adding assoc_legendre: legenp in mpmath_translation in lambdify.py

[harshkasat]

References to other Issues or PRs

Partial fix for: #15795

Brief description of what is fixed or changed

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Release Notes

• core
  • Added "assoc_legendre" ("legenp") support in lambdify.py mpmath translation. Included a required test case for robust functionality.

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  • Added "assoc_legendre" ("legenp") support in lambdify.py mpmath translation. Included a required test case for robust functionality. (#26117 by @harshkasat)

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Partial fix for: #15795 

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  * Added "assoc_legendre" ("legenp") support in lambdify.py mpmath translation. Included a required test case for robust functionality.
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[oscarbenjamin]

Note my comment here #15795 (comment)

Anyone looking to open such a PR should check that the definition SymPy's assoc_legendre matches with mpmath's legenp. I have only done a simple test:

The documented definitions should be compared and more values (including negative, non-real etc) should be checked.

[harshkasat]

• The definition of Sympy and mpmath are mostly likely same.
• The different name use in
  • scipy :- scipy.special.legendre
  • numpy:- numpy.polynomial.legendre.Legendre

[github-actions]

Benchmark results from GitHub Actions

Lower numbers are good, higher numbers are bad. A ratio less than 1
means a speed up and greater than 1 means a slowdown. Green lines
beginning with + are slowdowns (the PR is slower then master or
master is slower than the previous release). Red lines beginning
with - are speedups.

Significantly changed benchmark results (PR vs master)

Significantly changed benchmark results (master vs previous release)

| Change   | Before [a00718ba]    | After [8824e3ad]    |   Ratio | Benchmark (Parameter)                                                |
|----------|----------------------|---------------------|---------|----------------------------------------------------------------------|
| -        | 70.3±1ms             | 45.3±1ms            |    0.64 | integrate.TimeIntegrationRisch02.time_doit(10)                       |
| -        | 68.2±0.7ms           | 43.7±0.5ms          |    0.64 | integrate.TimeIntegrationRisch02.time_doit_risch(10)                 |
| +        | 18.4±0.1μs           | 30.4±0.1μs          |    1.65 | integrate.TimeIntegrationRisch03.time_doit(1)                        |
| -        | 5.42±0.06ms          | 2.86±0.03ms         |    0.53 | logic.LogicSuite.time_load_file                                      |
| -        | 73.0±1ms             | 29.3±0.1ms          |    0.4  | polys.TimeGCD_GaussInt.time_op(1, 'dense')                           |
| -        | 25.9±0.2ms           | 16.9±0.1ms          |    0.65 | polys.TimeGCD_GaussInt.time_op(1, 'expr')                            |
| -        | 74.2±0.5ms           | 29.0±0.2ms          |    0.39 | polys.TimeGCD_GaussInt.time_op(1, 'sparse')                          |
| -        | 255±1ms              | 126±0.4ms           |    0.49 | polys.TimeGCD_GaussInt.time_op(2, 'dense')                           |
| -        | 256±1ms              | 126±1ms             |    0.49 | polys.TimeGCD_GaussInt.time_op(2, 'sparse')                          |
| -        | 660±3ms              | 372±2ms             |    0.56 | polys.TimeGCD_GaussInt.time_op(3, 'dense')                           |
| -        | 655±6ms              | 371±0.9ms           |    0.57 | polys.TimeGCD_GaussInt.time_op(3, 'sparse')                          |
| -        | 500±10μs             | 286±1μs             |    0.57 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(1, 'dense')            |
| -        | 1.78±0.01ms          | 1.04±0.01ms         |    0.58 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(2, 'dense')            |
| -        | 5.84±0.03ms          | 3.05±0.03ms         |    0.52 | polys.TimeGCD_LinearDenseQuadraticGCD.time_op(3, 'dense')            |
| -        | 445±4μs              | 232±2μs             |    0.52 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(1, 'dense')               |
| -        | 1.46±0.01ms          | 672±8μs             |    0.46 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(2, 'dense')               |
| -        | 4.87±0.01ms          | 1.65±0.01ms         |    0.34 | polys.TimeGCD_QuadraticNonMonicGCD.time_op(3, 'dense')               |
| -        | 376±2μs              | 206±0.9μs           |    0.55 | polys.TimeGCD_SparseGCDHighDegree.time_op(1, 'dense')                |
| -        | 2.45±0.01ms          | 1.24±0.01ms         |    0.51 | polys.TimeGCD_SparseGCDHighDegree.time_op(3, 'dense')                |
| -        | 10.0±0.05ms          | 4.33±0.04ms         |    0.43 | polys.TimeGCD_SparseGCDHighDegree.time_op(5, 'dense')                |
| -        | 360±6μs              | 168±1μs             |    0.47 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(1, 'dense')            |
| -        | 2.49±0.01ms          | 888±10μs            |    0.36 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(3, 'dense')            |
| -        | 9.49±0.04ms          | 2.67±0.01ms         |    0.28 | polys.TimeGCD_SparseNonMonicQuadratic.time_op(5, 'dense')            |
| -        | 1.03±0.01ms          | 428±3μs             |    0.41 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(3, 'dense')           |
| -        | 1.76±0.03ms          | 499±1μs             |    0.28 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(3, 'sparse')          |
| -        | 6.05±0.2ms           | 1.79±0.02ms         |    0.3  | polys.TimePREM_LinearDenseQuadraticGCD.time_op(5, 'dense')           |
| -        | 8.49±0.07ms          | 1.49±0ms            |    0.18 | polys.TimePREM_LinearDenseQuadraticGCD.time_op(5, 'sparse')          |
| -        | 285±1μs              | 64.5±0.6μs          |    0.23 | polys.TimePREM_QuadraticNonMonicGCD.time_op(1, 'sparse')             |
| -        | 3.46±0.03ms          | 394±2μs             |    0.11 | polys.TimePREM_QuadraticNonMonicGCD.time_op(3, 'dense')              |
| -        | 4.05±0.01ms          | 281±2μs             |    0.07 | polys.TimePREM_QuadraticNonMonicGCD.time_op(3, 'sparse')             |
| -        | 7.02±0.08ms          | 1.26±0ms            |    0.18 | polys.TimePREM_QuadraticNonMonicGCD.time_op(5, 'dense')              |
| -        | 8.66±0.05ms          | 832±9μs             |    0.1  | polys.TimePREM_QuadraticNonMonicGCD.time_op(5, 'sparse')             |
| -        | 5.00±0.02ms          | 2.99±0.02ms         |    0.6  | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(2, 'sparse') |
| -        | 12.3±0.2ms           | 6.47±0.02ms         |    0.53 | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(3, 'dense')  |
| -        | 22.3±0.1ms           | 9.04±0.03ms         |    0.4  | polys.TimeSUBRESULTANTS_LinearDenseQuadraticGCD.time_op(3, 'sparse') |
| -        | 5.20±0.02ms          | 873±4μs             |    0.17 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(1, 'sparse')    |
| -        | 12.8±0.1ms           | 7.07±0.04ms         |    0.55 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(2, 'sparse')    |
| -        | 104±0.5ms            | 25.6±0.1ms          |    0.25 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(3, 'dense')     |
| -        | 166±0.6ms            | 53.7±0.2ms          |    0.32 | polys.TimeSUBRESULTANTS_QuadraticNonMonicGCD.time_op(3, 'sparse')    |
| -        | 174±0.9μs            | 113±0.8μs           |    0.65 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(1, 'dense')      |
| -        | 358±0.5μs            | 215±1μs             |    0.6  | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(1, 'sparse')     |
| -        | 4.35±0.08ms          | 861±3μs             |    0.2  | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(3, 'dense')      |
| -        | 5.36±0.03ms          | 382±3μs             |    0.07 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(3, 'sparse')     |
| -        | 20.2±0.09ms          | 2.75±0.01ms         |    0.14 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(5, 'dense')      |
| -        | 23.0±0.5ms           | 625±1μs             |    0.03 | polys.TimeSUBRESULTANTS_SparseGCDHighDegree.time_op(5, 'sparse')     |
| -        | 480±5μs              | 135±1μs             |    0.28 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(1, 'sparse') |
| -        | 4.75±0.01ms          | 619±1μs             |    0.13 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(3, 'dense')  |
| -        | 5.29±0.04ms          | 139±0.7μs           |    0.03 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(3, 'sparse') |
| -        | 13.2±0.08ms          | 1.29±0ms            |    0.1  | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(5, 'dense')  |
| -        | 13.7±0.08ms          | 142±0.4μs           |    0.01 | polys.TimeSUBRESULTANTS_SparseNonMonicQuadratic.time_op(5, 'sparse') |
| -        | 133±1μs              | 75.6±0.6μs          |    0.57 | solve.TimeMatrixOperations.time_rref(3, 0)                           |
| -        | 247±0.3μs            | 89.3±0.4μs          |    0.36 | solve.TimeMatrixOperations.time_rref(4, 0)                           |
| -        | 23.9±0.1ms           | 10.2±0.04ms         |    0.43 | solve.TimeSolveLinSys189x49.time_solve_lin_sys                       |
| -        | 28.5±0.3ms           | 15.5±0.2ms          |    0.54 | solve.TimeSparseSystem.time_linsolve_Aaug(20)                        |
| -        | 55.1±0.4ms           | 25.1±0.2ms          |    0.46 | solve.TimeSparseSystem.time_linsolve_Aaug(30)                        |
| -        | 28.1±0.2ms           | 15.3±0.07ms         |    0.54 | solve.TimeSparseSystem.time_linsolve_Ab(20)                          |
| -        | 54.5±0.2ms           | 24.7±0.1ms          |    0.45 | solve.TimeSparseSystem.time_linsolve_Ab(30)                          |

Full benchmark results can be found as artifacts in GitHub Actions
(click on checks at the top of the PR).

[oscarbenjamin]

• The definition of Sympy and mpmath are mostly likely same.

What does "most likely" mean?

It is fine if you do not know how to check this but then just say so clearly. If you haven't confirmed this then someone else will need to or otherwise this cannot be merged.

[harshkasat]

Dear @oscarbenjamin, please accept my sincere apologies for my previous behaviour. In the future, I will ensure that my messages are clear, concise, and respectful at all times. I understand that my previous conduct was unbecoming of the open-source community.

About associate legendre from sympy and mpmath:

Sympy:-

• The assoc_legendre function calculates the associated Legendre polynomial of degree n and order m for a given variable x. It returns a polynomial /numerical expression.

Mpmath:-

• The mpmath.legenp function calculates the (associated) Legendre function of the first kind of degree n and order m for a given complex number z. It returns a numerical value.

• In summary while there may be variations in the specific details of the implementation, the underlying mathematical concept of associated Legendre functions is consistent between SymPy and mpmath.

About associate legendre name from other library are :

Scipy:-

• scipy.special.legendre(n, monic=False)

from scipy.special import legendre
legendre(3)
poly1d([ 2.5,  0. , -1.5,  0. ])

Mpmath:-

• mpmath.legenp(n, m, z, type=2)

from mpmath import legenp
legenp(2, 0, 10)

Numpy

• numpy.polynomial.legendre.Legendre(coef, domain=None, window=None, symbol='x')

• If I have missed anything, please update me. And if I have made any mistakes, please let me know. Sorry for Incovince.

P.S

• Based on the observation mentioned above, I have concluded that mpmath.legenp and sympy.functions.special.polynomials.assoc_legendre have the same definition.

[harshkasat]

I tried to add a test case with a pull request, but I keep getting merge conflicts. I even pulled master into my branch, but I still got merge conflicts. Can you please guide me?