Add DAE signature support

[yizhang-yiz]

This PR addresses #1091.

new integration tests:

• test/integration/good/dae_good.stan
• test/integration/bad/variadic_dae/*.stan

Submission Checklist

• Run unit tests
• Documentation
  • If a user-facing facing change was made, the documentation PR is here: DAE doc docs#467
  • OR, no user-facing changes were made

Release notes

Support Differential-Algebraic Equation solver.

Copyright and Licensing

By submitting this pull request, the copyright holder is agreeing to
license the submitted work under the BSD 3-clause license (https://opensource.org/licenses/BSD-3-Clause)

[WardBrian]

Do you have any sense of a timeline on stan-dev/math#2644?

#1027 is nearing completion, and whichever one of these merges first will create problems for the other (the typechecking and code gen of variadic functions is more or less completely rewritten in #1027)

[yizhang-yiz]

Do you have any sense of a timeline on stan-dev/math#2644?

From the initial feedback from @charlesm93 it should be soon. Of course the final decision is his.

[WardBrian]

One minor comment, otherwise LGTM

[WardBrian]

I think these comments in Stan_math_signatures provide more clutter than illumination.

Other than that, LGTM. I will approve/merge when the stan::math PR is merged and docs are approved

[WardBrian]

Looks like you'll need to update to the new pattern introduced in #1004 - should be pretty easy to see based on the variadic ode, again

[WardBrian]

Should be good to go after the math branch is merged

[rok-cesnovar]

Tests pass so feel free to merge.

[WardBrian]

Doc is under review still, but I think it will definitely get in before the release so we can merge