Using the Convenience Classes
=============================
The convenience classes provided by the polynomial package are:
============ ================
Name Provides
============ ================
Polynomial Power series
Chebyshev Chebyshev series
Legendre Legendre series
Laguerre Laguerre series
Hermite Hermite series
HermiteE HermiteE series
============ ================
The series in this context are finite sums of the corresponding polynomial
basis functions multiplied by coefficients. For instance, a power series
looks like
.. math:: p(x) = 1 + 2x + 3x^2
and has coefficients :math:`[1, 2, 3]`. The Chebyshev series with the
same coefficients looks like
.. math:: p(x) = 1 T_0(x) + 2 T_1(x) + 3 T_2(x)
and more generally
.. math:: p(x) = \sum_{i=0}^n c_i T_i(x)
where in this case the :math:`T_n` are the Chebyshev functions of
degree :math:`n`, but could just as easily be the basis functions of
any of the other classes. The convention for all the classes is that
the coefficient :math:`c[i]` goes with the basis function of degree i.
All of the classes have the same methods, and especially they implement the
Python numeric operators +, -, \*, //, %, divmod, \*\*, ==,
and !=. The last two can be a bit problematic due to floating point
roundoff errors. We now give a quick demonstration of the various
operations using NumPy version 1.7.0.
Basics
------
First we need a polynomial class and a polynomial instance to play with.
The classes can be imported directly from the polynomial package or from
the module of the relevant type. Here we import from the package and use
the conventional Polynomial class because of its familiarity::
>>> from numpy.polynomial import Polynomial as P
>>> p = P([1,2,3])
>>> p
Polynomial([ 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
Note that there are three parts to the long version of the printout. The
first is the coefficients, the second is the domain, and the third is the
window::
>>> p.coef
array([ 1., 2., 3.])
>>> p.domain
array([-1., 1.])
>>> p.window
array([-1., 1.])
Printing a polynomial yields a shorter form without the domain
and window::
>>> print p
poly([ 1. 2. 3.])
We will deal with the domain and window when we get to fitting, for the moment
we ignore them and run through the basic algebraic and arithmetic operations.
Addition and Subtraction::
>>> p + p
Polynomial([ 2., 4., 6.], domain=[-1, 1], window=[-1, 1])
>>> p - p
Polynomial([ 0.], domain=[-1, 1], window=[-1, 1])
Multiplication::
>>> p * p
Polynomial([ 1., 4., 10., 12., 9.], domain=[-1, 1], window=[-1, 1])
Powers::
>>> p**2
Polynomial([ 1., 4., 10., 12., 9.], domain=[-1, 1], window=[-1, 1])
Division:
Floor division, '//', is the division operator for the polynomial classes,
polynomials are treated like integers in this regard. For Python versions <
3.x the '/' operator maps to '//', as it does for Python, for later
versions the '/' will only work for division by scalars. At some point it
will be deprecated::
>>> p // P([-1, 1])
Polynomial([ 5., 3.], domain=[-1, 1], window=[-1, 1])
Remainder::
>>> p % P([-1, 1])
Polynomial([ 6.], domain=[-1, 1], window=[-1, 1])
Divmod::
>>> quo, rem = divmod(p, P([-1, 1]))
>>> quo
Polynomial([ 5., 3.], domain=[-1, 1], window=[-1, 1])
>>> rem
Polynomial([ 6.], domain=[-1, 1], window=[-1, 1])
Evaluation::
>>> x = np.arange(5)
>>> p(x)
array([ 1., 6., 17., 34., 57.])
>>> x = np.arange(6).reshape(3,2)
>>> p(x)
array([[ 1., 6.],
[ 17., 34.],
[ 57., 86.]])
Substitution:
Substitute a polynomial for x and expand the result. Here we substitute
p in itself leading to a new polynomial of degree 4 after expansion. If
the polynomials are regarded as functions this is composition of
functions::
>>> p(p)
Polynomial([ 6., 16., 36., 36., 27.], domain=[-1, 1], window=[-1, 1])
Roots::
>>> p.roots()
array([-0.33333333-0.47140452j, -0.33333333+0.47140452j])
It isn't always convenient to explicitly use Polynomial instances, so
tuples, lists, arrays, and scalars are automatically cast in the arithmetic
operations::
>>> p + [1, 2, 3]
Polynomial([ 2., 4., 6.], domain=[-1, 1], window=[-1, 1])
>>> [1, 2, 3] * p
Polynomial([ 1., 4., 10., 12., 9.], domain=[-1, 1], window=[-1, 1])
>>> p / 2
Polynomial([ 0.5, 1. , 1.5], domain=[-1, 1], window=[-1, 1])
Polynomials that differ in domain, window, or class can't be mixed in
arithmetic::
>>> from numpy.polynomial import Chebyshev as T
>>> p + P([1], domain=[0,1])
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<string>", line 213, in __add__
TypeError: Domains differ
>>> p + P([1], window=[0,1])
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<string>", line 215, in __add__
TypeError: Windows differ
>>> p + T([1])
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<string>", line 211, in __add__
TypeError: Polynomial types differ
But different types can be used for substitution. In fact, this is how
conversion of Polynomial classes among themselves is done for type, domain,
and window casting::
>>> p(T([0, 1]))
Chebyshev([ 2.5, 2. , 1.5], domain=[-1, 1], window=[-1, 1])
Which gives the polynomial `p` in Chebyshev form. This works because
:math:`T_1(x) = x` and substituting :math:`x` for :math:`x` doesn't change
the original polynomial. However, all the multiplications and divisions
will be done using Chebyshev series, hence the type of the result.
Calculus
--------
Polynomial instances can be integrated and differentiated.::
>>> from numpy.polynomial import Polynomial as P
>>> p = P([2, 6])
>>> p.integ()
Polynomial([ 0., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> p.integ(2)
Polynomial([ 0., 0., 1., 1.], domain=[-1, 1], window=[-1, 1])
The first example integrates `p` once, the second example integrates it
twice. By default, the lower bound of the integration and the integration
constant are 0, but both can be specified.::
>>> p.integ(lbnd=-1)
Polynomial([-1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> p.integ(lbnd=-1, k=1)
Polynomial([ 0., 2., 3.], domain=[-1, 1], window=[-1, 1])
In the first case the lower bound of the integration is set to -1 and the
integration constant is 0. In the second the constant of integration is set
to 1 as well. Differentiation is simpler since the only option is the
number of times the polynomial is differentiated::
>>> p = P([1, 2, 3])
>>> p.deriv(1)
Polynomial([ 2., 6.], domain=[-1, 1], window=[-1, 1])
>>> p.deriv(2)
Polynomial([ 6.], domain=[-1, 1], window=[-1, 1])
Other Polynomial Constructors
-----------------------------
Constructing polynomials by specifying coefficients is just one way of
obtaining a polynomial instance, they may also be created by specifying
their roots, by conversion from other polynomial types, and by least
squares fits. Fitting is discussed in its own section, the other methods
are demonstrated below::
>>> from numpy.polynomial import Polynomial as P
>>> from numpy.polynomial import Chebyshev as T
>>> p = P.fromroots([1, 2, 3])
>>> p
Polynomial([ -6., 11., -6., 1.], domain=[-1, 1], window=[-1, 1])
>>> p.convert(kind=T)
Chebyshev([ -9. , 11.75, -3. , 0.25], domain=[-1, 1], window=[-1, 1])
The convert method can also convert domain and window::
>>> p.convert(kind=T, domain=[0, 1])
Chebyshev([-2.4375 , 2.96875, -0.5625 , 0.03125], [ 0., 1.], [-1., 1.])
>>> p.convert(kind=P, domain=[0, 1])
Polynomial([-1.875, 2.875, -1.125, 0.125], [ 0., 1.], [-1., 1.])
In numpy versions >= 1.7.0 the `basis` and `cast` class methods are also
available. The cast method works like the convert method while the basis
method returns the basis polynomial of given degree::
>>> P.basis(3)
Polynomial([ 0., 0., 0., 1.], domain=[-1, 1], window=[-1, 1])
>>> T.cast(p)
Chebyshev([ -9. , 11.75, -3. , 0.25], domain=[-1, 1], window=[-1, 1])
Conversions between types can be useful, but it is *not* recommended
for routine use. The loss of numerical precision in passing from a
Chebyshev series of degree 50 to a Polynomial series of the same degree
can make the results of numerical evaluation essentially random.
Fitting
-------
Fitting is the reason that the `domain` and `window` attributes are part of
the convenience classes. To illustrate the problem, the values of the Chebyshev
polynomials up to degree 5 are plotted below.
.. plot::
>>> import matplotlib.pyplot as plt
>>> from numpy.polynomial import Chebyshev as T
>>> x = np.linspace(-1, 1, 100)
>>> for i in range(6): ax = plt.plot(x, T.basis(i)(x), lw=2, label="$T_%d$"%i)
...
>>> plt.legend(loc="upper left")
<matplotlib.legend.Legend object at 0x3b3ee10>
>>> plt.show()
In the range -1 <= `x` <= 1 they are nice, equiripple functions lying between +/- 1.
The same plots over the range -2 <= `x` <= 2 look very different:
.. plot::
>>> import matplotlib.pyplot as plt
>>> from numpy.polynomial import Chebyshev as T
>>> x = np.linspace(-2, 2, 100)
>>> for i in range(6): ax = plt.plot(x, T.basis(i)(x), lw=2, label="$T_%d$"%i)
...
>>> plt.legend(loc="lower right")
<matplotlib.legend.Legend object at 0x3b3ee10>
>>> plt.show()
As can be seen, the "good" parts have shrunk to insignificance. In using
Chebyshev polynomials for fitting we want to use the region where `x` is
between -1 and 1 and that is what the `window` specifies. However, it is
unlikely that the data to be fit has all its data points in that interval,
so we use `domain` to specify the interval where the data points lie. When
the fit is done, the domain is first mapped to the window by a linear
transformation and the usual least squares fit is done using the mapped
data points. The window and domain of the fit are part of the returned series
and are automatically used when computing values, derivatives, and such. If
they aren't specified in the call the fitting routine will use the default
window and the smallest domain that holds all the data points. This is
illustrated below for a fit to a noisy sine curve.
.. plot::
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from numpy.polynomial import Chebyshev as T
>>> np.random.seed(11)
>>> x = np.linspace(0, 2*np.pi, 20)
>>> y = np.sin(x) + np.random.normal(scale=.1, size=x.shape)
>>> p = T.fit(x, y, 5)
>>> plt.plot(x, y, 'o')
[<matplotlib.lines.Line2D object at 0x2136c10>]
>>> xx, yy = p.linspace()
>>> plt.plot(xx, yy, lw=2)
[<matplotlib.lines.Line2D object at 0x1cf2890>]
>>> p.domain
array([ 0. , 6.28318531])
>>> p.window
array([-1., 1.])
>>> plt.show()