All about orthogonal polynomials.

``` on your machine and you're good to go. For commercial use, please contact support@mondaytech.com. ### Basic usage 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 # 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 ```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 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)<sup>α</sup> (1+x)<sup>β</sup> | | | | | :---------------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------------: | | 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. ```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. ```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.) ```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](https://en.wikipedia.org/wiki/Barycentric_coordinate_system), 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](https://en.wikipedia.org/wiki/Zernike_polynomials) | Zernike 2 | orthopy contains several families of orthogonal polynomials on the unit disk: After [Xu](https://arxiv.org/abs/1701.02709), [Zernike](https://en.wikipedia.org/wiki/Zernike_polynomials), 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 (_U<sub>3</sub>_) Complex-valued _spherical harmonics,_ (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. ```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" ) ``` ### _n_-Cube (_C<sub>n</sub>_) | | | | | :---------------------------------------------------------------------------------------: | :---------------------------------------------------------------------------------------: | :---------------------------------------------------------------------------------------: | | C<sub>1</sub> (Legendre) | C<sub>2</sub> | C<sub>3</sub> | 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(-r<sup>2</sup>) (_E<sub>n</sub><sup>r2</sup>_) | | | | | :-----------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------: | | _E<sub>1</sub><sup>r2</sup>_ | _E<sub>2</sub><sup>r2</sup>_ | _E<sub>3</sub><sup>r2</sup>_ | 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](https://github.com/nschloe/orthopy/wiki/Generating-1D-recurrence-coefficients-for-a-given-weight#stieltjes), [Golub-Welsch](https://github.com/nschloe/orthopy/wiki/Generating-1D-recurrence-coefficients-for-a-given-weight#golub-welsch), [Chebyshev](https://github.com/nschloe/orthopy/wiki/Generating-1D-recurrence-coefficients-for-a-given-weight#chebyshev), and [modified Chebyshev](https://github.com/nschloe/orthopy/wiki/Generating-1D-recurrence-coefficients-for-a-given-weight#modified-chebyshev). - The the sanity of recurrence coefficients with test 3 from [Gautschi's article](https://doi.org/10.1007/BF02218441): computing the weighted sum of orthogonal polynomials: ```python orthopy.tools.gautschi_test_3(moments, alpha, beta) ``` - [Clenshaw algorithm](https://en.wikipedia.org/wiki/Clenshaw_algorithm) for computing the weighted sum of orthogonal polynomials: ```python vals = orthopy.c1.clenshaw(a, alpha, beta, t) ``` ### 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](https://doi.org/10.1145/1644001.1644006) - [Abedallah Rababah, Recurrence Relations for Orthogonal Polynomials on Triangular Domains, MDPI Mathematics 2016, 4(2)](https://doi.org/10.3390/math4020025) - [Yuan Xu, Orthogonal polynomials of several variables, arxiv.org, January 2017](https://arxiv.org/abs/1701.02709)