└── README.md


/README.md:
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  1 | <p align="center">
  2 |   <a href="https://github.com/nschloe/orthopy"><img alt="orthopy" src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/orthopy-logo-with-text.png" width="30%"></a>
  3 |   <p align="center">All about orthogonal polynomials.</p>
  4 | </p>
  5 | 
  6 | [![PyPi Version](https://img.shields.io/pypi/v/orthopy.svg?style=flat-square)](https://pypi.org/project/orthopy)
  7 | [![PyPI pyversions](https://img.shields.io/pypi/pyversions/orthopy.svg?style=flat-square)](https://pypi.org/pypi/orthopy/)
  8 | [![GitHub stars](https://img.shields.io/github/stars/nschloe/orthopy.svg?style=flat-square&logo=github&label=Stars&logoColor=white)](https://github.com/nschloe/orthopy)
  9 | [![Downloads](https://pepy.tech/badge/orthopy/month?style=flat-square)](https://pepy.tech/project/orthopy)
 10 | 
 11 | <!--[![PyPi downloads](https://img.shields.io/pypi/dm/orthopy.svg?style=flat-square)](https://pypistats.org/packages/orthopy)-->
 12 | 
 13 | [![Discord](https://img.shields.io/static/v1?logo=discord&logoColor=white&label=chat&message=on%20discord&color=7289da&style=flat-square)](https://discord.gg/hnTJ5MRX2Y)
 14 | [![orthogonal](https://img.shields.io/badge/orthogonal-yes-ff69b4.svg?style=flat-square)](https://github.com/nschloe/orthopy)
 15 | 
 16 | orthopy provides various orthogonal polynomial classes for
 17 | [lines](#line-segment--1-1-with-weight-function-1-x%CE%B1-1-x%CE%B2),
 18 | [triangles](#triangle-42),
 19 | [disks](#disk-s2),
 20 | [spheres](#sphere-u2),
 21 | [n-cubes](#n-cube-cn),
 22 | [the nD space with weight function exp(-r<sup>2</sup>)](#nd-space-with-weight-function-exp-r2-enr2)
 23 | and more.
 24 | All computations are done using numerically stable recurrence schemes. Furthermore, all
 25 | functions are fully vectorized and can return results in _exact arithmetic_.
 26 | 
 27 | ### Installation
 28 | 
 29 | Install orthopy [from PyPI](https://pypi.org/project/orthopy/) with
 30 | 
 31 | ```
 32 | pip install orthopy
 33 | ```
 34 | 
 35 | ### How to get a license
 36 | 
 37 | Licenses for personal and academic use can be purchased
 38 | [here](https://buy.stripe.com/aEUg1H38OgDw5qMfZ3).
 39 | You'll receive a confirmation email with a license key.
 40 | Install the key with
 41 | 
 42 | ```
 43 | plm add <your-license-key>
 44 | ```
 45 | 
 46 | on your machine and you're good to go.
 47 | 
 48 | For commercial use, please contact support@mondaytech.com.
 49 | 
 50 | ### Basic usage
 51 | 
 52 | The main function of all submodules is the iterator `Eval` which evaluates the series of
 53 | orthogonal polynomials with increasing degree at given points using a recurrence
 54 | relation, e.g.,
 55 | 
 56 | ```python
 57 | import orthopy
 58 | 
 59 | x = 0.5
 60 | 
 61 | evaluator = orthopy.c1.legendre.Eval(x, "classical")
 62 | for _ in range(5):
 63 |      print(next(evaluator))
 64 | ```
 65 | 
 66 | ```python
 67 | 1.0          # P_0(0.5)
 68 | 0.5          # P_1(0.5)
 69 | -0.125       # P_2(0.5)
 70 | -0.4375      # P_3(0.5)
 71 | -0.2890625   # P_4(0.5)
 72 | ```
 73 | 
 74 | Other ways of getting the first `n` items are
 75 | 
 76 | <!--pytest.mark.skip-->
 77 | 
 78 | ```python
 79 | evaluator = Eval(x, "normal")
 80 | vals = [next(evaluator) for _ in range(n)]
 81 | 
 82 | import itertools
 83 | vals = list(itertools.islice(Eval(x, "normal"), n))
 84 | ```
 85 | 
 86 | Instead of evaluating at only one point, you can provide any array for `x`; the
 87 | polynomials will then be evaluated for all points at once. You can also use sympy for
 88 | symbolic computation:
 89 | 
 90 | ```python
 91 | import itertools
 92 | import orthopy
 93 | import sympy
 94 | 
 95 | x = sympy.Symbol("x")
 96 | 
 97 | evaluator = orthopy.c1.legendre.Eval(x, "classical")
 98 | for val in itertools.islice(evaluator, 5):
 99 |      print(sympy.expand(val))
100 | ```
101 | 
102 | ```
103 | 1
104 | x
105 | 3*x**2/2 - 1/2
106 | 5*x**3/2 - 3*x/2
107 | 35*x**4/8 - 15*x**2/4 + 3/8
108 | ```
109 | 
110 | All `Eval` methods have a `scaling` argument which can have three values:
111 | 
112 | - `"monic"`: The leading coefficient is 1.
113 | - `"classical"`: The maximum value is 1 (or (n+alpha over n)).
114 | - `"normal"`: The integral of the squared function over the domain is 1.
115 | 
116 | For univariate ("one-dimensional") integrals, every new iteration contains one function.
117 | For bivariate ("two-dimensional") domains, every level will contain one function more
118 | than the previous, and similarly for multivariate families. See the tree plots below.
119 | 
120 | ### Line segment (-1, +1) with weight function (1-x)<sup>α</sup> (1+x)<sup>β</sup>
121 | 
122 | | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/legendre.svg" width="100%"> | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/chebyshev1.svg" width="100%"> | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/chebyshev2.svg" width="100%"> |
123 | | :---------------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------------: |
124 | |                                            Legendre                                             |                                            Chebyshev 1                                            |                                            Chebyshev 2                                            |
125 | 
126 | Jacobi, Gegenbauer (α=β), Chebyshev 1 (α=β=-1/2), Chebyshev 2 (α=β=1/2), Legendre
127 | (α=β=0) polynomials.
128 | 
129 | <!--pytest.mark.skip-->
130 | 
131 | ```python
132 | import orthopy
133 | 
134 | orthopy.c1.legendre.Eval(x, "normal")
135 | orthopy.c1.chebyshev1.Eval(x, "normal")
136 | orthopy.c1.chebyshev2.Eval(x, "normal")
137 | orthopy.c1.gegenbauer.Eval(x, "normal", lmbda)
138 | orthopy.c1.jacobi.Eval(x, "normal", alpha, beta)
139 | ```
140 | 
141 | The plots above are generated with
142 | 
143 | ```python
144 | import orthopy
145 | 
146 | orthopy.c1.jacobi.show(5, "normal", 0.0, 0.0)
147 | # plot, savefig also exist
148 | ```
149 | 
150 | Recurrence coefficients can be explicitly retrieved by
151 | 
152 | ```python
153 | import orthopy
154 | 
155 | rc = orthopy.c1.jacobi.RecurrenceCoefficients(
156 |     "monic",  # or "classical", "normal"
157 |     alpha=0, beta=0, symbolic=True
158 | )
159 | print(rc.p0)
160 | for k in range(5):
161 |     print(rc[k])
162 | ```
163 | 
164 | ```
165 | 1
166 | (1, 0, None)
167 | (1, 0, 1/3)
168 | (1, 0, 4/15)
169 | (1, 0, 9/35)
170 | (1, 0, 16/63)
171 | ```
172 | 
173 | ### 1D half-space with weight function x<sup>α</sup> exp(-r)
174 | 
175 | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/e1r.svg" width="45%">
176 | 
177 | (Generalized) Laguerre polynomials.
178 | 
179 | <!--pytest.mark.skip-->
180 | 
181 | ```python
182 | evaluator = orthopy.e1r.Eval(x, alpha=0, scaling="normal")
183 | ```
184 | 
185 | ### 1D space with weight function exp(-r<sup>2</sup>)
186 | 
187 | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/e1r2.svg" width="45%">
188 | 
189 | Hermite polynomials come in two standardizations:
190 | 
191 | - `"physicists"` (against the weight function `exp(-x ** 2)`
192 | - `"probabilists"` (against the weight function `1 / sqrt(2 * pi) * exp(-x ** 2 / 2)`
193 | 
194 | <!--pytest.mark.skip-->
195 | 
196 | ```python
197 | evaluator = orthopy.e1r2.Eval(
198 |     x,
199 |     "probabilists",  # or "physicists"
200 |     "normal"
201 | )
202 | ```
203 | 
204 | #### Associated Legendre "polynomials"
205 | 
206 | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/associated-legendre.svg" width="45%">
207 | 
208 | Not all of those are polynomials, so they should really be called associated Legendre
209 | _functions_. The <i>k</i>th iteration contains _2k+1_ functions, indexed from
210 | _-k_ to _k_. (See the color grouping in the above plot.)
211 | 
212 | <!--pytest.mark.skip-->
213 | 
214 | ```python
215 | evaluator = orthopy.c1.associated_legendre.Eval(
216 |     x, phi=None, standardization="natural", with_condon_shortley_phase=True
217 | )
218 | ```
219 | 
220 | ### Triangle (_T<sub>2</sub>_)
221 | 
222 | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/triangle-tree.png" width="40%">
223 | 
224 | orthopy's triangle orthogonal polynomials are evaluated in terms of [barycentric
225 | coordinates](https://en.wikipedia.org/wiki/Barycentric_coordinate_system), so the
226 | `X.shape[0]` has to be 3.
227 | 
228 | ```python
229 | import orthopy
230 | 
231 | bary = [0.1, 0.7, 0.2]
232 | evaluator = orthopy.t2.Eval(bary, "normal")
233 | ```
234 | 
235 | ### Disk (_S<sub>2</sub>_)
236 | 
237 | | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/disk-xu-tree.png" width="70%"> | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/disk-zernike-tree.png" width="70%"> | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/disk-zernike2-tree.png" width="70%"> |
238 | | :------------------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------------------: | :------------------------------------------------------------------------------------------------------: |
239 | |                                                 Xu                                                 |                      [Zernike](https://en.wikipedia.org/wiki/Zernike_polynomials)                       |                                                Zernike 2                                                 |
240 | 
241 | orthopy contains several families of orthogonal polynomials on the unit disk: After
242 | [Xu](https://arxiv.org/abs/1701.02709),
243 | [Zernike](https://en.wikipedia.org/wiki/Zernike_polynomials), and a simplified version
244 | of Zernike polynomials.
245 | 
246 | ```python
247 | import orthopy
248 | 
249 | x = [0.1, -0.3]
250 | 
251 | evaluator = orthopy.s2.xu.Eval(x, "normal")
252 | # evaluator = orthopy.s2.zernike.Eval(x, "normal")
253 | # evaluator = orthopy.s2.zernike2.Eval(x, "normal")
254 | ```
255 | 
256 | ### Sphere (_U<sub>3</sub>_)
257 | 
258 | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/sph-tree.png" width="50%">
259 | 
260 | Complex-valued _spherical harmonics,_ (black=zero, green=real positive,
261 | pink=real negative, blue=imaginary positive, yellow=imaginary negative). The
262 | functions in the middle are real-valued. The complex angle takes _n_ turns on
263 | the <i>n</i>th level.
264 | 
265 | <!--pytest.mark.skip-->
266 | 
267 | ```python
268 | evaluator = orthopy.u3.EvalCartesian(
269 |     x,
270 |     scaling="quantum mechanic"  # or "acoustic", "geodetic", "schmidt"
271 | )
272 | 
273 | evaluator = orthopy.u3.EvalSpherical(
274 |     theta_phi,  # polar, azimuthal angles
275 |     scaling="quantum mechanic"  # or "acoustic", "geodetic", "schmidt"
276 | )
277 | ```
278 | 
279 | <!-- To generate the above plot, write the tree mesh to a file -->
280 | <!---->
281 | <!-- ```python -->
282 | <!-- import orthopy -->
283 | <!---->
284 | <!-- orthopy.u3.write_tree("u3.vtk", 5, "quantum mechanic") -->
285 | <!-- ``` -->
286 | <!---->
287 | <!-- and open it with [ParaView](https://www.paraview.org/). Select the _srgb1_ data set and -->
288 | <!-- turn off _Map Scalars_. -->
289 | 
290 | ### _n_-Cube (_C<sub>n</sub>_)
291 | 
292 | | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/c1.svg" width="100%"> | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/c2.png" width="100%"> | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/c3.png" width="100%"> |
293 | | :---------------------------------------------------------------------------------------: | :---------------------------------------------------------------------------------------: | :---------------------------------------------------------------------------------------: |
294 | |                                 C<sub>1</sub> (Legendre)                                  |                                       C<sub>2</sub>                                       |                                       C<sub>3</sub>                                       |
295 | 
296 | Jacobi product polynomials.
297 | All polynomials are normalized on the n-dimensional cube. The dimensionality is
298 | determined by `X.shape[0]`.
299 | 
300 | <!--pytest.mark.skip-->
301 | 
302 | ```python
303 | evaluator = orthopy.cn.Eval(X, alpha=0, beta=0)
304 | values, degrees = next(evaluator)
305 | ```
306 | 
307 | ### <i>n</i>D space with weight function exp(-r<sup>2</sup>) (_E<sub>n</sub><sup>r<sup>2</sup></sup>_)
308 | 
309 | | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/e1r2.svg" width="100%"> | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/e2r2.png" width="100%"> | <img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/e3r2.png" width="100%"> |
310 | | :-----------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------: |
311 | |                           _E<sub>1</sub><sup>r<sup>2</sup></sup>_                           |                           _E<sub>2</sub><sup>r<sup>2</sup></sup>_                           |                           _E<sub>3</sub><sup>r<sup>2</sup></sup>_                           |
312 | 
313 | Hermite product polynomials.
314 | All polynomials are normalized over the measure. The dimensionality is determined by
315 | `X.shape[0]`.
316 | 
317 | <!--pytest.mark.skip-->
318 | 
319 | ```python
320 | evaluator = orthopy.enr2.Eval(
321 |     x,
322 |     standardization="probabilists"  # or "physicists"
323 | )
324 | values, degrees = next(evaluator)
325 | ```
326 | 
327 | ### Other tools
328 | 
329 | - Generating recurrence coefficients for 1D domains with
330 |   [Stieltjes](https://github.com/nschloe/orthopy/wiki/Generating-1D-recurrence-coefficients-for-a-given-weight#stieltjes),
331 |   [Golub-Welsch](https://github.com/nschloe/orthopy/wiki/Generating-1D-recurrence-coefficients-for-a-given-weight#golub-welsch),
332 |   [Chebyshev](https://github.com/nschloe/orthopy/wiki/Generating-1D-recurrence-coefficients-for-a-given-weight#chebyshev), and
333 |   [modified
334 |   Chebyshev](https://github.com/nschloe/orthopy/wiki/Generating-1D-recurrence-coefficients-for-a-given-weight#modified-chebyshev).
335 | 
336 | - The the sanity of recurrence coefficients with test 3 from [Gautschi's article](https://doi.org/10.1007/BF02218441):
337 |   computing the weighted sum of orthogonal polynomials:
338 |   <!--pytest.mark.skip-->
339 | 
340 |   ```python
341 |   orthopy.tools.gautschi_test_3(moments, alpha, beta)
342 |   ```
343 | 
344 | - [Clenshaw algorithm](https://en.wikipedia.org/wiki/Clenshaw_algorithm) for
345 |   computing the weighted sum of orthogonal polynomials:
346 |   <!--pytest.mark.skip-->
347 |   ```python
348 |   vals = orthopy.c1.clenshaw(a, alpha, beta, t)
349 |   ```
350 | 
351 | ### Relevant publications
352 | 
353 | - [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)
354 | - [Abedallah Rababah, Recurrence Relations for Orthogonal Polynomials on Triangular Domains, MDPI Mathematics 2016, 4(2)](https://doi.org/10.3390/math4020025)
355 | - [Yuan Xu, Orthogonal polynomials of several variables, arxiv.org, January 2017](https://arxiv.org/abs/1701.02709)
356 | 


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