Orthogonal polynomials for Python
All about orthogonal polynomials.
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.
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
import orthopy x = 0.5 evaluator = orthopy.c1.legendre.Eval(x, "classical") for _ in range(5): print(next(evaluator))
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
polynomials will then be evaluated for all points at once. You can also use sympy for
import itertools import orthopy import sympy x = sympy.Symbol("x") evaluator = orthopy.c1.legendre.Eval(x, "classical") for val in itertools.islice(evaluator, 5): print(sympy.expand(val))
1 x 3*x**2/2 - 1/2 5*x**3/2 - 3*x/2 35*x**4/8 - 15*x**2/4 + 3/8
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 orthopy.c1.jacobi.show(5, "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 ) print(rc.p0) for k in range(5): print(rc[k])
1 (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( x, "probabilists", # or "physicists" "normal" )
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 )
orthopy's triangle orthogonal polynomials are evaluated in terms of barycentric
coordinates, so the
X.shape has to be 3.
import orthopy bary = [0.1, 0.7, 0.2] evaluator = orthopy.t2.Eval(bary, "normal")
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")
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( x, 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.
Jacobi product polynomials.
All polynomials are normalized on the n-dimensional cube. The dimensionality is
evaluator = orthopy.cn.Eval(X, alpha=0, beta=0) values, degrees = next(evaluator)
nD space with weight function exp(-r2) (Enr2)
Hermite product polynomials.
All polynomials are normalized over the measure. The dimensionality is determined by
evaluator = orthopy.enr2.Eval( x, standardization="probabilists" # or "physicists" ) values, degrees = next(evaluator)
The the sanity of recurrence coefficients with test 3 from Gautschi's article: computing the weighted sum of orthogonal polynomials:
orthopy.tools.gautschi_test_3(moments, alpha, beta)
Clenshaw algorithm for computing the weighted sum of orthogonal polynomials:
vals = orthopy.c1.clenshaw(a, alpha, beta, t)
- Robert C. Kirby, Singularity-free evaluation of collapsed-coordinate orthogonal polynomials, ACM Transactions on Mathematical Software (TOMS), Volume 37, Issue 1, January 2010
- Abedallah Rababah, Recurrence Relations for Orthogonal Polynomials on Triangular Domains, MDPI Mathematics 2016, 4(2)
- Yuan Xu, Orthogonal polynomials of several variables, arxiv.org, January 2017
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