Added Jacobian Features in Calculus functions.py

[Kritika75]

Enhanced the jacobian function to include additional features: computation of Hessian matrices for single functions, optional sparse matrix representation for large systems, and the ability to calculate the rank of the Jacobian matrix.

Fixes #39218

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

[vincentmacri]

@tscrim (since I know you have the permission to approve workflows from first-time contributors) would you be willing to take a look at this? Or approve the CI workflows so I can review it?

[github-actions]

Documentation preview for this PR (built with commit ec51fc1; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[tscrim]

@vincentmacri It has been done by someone else. Please go ahead and do the review. I can re-approve workflows as needed.

[vincentmacri]

I think you meant:

Suggested change

	jacobian_matrix([[diff(f, v) for v in variables] for f in functions])
	jacobian_matrix = matrix([[diff(f, v) for v in variables] for f in functions])

[vincentmacri]

But with the other changes I suggested you should just revert the jacobian function back to how it was, and add a hessian function for that functionality.

[vincentmacri]

I think that hessian should be its own function unless you have a good reason for it not to be separate.

I don't think we need the rank option since the returns matrix already has a .rank() method that can be called on it.

[vincentmacri]

I also don't see why the sparse option is needed when you could just call .sparse_matrix() on the returned matrix.

[vincentmacri]

@Kritika75 In summary, unless there is some good reason I'm missing for why you implemented things the way you did, can you make these three changes?

1. Remove the hessian option and add a hessian function instead.
2. Remove the rank option, it's unnecessary since matrices already have a .rank() method. It also complicates type checking if the function can possibly return two different types (either a matrix or a tuple).
3. Remove the sparse option, it's unnecessary since matrices already have a .sparse_matrix() method.

Since you're a first-time contributor, GitHub is kind of annoying about running the CI tests on your PR and someone needs to approve every test run manually. To speed things up, please test things locally and make sure your tests pass before requesting another review.

[tscrim]

I also don't see why the sparse option is needed when you could just call .sparse_matrix() on the returned matrix.

You don't want to create a dense matrix just to make it sparse. This is wasteful in both memory and computation time.

[tscrim]

That being said, the code itself should directly create thr sparse matrix, otherwise it should not have such an option as @vincentmacri said.

Although in nearly all cases, this will not be a sparse matrix. So there might not be much practicality in having it (and hence, the maintenance burden of such code).

[Kritika75]

Apologies for the delay in getting back to you. I had my exams, but I completely understand your concern and will address it promptly

[vincentmacri]

One small change request. Looks good otherwise!

[vincentmacri]

@Kritika75 also make sure to pull the latest changes from develop, otherwise git won't let us merge this.

[vincentmacri]

@Kritika75 you need to merge the changes from develop into your patch-1 branch.

[vincentmacri]

If you need a quick git tutorial.

1. Make sure the develop branch of your fork is up to date with the sage develop branch
2. On your machine, from the develop branch run git pull. Then change to your patch-1 branch and run git merge develop. Then git push.