orthopy 0.10.4
Orthogonal polynomials for Python
Requires Python: >=3.10

<p align="center">
  <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>
  <p align="center">All about orthogonal polynomials.</p>
</p>

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orthopy provides various orthogonal polynomial classes for
[lines](#line-segment--1-1-with-weight-function-1-x%CE%B1-1-x%CE%B2),
[triangles](#triangle-42),
[disks](#disk-s2),
[spheres](#sphere-u2),
[n-cubes](#n-cube-cn),
[the nD space with weight function exp(-r<sup>2</sup>)](#nd-space-with-weight-function-exp-r2-enr2)
and more.
All computations are done using numerically stable recurrence schemes. Furthermore, all
functions are fully vectorized and can return results in _exact arithmetic_.

### Installation

Install orthopy [from PyPI](https://pypi.org/project/orthopy/) with

```
pip install orthopy
```

### How to get a license

Licenses for personal and academic use can be purchased
[here](https://buy.stripe.com/aEUg1H38OgDw5qMfZ3).
You'll receive a confirmation email with a license key.
Install the key with

```
plm add <your-license-key>
```

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

<!--pytest.mark.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
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>

| <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%"> |
| :---------------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------------: |
|                                            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.mark.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)

<img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/e1r.svg" width="45%">

(Generalized) Laguerre polynomials.

<!--pytest.mark.skip-->

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

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

<img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/e1r2.svg" width="45%">

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)`

<!--pytest.mark.skip-->

```python
evaluator = orthopy.e1r2.Eval(
    x,
    "probabilists",  # or "physicists"
    "normal"
)
```

#### Associated Legendre "polynomials"

<img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/associated-legendre.svg" width="45%">

Not all of those are polynomials, so they should really be called associated Legendre
_functions_. The <i>k</i>th iteration contains _2k+1_ functions, indexed from
_-k_ to _k_. (See the color grouping in the above plot.)

<!--pytest.mark.skip-->

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

### Triangle (_T<sub>2</sub>_)

<img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/triangle-tree.png" width="40%">

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>_)

| <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%"> |
| :------------------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------------------: | :------------------------------------------------------------------------------------------------------: |
|                                                 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>_)

<img src="https://raw.githubusercontent.com/sigma-py/orthopy/assets/sph-tree.png" width="50%">

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 <i>n</i>th level.

<!--pytest.mark.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.write_tree("u3.vtk", 5, "quantum mechanic") -->
<!-- ``` -->
<!---->
<!-- and open it with [ParaView](https://www.paraview.org/). Select the _srgb1_ data set and -->
<!-- turn off _Map Scalars_. -->

### _n_-Cube (_C<sub>n</sub>_)

| <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%"> |
| :---------------------------------------------------------------------------------------: | :---------------------------------------------------------------------------------------: | :---------------------------------------------------------------------------------------: |
|                                 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]`.

<!--pytest.mark.skip-->

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

### <i>n</i>D space with weight function exp(-r<sup>2</sup>) (_E<sub>n</sub><sup>r<sup>2</sup></sup>_)

| <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%"> |
| :-----------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------------: |
|                           _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>_                           |

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

<!--pytest.mark.skip-->

```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:
  <!--pytest.mark.skip-->

  ```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:
  <!--pytest.mark.skip-->
  ```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)