Support for hypergeometric functions #1383

aravindh-krishnamoorthy · 2025-02-12T13:36:51Z

Note

This PR is still under development.

Introduction

This PR includes support for the following hypergeometric functions:

Hypergeometric1F1
HypergeometricPFQ
MeijerG
HypergeometricU

Checklist

Functionality and output (see tests)
Documentation (valid / in line with MMA)
Function attributes (valid / in line with MMA)
Special cases (valid / in line with MMA)
~~Special behaviour where z_?RealQ leads to a numerical evaluation (as an earlier MMA alternative to N[]).~~ (z_?MachineNumberQ is used instead of RealQ.)
Functions work with Plot[]

Tests

TBD

mathics/builtin/specialfns/hypergeom.py

Co-authored-by: R. Bernstein <[email protected]>

aravindh-krishnamoorthy · 2025-02-12T23:19:12Z

I am actually thinking about changing this to a sympy based implementation.

rocky · 2025-02-12T23:54:46Z

I am actually thinking about changing this to a sympy based implementation.

SymPy does import mpmath. So in theory SymPy could/should fall back to mpmath. (I don't know if that was always the case in the past when this project first started out.)

I don't know enough about the pros and cons of doing using SymPy over mpmath. Please educate me.

aravindh-krishnamoorthy · 2025-02-13T11:23:48Z

It'd possibly allow for things like Limit[Hypergeometric1F1[a,b,z], a->0] and hyperexpand to simplify expression involving hypergeometric functions. OTOH, the mpmath version only works with numerical (fp) arguments. I'll check the details out and get back to you soon...

If I do it, I'll also change the existing HypergeometricU function.

mathics/builtin/specialfns/hypergeom.py

rocky · 2025-02-14T00:05:55Z

It'd possibly allow for things like Limit[Hypergeometric1F1[a,b,z], a->0] and hyperexpand to simplify expression involving hypergeometric functions. OTOH, the mpmath version only works with numerical (fp) arguments. I'll check the details out and get back to you soon...

While I think adding a SymPy method for Hypergeometric1F1 when a, b, and z are vectors or are symbolic is a great idea, I think in this situation it is better left as a follow-on PR. Here is why...

If doing that would cause the stuff here to be drastically ripped out or changed, then it would make sense to wait. But as best as I can tell that's not likely to happen here. Right?

This PR, I think, is complete by itself and a useful step forward. It is also getting large.

Many small and quicker PRs (assuming we are not just redoing the previous ones because we haven't thought about the bigger picture), are better than one large PR that spans a long time period. Here, I think you outlined the bigger picture, and if I have it right there are two reasonably separable steps to this.

If I do it, I'll also change the existing HypergeometricU function.

Thanks.

Co-authored-by: R. Bernstein <[email protected]>

rocky · 2025-02-14T13:18:02Z

mathics/builtin/specialfns/bessel.py

-    summary_text = "Tricomi confluent hypergeometric function"
-    sympy_name = ""
-
-


Please don't revise functions until we've mastered the existing ones.

rocky · 2025-02-14T13:18:45Z

mathics/builtin/specialfns/hypergeom.py

@@ -117,3 +117,101 @@ class Hypergeometric1F1(MPMathFunction):
    }
    summary_text = "compute Kummer confluent hypergeometric function"
    sympy_name = ""
+
+
+class MeijerG(MPMathFunction):


Please do not start more functions until we have mastered the ones in progress.

Rocky, please kindly wait with the reviews. I'll let you know once I'm done with this PR and put it out of draft mode. Otherwise, my development flow may be confusing to you.

This is already too large. Please reduce the scope.

If you don't want folks to see what's up. People get notifications even when working in draft mode.

Instead, fork the code and work in your private fork. Thanks.

rocky · 2025-02-14T13:23:37Z

235cb2f improves things but also has some serious problems. WMA for Hypergeometric functions gives numeric results when numeric arguments are given. So we need to do the same thing.

aravindh-krishnamoorthy · 2025-02-14T13:27:17Z

235cb2f improves things but also has some serious problems. WMA for Hypergeometric functions gives numeric results when numeric arguments are given. So we need to do the same thing.

Thank you for brining this up. Indeed, I've this on my radar and take care of the compatibility and the flags. Once I'm done, I will flag this PR for review.

rocky · 2025-02-14T13:27:39Z

mathics/builtin/specialfns/hypergeom.py

+    Here, plot explicitly requests a numerical evaluation.
+    """
+
+    attributes = A_LISTABLE | A_NUMERIC_FUNCTION | A_PROTECTED | A_READ_PROTECTED


@rocky I've these changes on my radar. Kindly wait until I've marked this PR for review. Before that, I'm just focusing on the important functional aspects. But, I'll fix the documentation and the attributes before completion.

For now, I'll limit myself to these four functions, which I will also be needing at work right now. But, eventually, other hypergeometric functions can be implemented just via rules.

I'll do that later in another PR.

rocky · 2025-02-14T13:53:01Z

Note

This PR is still under development.

Introduction

This PR includes support for the following hypergeometric functions:

Hypergeometric1F1

HypergeometricPFQ

MeijerG

HypergeometricU

Why include MeijerG and HypergeometricU? What essential features do each of these add above Hypergeometric1F1 and HypegeomtericPFQ?

Also note there is Hypergeometric0F1 and Hypergeometric2F1 which might easily be added later.

aravindh-krishnamoorthy · 2025-02-14T14:24:34Z

Note
This PR is still under development.

Introduction

This PR includes support for the following hypergeometric functions:

Hypergeometric1F1

HypergeometricPFQ

MeijerG

HypergeometricU

Why include MeijerG and HypergeometricU? What essential features do each of these add above Hypergeometric1F1 and HypegeomtericPFQ?

Also note there is Hypergeometric0F1 and Hypergeometric2F1 which might easily be added later.

In fact, as rightly identified by SymPy, good routines for HypergeometricPFQ and MeijerG are sufficient to implement other hypergeometric functions.

Unfortunately, I need Hypergeometric1F1 and HypergeometricU for my work right now and HypergeometricU is symbolically supported in Sympy only via MeijerG.

But then I realised that mpmath has a specialised routine for HypergeometricU. So, am using the current implementation to compare the numerical performance of the mpmath version to that via N[MeijerG]. If I find them to be equivalent, we'll only need HypergeometricPFQ and MeijerG (sympy + mpmath) and the remainder can be implemented via rules.

rocky · 2025-02-14T14:32:00Z

In fact, as rightly identified by SymPy, good routines for HypergeometricPFQ and MeijerG are sufficient to implement other hypergeometric functions.

Unfortunately, I need Hypergeometric1F1 and HypergeometricU for my work right now and HypergeometricU is symbolically supported in Sympy only via MeijerG.

But then I realised that mpmath has a specialised routine for HypergeometricU. So, am using the current implementation to compare the numerical performance of the mpmath version to that via N[MeijerG]. If I find them to be equivalent, we'll only need HypergeometricPFQ and MeijerG (sympy + mpmath) and the remainder can be implemented via rules.

Ok - Thanks for the information and explanation.

Aravindh Krishnamoorthy added 2 commits February 12, 2025 13:33

Hypergeometric1F1

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Black and isort: hypergeom.py

bbecd08

rocky reviewed Feb 12, 2025

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Apply suggestions from code review

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Co-authored-by: R. Bernstein <[email protected]>

Aravindh Krishnamoorthy added 2 commits February 13, 2025 20:18

Hypergeometric functions based on sympy and mmpmath.

2d369c7

Black and isort on hypergeom.py

6eb4e20