Polynomials#

Polynomials in NumPy can be created, manipulated, and even fitted using the convenience classes of the numpy.polynomial package, introduced in NumPy 1.4.

Prior to NumPy 1.4, numpy.poly1d was the class of choice and it is still available in order to maintain backward compatibility. However, the newer polynomial package is more complete and its convenience classes provide a more consistent, better-behaved interface for working with polynomial expressions. Therefore numpy.polynomial is recommended for new coding.

Note

Terminology

The term polynomial module refers to the old API defined in numpy.lib.polynomial, which includes the numpy.poly1d class and the polynomial functions prefixed with poly accessible from the numpy namespace (e.g. numpy.polyadd, numpy.polyval, numpy.polyfit, etc.).

The term polynomial package refers to the new API defined in numpy.polynomial, which includes the convenience classes for the different kinds of polynomials (Polynomial, Chebyshev, etc.).

Transitioning from numpy.poly1d to numpy.polynomial#

As noted above, the poly1d class and associated functions defined in numpy.lib.polynomial, such as numpy.polyfit and numpy.poly, are considered legacy and should not be used in new code. Since NumPy version 1.4, the numpy.polynomial package is preferred for working with polynomials.

Quick Reference#

The following table highlights some of the main differences between the legacy polynomial module and the polynomial package for common tasks. The Polynomial class is imported for brevity:

from numpy.polynomial import Polynomial

How to…

Legacy (numpy.poly1d)

numpy.polynomial

Create a polynomial object from coefficients [1]

p = np.poly1d([1, 2, 3])

p = Polynomial([3, 2, 1])

Create a polynomial object from roots

r = np.poly([-1, 1]) p = np.poly1d(r)

p = Polynomial.fromroots([-1, 1])

Fit a polynomial of degree deg to data

np.polyfit(x, y, deg)

Polynomial.fit(x, y, deg)

Transition Guide#

There are significant differences between numpy.lib.polynomial and numpy.polynomial. The most significant difference is the ordering of the coefficients for the polynomial expressions. The various routines in numpy.polynomial all deal with series whose coefficients go from degree zero upward, which is the reverse order of the poly1d convention. The easy way to remember this is that indices correspond to degree, i.e., coef[i] is the coefficient of the term of degree i.

Though the difference in convention may be confusing, it is straightforward to convert from the legacy polynomial API to the new. For example, the following demonstrates how you would convert a numpy.poly1d instance representing the expression \(x^{2} + 2x + 3\) to a Polynomial instance representing the same expression:

>>> import numpy as np

>>> p1d = np.poly1d([1, 2, 3])
>>> p = np.polynomial.Polynomial(p1d.coef[::-1])

In addition to the coef attribute, polynomials from the polynomial package also have domain and window attributes. These attributes are most relevant when fitting polynomials to data, though it should be noted that polynomials with different domain and window attributes are not considered equal, and can’t be mixed in arithmetic:

>>> p1 = np.polynomial.Polynomial([1, 2, 3])
>>> p1
Polynomial([1., 2., 3.], domain=[-1.,  1.], window=[-1.,  1.], symbol='x')
>>> p2 = np.polynomial.Polynomial([1, 2, 3], domain=[-2, 2])
>>> p1 == p2
False
>>> p1 + p2
Traceback (most recent call last):
    ...
TypeError: Domains differ

See the documentation for the convenience classes for further details on the domain and window attributes.

Another major difference between the legacy polynomial module and the polynomial package is polynomial fitting. In the old module, fitting was done via the polyfit function. In the polynomial package, the fit class method is preferred. For example, consider a simple linear fit to the following data:

In [1]: rng = np.random.default_rng()

In [2]: x = np.arange(10)

In [3]: y = np.arange(10) + rng.standard_normal(10)

With the legacy polynomial module, a linear fit (i.e. polynomial of degree 1) could be applied to these data with polyfit:

In [4]: np.polyfit(x, y, deg=1)
Out[4]: array([0.83145375, 0.99125412])

With the new polynomial API, the fit class method is preferred:

In [5]: p_fitted = np.polynomial.Polynomial.fit(x, y, deg=1)

In [6]: p_fitted
Out[6]: Polynomial([4.73279597, 3.74154186], domain=[0., 9.], window=[-1.,  1.], symbol='x')

Note that the coefficients are given in the scaled domain defined by the linear mapping between the window and domain. convert can be used to get the coefficients in the unscaled data domain.

In [7]: p_fitted.convert()
Out[7]: Polynomial([0.99125412, 0.83145375], domain=[-1.,  1.], window=[-1.,  1.], symbol='x')

Documentation for the polynomial package#

In addition to standard power series polynomials, the polynomial package provides several additional kinds of polynomials including Chebyshev, Hermite (two subtypes), Laguerre, and Legendre polynomials. Each of these has an associated convenience class available from the numpy.polynomial namespace that provides a consistent interface for working with polynomials regardless of their type.

Documentation pertaining to specific functions defined for each kind of polynomial individually can be found in the corresponding module documentation:

Documentation for legacy polynomials#

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