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]ÐKc @ s% d Z d d k Z e i d ƒ Z d S( s4 Template for the Chebyshev and Polynomial classes.
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from __future__ import division
import polyutils as pu
import numpy as np
class $name(pu.PolyBase) :
"""A $name series class.
Parameters
----------
coef : array_like
$name coefficients, in increasing order. For example,
``(1, 2, 3)`` implies ``P_0 + 2P_1 + 3P_2`` where the
``P_i`` are a graded polynomial basis.
domain : (2,) array_like
Domain to use. The interval ``[domain[0], domain[1]]`` is mapped to
the interval ``$domain`` by shifting and scaling.
Attributes
----------
coef : (N,) array
$name coefficients, from low to high.
domain : (2,) array_like
Domain that is mapped to ``$domain``.
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.
Notes
-----
It is important to specify the domain for many uses of graded polynomial,
for instance in fitting data. This is because many of the important
properties of the polynomial basis only hold in a specified interval and
thus the data must be mapped into that domain in order to benefit.
Examples
--------
"""
# Limit runaway size. T_n^m has degree n*2^m
maxpower = 16
# Default domain
domain = np.array($domain)
# Don't let participate in array operations. Value doesn't matter.
__array_priority__ = 0
def __init__(self, coef, domain=$domain) :
[coef, domain] = pu.as_series([coef, domain], trim=False)
if len(domain) != 2 :
raise ValueError("Domain has wrong number of elements.")
self.coef = coef
self.domain = domain
def __repr__(self):
format = "%s(%s, %s)"
coef = repr(self.coef)[6:-1]
domain = repr(self.domain)[6:-1]
return format % ('$name', coef, domain)
def __str__(self) :
format = "%s(%s, %s)"
return format % ('$nick', str(self.coef), str(self.domain))
# Pickle and copy
def __getstate__(self) :
ret = self.__dict__.copy()
ret['coef'] = self.coef.copy()
ret['domain'] = self.domain.copy()
return ret
def __setstate__(self, dict) :
self.__dict__ = dict
# Call
def __call__(self, arg) :
off, scl = pu.mapparms(self.domain, $domain)
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)
def __pos__(self) :
return self
def __add__(self, other) :
"""Returns sum"""
if isinstance(other, self.__class__) :
if np.all(self.domain == other.domain) :
coef = ${nick}add(self.coef, other.coef)
else :
raise PolyDomainError()
else :
try :
coef = ${nick}add(self.coef, other)
except :
return NotImplemented
return self.__class__(coef, self.domain)
def __sub__(self, other) :
"""Returns difference"""
if isinstance(other, self.__class__) :
if np.all(self.domain == other.domain) :
coef = ${nick}sub(self.coef, other.coef)
else :
raise PolyDomainError()
else :
try :
coef = ${nick}sub(self.coef, other)
except :
return NotImplemented
return self.__class__(coef, self.domain)
def __mul__(self, other) :
"""Returns product"""
if isinstance(other, self.__class__) :
if np.all(self.domain == other.domain) :
coef = ${nick}mul(self.coef, other.coef)
else :
raise PolyDomainError()
else :
try :
coef = ${nick}mul(self.coef, other)
except :
return NotImplemented
return self.__class__(coef, self.domain)
def __div__(self, other):
# set to __floordiv__ /.
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 isinstance(other, self.__class__) :
if len(other.coef) == 1 :
coef = div(self.coef, other.coef)
else :
return NotImplemented
elif np.isscalar(other) :
coef = self.coef/other
else :
return NotImplemented
return self.__class__(coef, self.domain)
def __floordiv__(self, other) :
"""Returns the quotient."""
if isinstance(other, self.__class__) :
if np.all(self.domain == other.domain) :
quo, rem = ${nick}div(self.coef, other.coef)
else :
raise PolyDomainError()
else :
try :
quo, rem = ${nick}div(self.coef, other)
except :
return NotImplemented
return self.__class__(quo, self.domain)
def __mod__(self, other) :
"""Returns the remainder."""
if isinstance(other, self.__class__) :
if np.all(self.domain == other.domain) :
quo, rem = ${nick}div(self.coef, other.coef)
else :
raise PolyDomainError()
else :
try :
quo, rem = ${nick}div(self.coef, other)
except :
return NotImplemented
return self.__class__(rem, self.domain)
def __divmod__(self, other) :
"""Returns quo, remainder"""
if isinstance(other, self.__class__) :
if np.all(self.domain == other.domain) :
quo, rem = ${nick}div(self.coef, other.coef)
else :
raise PolyDomainError()
else :
try :
quo, rem = ${nick}div(self.coef, other)
except :
return NotImplemented
return self.__class__(quo, self.domain), self.__class__(rem, self.domain)
def __pow__(self, other) :
try :
coef = ${nick}pow(self.coef, other, maxpower = self.maxpower)
except :
raise
return self.__class__(coef, self.domain)
def __radd__(self, other) :
try :
coef = ${nick}add(other, self.coef)
except :
return NotImplemented
return self.__class__(coef, self.domain)
def __rsub__(self, other):
try :
coef = ${nick}sub(other, self.coef)
except :
return NotImplemented
return self.__class__(coef, self.domain)
def __rmul__(self, other) :
try :
coef = ${nick}mul(other, self.coef)
except :
return NotImplemented
return self.__class__(coef, self.domain)
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)
def __rfloordiv__(self, other) :
try :
quo, rem = ${nick}div(other, self.coef)
except :
return NotImplemented
return self.__class__(quo, self.domain)
def __rmod__(self, other) :
try :
quo, rem = ${nick}div(other, self.coef)
except :
return NotImplemented
return self.__class__(rem, self.domain)
def __rdivmod__(self, other) :
try :
quo, rem = ${nick}div(other, self.coef)
except :
return NotImplemented
return self.__class__(quo, self.domain), self.__class__(rem, self.domain)
# Enhance me
# some augmented arithmetic operations could be added here
def __eq__(self, other) :
res = isinstance(other, self.__class__) and len(self.coef) == len(other.coef) and np.all(self.domain == other.domain) and np.all(self.coef == other.coef)
return res
def __ne__(self, other) :
return not self.__eq__(other)
#
# Extra numeric functions.
#
def convert(self, domain=None, kind=None) :
"""Convert to different class and/or domain.
Parameters:
-----------
domain : {None, array_like}
The domain of the new series type instance. If the value is is
``None``, then the default domain of `kind` is used.
kind : {None, class}
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.
Returns:
--------
new_series_instance : `kind`
The returned class can be of different type than the current
instance and/or have a different domain.
Examples:
---------
Notes:
------
Conversion between domains and class types can result in
numerically ill defined series.
"""
if kind is None :
kind = $name
if domain is None :
domain = kind.domain
return self(kind.identity(domain))
def mapparms(self) :
"""Return the mapping parameters.
The returned values define a linear map ``off + scl*x`` that is
applied to the input arguments before the series is evaluated. The
of the map depend on the domain; if the current domain is equal to
the default domain ``$domain`` the resulting map is the identity.
If the coeffients of the ``$name`` instance are to be used
separately, 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 default
interval 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, $domain)
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.
"""
return self.__class__(pu.trimcoef(self.coef, tol), self.domain)
def truncate(self, size) :
"""Truncate series by discarding trailing coefficients.
Reduce the $name series to length `size` by removing trailing
coefficients. The value of `size` must be greater than zero. This
is most likely to be useful in least squares fits when the high
order coefficients are very small.
Parameters:
-----------
size : int
The series is reduced to length `size` by discarding trailing
coefficients. The value of `size` must be greater than zero.
Returns:
-------
new_instance : $name
New instance of $name with truncated coefficients.
"""
if size < 1 :
raise ValueError("size must be > 0")
if size >= len(self.coef) :
return self.__class__(self.coef, self.domain)
else :
return self.__class__(self.coef[:size], self.domain)
def copy(self) :
"""Return a copy.
A new instance of $name is returned that has the same
coefficients and domain as the current instance.
Returns:
--------
new_instance : $name
New instance of $name with the same coefficients and domain.
"""
return self.__class__(self.coef, self.domain)
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 : positive integer
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 : $name
The integral of the original series defined with the same
domain.
See Also
--------
`${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)
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 : positive integer
The number of integrations to perform.
Returns:
--------
derivative : $name
The derivative of the original series defined with the same
domain.
See Also
--------
`${nick}der` : similar function.
`${nick}int` : similar function for integration.
"""
off, scl = self.mapparms()
coef = ${nick}der(self.coef, m, scl)
return self.__class__(coef, self.domain)
def roots(self) :
"""Return list of roots.
Return ndarray of roots for this series. See `${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, $domain, self.domain)
@staticmethod
def fit(x, y, deg, domain=$domain, rcond=None, full=False) :
"""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. 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 $name instance. If ``None``,
then a minimal domain that covers the points `x` is chosen. The
default value is ``$domain``.
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.
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)
xnew = pu.mapdomain(x, domain, $domain)
res = ${nick}fit(xnew, y, deg, rcond=None, full=full)
if full :
[coef, status] = res
return $name(coef, domain=domain), status
else :
coef = res
return $name(coef, domain=domain)
@staticmethod
def fromroots(roots, domain=$domain) :
"""Return $name object with specified roots.
See ${nick}fromroots for full documentation.
See Also
--------
${nick}fromroots : equivalent function
"""
if domain is None :
domain = pu.getdomain(roots)
rnew = pu.mapdomain(roots, domain, $domain)
coef = ${nick}fromroots(rnew)
return $name(coef, domain=domain)
@staticmethod
def identity(domain=$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 if the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the domain.
Returns:
--------
identity : $name object
"""
off, scl = pu.mapparms($domain, domain)
coef = ${nick}line(off, scl)
return $name(coef, domain)
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