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., ```python import orthopy

x = 0.5

evaluator = orthopy.c1.legendre.Eval(x, "classical") for _ in range(5): print(next(evaluator)) python 1.0 # P0(0.5) 0.5 # P1(0.5) -0.125 # P2(0.5) -0.4375 # P3(0.5) -0.2890625 # P_4(0.5) Other ways of getting the first `n` items are <!--pytest-codeblocks:skip--> python 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: python 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 3x2/2 - 1/2 5x3/2 - 3x/2 35x4/8 - 15x*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. <!--pytest-codeblocks:skip--> ```python 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 ```python import orthopy

orthopy.c1.jacobi.show(5, "normal", 0.0, 0.0)

plot, savefig also exist

```

Recurrence coefficients can be explicitly retrieved by ```python 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 x<sup>α</sup> exp(-r)

(Generalized) Laguerre polynomials. <!--pytest-codeblocks:skip--> python evaluator = orthopy.e1r.Eval(x, alpha=0, scaling="normal")

1D space with weight function exp(-r<sup>2</sup>)

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)

python 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.) <!--pytest-codeblocks:skip--> python evaluator = orthopy.c1.associated_legendre.Eval( x, phi=None, standardization="natural", with_condon_shortley_phase=True )

Triangle (T<sub>2</sub>)

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

```python import orthopy

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

Disk (S<sub>2</sub>)

| | :------------:|:-----------------:|:-----------:| 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.

```python 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. <!--pytest-codeblocks:skip--> ```python 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 python import orthopy

orthopy.u3.writetree("u3.vtk", 5, "quantum mechanic") ``` and open it with ParaView. Select the _srgb1 data set and turn off Map Scalars.

n-Cube (C_n)

| | :-------------------------:|:------------------:|:---------------:| C₁ (Legendre) | C₂ | C₃ |

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

python evaluator = orthopy.cn.Eval(X, alpha=0, beta=0) values, degrees = next(evaluator)

nD space with weight function exp(-r²) (E_n^r2)

| | :-------------------------:|:------------------:|:---------------:| E₁^r2 | E₂^r2 | E₃^r2 |

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

python evaluator = orthopy.enr2.Eval( x, 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: <!--pytest-codeblocks:skip--> python orthopy.tools.gautschi_test_3(moments, alpha, beta)

• Clenshaw algorithm for computing the weighted sum of orthogonal polynomials: <!--pytest-codeblocks:skip--> python vals = orthopy.c1.clenshaw(a, alpha, beta, t)

Installation

orthopy is available from the Python Package Index, so use pip install orthopy to install.

Testing

To run the tests, simply check out this repository and run pytest

Relevant publications

• 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

License

This software is published under the GPLv3 license.