
Chris@87:             window = $domain
Chris@87: 
Chris@87:         xnew = pu.mapdomain(x, domain, window)
Chris@87:         res = ${nick}fit(xnew, y, deg, w=w, rcond=rcond, full=full)
Chris@87:         if full :
Chris@87:             [coef, status] = res
Chris@87:             return $name(coef, domain=domain, window=window), status
Chris@87:         else :
Chris@87:             coef = res
Chris@87:             return $name(coef, domain=domain, window=window)
Chris@87: 
Chris@87:     @staticmethod
Chris@87:     def fromroots(roots, domain=$domain, window=$domain) :
Chris@87:         """Return $name instance with specified roots.
Chris@87: 
Chris@87:         Returns an instance of $name representing the product
Chris@87:         ``(x - r[0])*(x - r[1])*...*(x - r[n-1])``, where ``r`` is the
Chris@87:         list of roots.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         roots : array_like
Chris@87:             List of roots.
Chris@87:         domain : {array_like, None}, optional
Chris@87:             Domain for the resulting instance of $name. If none the domain
Chris@87:             is the interval from the smallest root to the largest. The
Chris@87:             default is $domain.
Chris@87:         window : array_like, optional
Chris@87:             Window for the resulting instance of $name. The default value
Chris@87:             is $domain.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         object : $name instance
Chris@87:             Series with the specified roots.
Chris@87: 
Chris@87:         See Also
Chris@87:         --------
Chris@87:         ${nick}fromroots : equivalent function
Chris@87: 
Chris@87:         """
Chris@87:         [roots] = pu.as_series([roots], trim=False)
Chris@87:         if domain is None :
Chris@87:             domain = pu.getdomain(roots)
Chris@87:         deg = len(roots)
Chris@87:         off, scl = pu.mapparms(domain, window)
Chris@87:         rnew = off + scl*roots
Chris@87:         coef = ${nick}fromroots(rnew) / scl**deg
Chris@87:         return $name(coef, domain=domain, window=window)
Chris@87: 
Chris@87:     @staticmethod
Chris@87:     def identity(domain=$domain, window=$domain) :
Chris@87:         """Identity function.
Chris@87: 
Chris@87:         If ``p`` is the returned $name object, then ``p(x) == x`` for all
Chris@87:         values of x.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         domain : array_like
Chris@87:             The resulting array must be of the form ``[beg, end]``, where
Chris@87:             ``beg`` and ``end`` are the endpoints of the domain.
Chris@87:         window : array_like
Chris@87:             The resulting array must be if the form ``[beg, end]``, where
Chris@87:             ``beg`` and ``end`` are the endpoints of the window.
Chris@87: 
Chris@87:         Returns
Chris@87:         -------
Chris@87:         identity : $name instance
Chris@87: 
Chris@87:         """
Chris@87:         off, scl = pu.mapparms(window, domain)
Chris@87:         coef = ${nick}line(off, scl)
Chris@87:         return $name(coef, domain, window)
Chris@87: 
Chris@87:     @staticmethod
Chris@87:     def basis(deg, domain=$domain, window=$domain):
Chris@87:         """$name polynomial of degree `deg`.
Chris@87: 
Chris@87:         Returns an instance of the $name polynomial of degree `d`.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         deg : int
Chris@87:             Degree of the $name polynomial. Must be >= 0.
Chris@87:         domain : array_like
Chris@87:             The resulting array must be of the form ``[beg, end]``, where
Chris@87:             ``beg`` and ``end`` are the endpoints of the domain.
Chris@87:         window : array_like
Chris@87:             The resulting array must be if the form ``[beg, end]``, where
Chris@87:             ``beg`` and ``end`` are the endpoints of the window.
Chris@87: 
Chris@87:         Returns
Chris@87:         p : $name instance
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         .. versionadded:: 1.7.0
Chris@87: 
Chris@87:         """
Chris@87:         ideg = int(deg)
Chris@87:         if ideg != deg or ideg < 0:
Chris@87:             raise ValueError("deg must be non-negative integer")
Chris@87:         return $name([0]*ideg + [1], domain, window)
Chris@87: 
Chris@87:     @staticmethod
Chris@87:     def cast(series, domain=$domain, window=$domain):
Chris@87:         """Convert instance to equivalent $name series.
Chris@87: 
Chris@87:         The `series` is expected to be an instance of some polynomial
Chris@87:         series of one of the types supported by by the numpy.polynomial
Chris@87:         module, but could be some other class that supports the convert
Chris@87:         method.
Chris@87: 
Chris@87:         Parameters
Chris@87:         ----------
Chris@87:         series : series
Chris@87:             The instance series to be converted.
Chris@87:         domain : array_like
Chris@87:             The resulting array must be of the form ``[beg, end]``, where
Chris@87:             ``beg`` and ``end`` are the endpoints of the domain.
Chris@87:         window : array_like
Chris@87:             The resulting array must be if the form ``[beg, end]``, where
Chris@87:             ``beg`` and ``end`` are the endpoints of the window.
Chris@87: 
Chris@87:         Returns
Chris@87:         p : $name instance
Chris@87:             A $name series equal to the `poly` series.
Chris@87: 
Chris@87:         See Also
Chris@87:         --------
Chris@87:         convert -- similar instance method
Chris@87: 
Chris@87:         Notes
Chris@87:         -----
Chris@87:         .. versionadded:: 1.7.0
Chris@87: 
Chris@87:         """
Chris@87:         return series.convert(domain, $name, window)
Chris@87: 
Chris@87: ''')