package documentation

A sub-package for efficiently dealing with polynomials.

Within the documentation for this sub-package, a "finite power series," i.e., a polynomial (also referred to simply as a "series") is represented by a 1-D numpy array of the polynomial's coefficients, ordered from lowest order term to highest. For example, array([1,2,3]) represents P_0 + 2*P_1 + 3*P_2, where P_n is the n-th order basis polynomial applicable to the specific module in question, e.g., polynomial (which "wraps" the "standard" basis) or chebyshev. For optimal performance, all operations on polynomials, including evaluation at an argument, are implemented as operations on the coefficients. Additional (module-specific) information can be found in the docstring for the module of interest.

This package provides convenience classes for each of six different kinds of polynomials:

Name Provides
~polynomial.Polynomial Power series
~chebyshev.Chebyshev Chebyshev series
~legendre.Legendre Legendre series
~laguerre.Laguerre Laguerre series
~hermite.Hermite Hermite series
~hermite_e.HermiteE HermiteE series

These convenience classes provide a consistent interface for creating, manipulating, and fitting data with polynomials of different bases. The convenience classes are the preferred interface for the ~numpy.polynomial package, and are available from the numpy.polynomial namespace. This eliminates the need to navigate to the corresponding submodules, e.g. np.polynomial.Polynomial or np.polynomial.Chebyshev instead of np.polynomial.polynomial.Polynomial or np.polynomial.chebyshev.Chebyshev, respectively. The classes provide a more consistent and concise interface than the type-specific functions defined in the submodules for each type of polynomial. For example, to fit a Chebyshev polynomial with degree 1 to data given by arrays xdata and ydata, the class method:

>>> from numpy.polynomial import Chebyshev
>>> c =, ydata, deg=1)

is preferred over the chebyshev.chebfit function from the np.polynomial.chebyshev module:

>>> from numpy.polynomial.chebyshev import chebfit
>>> c = chebfit(xdata, ydata, deg=1)

See :doc:`routines.polynomials.classes` for more details.

Convenience Classes

The following lists the various constants and methods common to all of the classes representing the various kinds of polynomials. In the following, the term Poly represents any one of the convenience classes (e.g. ~polynomial.Polynomial, ~chebyshev.Chebyshev, ~hermite.Hermite, etc.) while the lowercase p represents an instance of a polynomial class.


  • Poly.domain -- Default domain
  • Poly.window -- Default window
  • Poly.basis_name -- String used to represent the basis
  • Poly.maxpower -- Maximum value n such that p**n is allowed
  • Poly.nickname -- String used in printing


Methods for creating polynomial instances.

  • Poly.basis(degree) -- Basis polynomial of given degree
  • Poly.identity() -- p where p(x) = x for all x
  •, y, deg) -- p of degree deg with coefficients determined by the least-squares fit to the data x, y
  • Poly.fromroots(roots) -- p with specified roots
  • p.copy() -- Create a copy of p


Methods for converting a polynomial instance of one kind to another.

  • p.cast(Poly) -- Convert p to instance of kind Poly
  • p.convert(Poly) -- Convert p to instance of kind Poly or map between domain and window


  • p.deriv() -- Take the derivative of p
  • p.integ() -- Integrate p


  • Poly.has_samecoef(p1, p2) -- Check if coefficients match
  • Poly.has_samedomain(p1, p2) -- Check if domains match
  • Poly.has_sametype(p1, p2) -- Check if types match
  • Poly.has_samewindow(p1, p2) -- Check if windows match


  • p.linspace() -- Return x, p(x) at equally-spaced points in domain
  • p.mapparms() -- Return the parameters for the linear mapping between domain and window.
  • p.roots() -- Return the roots of p.
  • p.trim() -- Remove trailing coefficients.
  • p.cutdeg(degree) -- Truncate p to given degree
  • p.truncate(size) -- Truncate p to given size
Module chebyshev ==================================================== Chebyshev Series (numpy.polynomial.chebyshev) ====================================================
Module hermite ============================================================== Hermite Series, "Physicists" (numpy.polynomial.hermite) ==============================================================
Module hermite​_e =================================================================== HermiteE Series, "Probabilists" (numpy.polynomial.hermite_e) ===================================================================
Module laguerre ================================================== Laguerre Series (numpy.polynomial.laguerre) ==================================================
Module legendre ================================================== Legendre Series (numpy.polynomial.legendre) ==================================================
Module polynomial ================================================= Power Series (numpy.polynomial.polynomial) =================================================
Module polyutils Utility classes and functions for the polynomial modules.
Module ​_polybase Abstract base class for the various polynomial Classes.
Module setup Undocumented
Package tests No package docstring; 8/9 modules documented


Function set​_default​_printstyle Set the default format for the string representation of polynomials.
Variable test Undocumented
def set_default_printstyle(style):

Set the default format for the string representation of polynomials.

Values for style must be valid inputs to __format__, i.e. 'ascii' or 'unicode'.


style : str
Format string for default printing style. Must be either 'ascii' or 'unicode'.


The default format depends on the platform: 'unicode' is used on Unix-based systems and 'ascii' on Windows. This determination is based on default font support for the unicode superscript and subscript ranges.


>>> p = np.polynomial.Polynomial([1, 2, 3])
>>> c = np.polynomial.Chebyshev([1, 2, 3])
>>> np.polynomial.set_default_printstyle('unicode')
>>> print(p)
1.0 + 2.0·x¹ + 3.0·x²
>>> print(c)
1.0 + 2.0·T₁(x) + 3.0·T₂(x)
>>> np.polynomial.set_default_printstyle('ascii')
>>> print(p)
1.0 + 2.0 x**1 + 3.0 x**2
>>> print(c)
1.0 + 2.0 T_1(x) + 3.0 T_2(x)
>>> # Formatting supersedes all class/package-level defaults
>>> print(f"{p:unicode}")
1.0 + 2.0·x¹ + 3.0·x²
test =