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Orthogonal polynomials for Python

Project description


All about orthogonal polynomials.

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orthopy provides various orthogonal polynomial classes for lines, triangles, disks, spheres, n-cubes, the nD space with weight function exp(-r2) and more. All computations are done using numerically stable recurrence schemes. Furthermore, all functions are fully vectorized and can return results in exact arithmetic.

Basic usage

Install orthopy from PyPi via

pip install orthopy

The main function of all submodules is the iterator Eval which evaluates the series of orthogonal polynomials with increasing degree at given points using a recurrence relation, e.g.,

import orthopy

x = 0.5

evaluator = orthopy.c1.legendre.Eval(x, "classical")
for _ in range(5):
1.0          # P_0(0.5)
0.5          # P_1(0.5)
-0.125       # P_2(0.5)
-0.4375      # P_3(0.5)
-0.2890625   # P_4(0.5)

Other ways of getting the first n items are

evaluator = Eval(x, "normal")
vals = [next(evaluator) for _ in range(n)]

import itertools
vals = list(itertools.islice(Eval(x, "normal"), n))

Instead of evaluating at only one point, you can provide any array for x; the polynomials will then be evaluated for all points at once. You can also use sympy for symbolic computation:

import itertools
import orthopy
import sympy

x = sympy.Symbol("x")

evaluator = orthopy.c1.legendre.Eval(x, "classical")
for val in itertools.islice(evaluator, 5):
3*x**2/2 - 1/2
5*x**3/2 - 3*x/2
35*x**4/8 - 15*x**2/4 + 3/8

All Eval methods have a scaling argument which can have three values:

  • "monic": The leading coefficient is 1.
  • "classical": The maximum value is 1 (or (n+alpha over n)).
  • "normal": The integral of the squared function over the domain is 1.

For univariate ("one-dimensional") integrals, every new iteration contains one function. For bivariate ("two-dimensional") domains, every level will contain one function more than the previous, and similarly for multivariate families. See the tree plots below.

Line segment (-1, +1) with weight function (1-x)α (1+x)β

Legendre Chebyshev 1 Chebyshev 2

Jacobi, Gegenbauer (α=β), Chebyshev 1 (α=β=-1/2), Chebyshev 2 (α=β=1/2), Legendre (α=β=0) polynomials.

import orthopy

orthopy.c1.legendre.Eval(x, "normal")
orthopy.c1.chebyshev1.Eval(x, "normal")
orthopy.c1.chebyshev2.Eval(x, "normal")
orthopy.c1.gegenbauer.Eval(x, "normal", lmbda)
orthopy.c1.jacobi.Eval(x, "normal", alpha, beta)

The plots above are generated with

import orthopy, "normal", 0.0, 0.0)
# plot, savefig also exist

Recurrence coefficients can be explicitly retrieved by

import orthopy

rc = orthopy.c1.jacobi.RecurrenceCoefficients(
    "monic",  # or "classical", "normal"
    alpha=0, beta=0, symbolic=True
for k in range(5):
(1, 0, None)
(1, 0, 1/3)
(1, 0, 4/15)
(1, 0, 9/35)
(1, 0, 16/63)

1D half-space with weight function xα exp(-r)

(Generalized) Laguerre polynomials.

evaluator = orthopy.e1r.Eval(x, alpha=0, scaling="normal")

1D space with weight function exp(-r2)

Hermite polynomials come in two standardizations:

  • "physicists" (against the weight function exp(-x ** 2)
  • "probabilists" (against the weight function 1 / sqrt(2 * pi) * exp(-x ** 2 / 2)
evaluator = orthopy.e1r2.Eval(
    "probabilists",  # or "physicists"

Associated Legendre "polynomials"

Not all of those are polynomials, so they should really be called associated Legendre functions. The kth iteration contains 2k+1 functions, indexed from -k to k. (See the color grouping in the above plot.)

evaluator = orthopy.c1.associated_legendre.Eval(
    x, phi=None, standardization="natural", with_condon_shortley_phase=True

Triangle (T2)

orthopy's triangle orthogonal polynomials are evaluated in terms of barycentric coordinates, so the X.shape[0] has to be 3.

import orthopy

bary = [0.1, 0.7, 0.2]
evaluator = orthopy.t2.Eval(bary, "normal")

Disk (S2)

Xu Zernike Zernike 2

orthopy contains several families of orthogonal polynomials on the unit disk: After Xu, Zernike, and a simplified version of Zernike polynomials.

import orthopy

x = [0.1, -0.3]

evaluator = orthopy.s2.xu.Eval(x, "normal")
# evaluator = orthopy.s2.zernike.Eval(x, "normal")
# evaluator = orthopy.s2.zernike2.Eval(x, "normal")

Sphere (U3)

Complex-valued spherical harmonics, plotted with cplot coloring (black=zero, green=real positive, pink=real negative, blue=imaginary positive, yellow=imaginary negative). The functions in the middle are real-valued. The complex angle takes n turns on the nth level.

evaluator = orthopy.u3.EvalCartesian(
    scaling="quantum mechanic"  # or "acoustic", "geodetic", "schmidt"

evaluator = orthopy.u3.EvalSpherical(
    theta_phi,  # polar, azimuthal angles
    scaling="quantum mechanic"  # or "acoustic", "geodetic", "schmidt"

To generate the above plot, write the tree mesh to a file

import orthopy

orthopy.u3.write_tree("u3.vtk", 5, "quantum mechanic")

and open it with ParaView. Select the srgb1 data set and turn off Map Scalars.

n-Cube (Cn)

C1 (Legendre) C2 C3

Jacobi product polynomials. All polynomials are normalized on the n-dimensional cube. The dimensionality is determined by X.shape[0].

evaluator =, alpha=0, beta=0)
values, degrees = next(evaluator)

nD space with weight function exp(-r2) (Enr2)

E1r2 E2r2 E3r2

Hermite product polynomials. All polynomials are normalized over the measure. The dimensionality is determined by X.shape[0].

evaluator = orthopy.enr2.Eval(
    standardization="probabilists"  # or "physicists"
values, degrees = next(evaluator)

Other tools

  • Generating recurrence coefficients for 1D domains with Stieltjes, Golub-Welsch, Chebyshev, and modified Chebyshev.

  • The the sanity of recurrence coefficients with test 3 from Gautschi's article: computing the weighted sum of orthogonal polynomials:, alpha, beta)
  • Clenshaw algorithm for computing the weighted sum of orthogonal polynomials:

    vals = orthopy.c1.clenshaw(a, alpha, beta, t)

Relevant publications

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