pow() for varmat

[SteveBronder]

Summary

This adds pow for var<Matrix> types. I thought it would be better as it's own PR since this is a vectorized binary functions that takes in all combinations of matrices and scalars (and std::vector<int>/std::vector<std::vector<int>>)

Tests

Tests added in mix for pow with varmat as well as all of the vectorizable input types.

./': ./runTests.py -j28 ./test/unit/math/mix/fun/ -f /pow

Side Effects

Nope!

Release notes

Replace this text with a short note on what will change if this pull request is merged in which case this will be included in the release notes.

Checklist

• Math issue How to add static matrix? #1805

• Copyright holder: Steve Bronder

  The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses:
  - Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
  - Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

• the basic tests are passing

  • unit tests pass (to run, use: ./runTests.py test/unit)
  • header checks pass, (make test-headers)
  • dependencies checks pass, (make test-math-dependencies)
  • docs build, (make doxygen)
  • code passes the built in C++ standards checks (make cpplint)

• [x the code is written in idiomatic C++ and changes are documented in the doxygen

• the new changes are tested

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_pois_regr/gp_pois_regr.stan	3.11	3.05	1.02	1.73% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	1.02	2.36% faster
eight_schools/eight_schools.stan	0.12	0.12	1.0	-0.23% slower
gp_regr/gp_regr.stan	0.16	0.16	1.01	1.12% faster
irt_2pl/irt_2pl.stan	5.88	5.95	0.99	-1.26% slower
performance.compilation	87.9	86.78	1.01	1.27% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.58	8.59	1.0	-0.15% slower
pkpd/one_comp_mm_elim_abs.stan	31.08	29.58	1.05	4.82% faster
sir/sir.stan	122.32	119.95	1.02	1.94% faster
gp_regr/gen_gp_data.stan	0.03	0.03	0.99	-0.91% slower
low_dim_gauss_mix/low_dim_gauss_mix.stan	3.0	2.99	1.0	0.18% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.42	0.4	1.03	3.22% faster
arK/arK.stan	2.53	2.54	1.0	-0.35% slower
arma/arma.stan	0.66	0.65	1.02	1.93% faster
garch/garch.stan	0.65	0.64	1.01	1.36% faster
Mean result: 1.0117451219

---

Jenkins Console Log
Blue Ocean
Commit hash: 74cab27

---

Machine information

ProductName: Mac OS X ProductVersion: 10.11.6 BuildVersion: 15G22010

CPU:
Intel(R) Xeon(R) CPU E5-1680 v2 @ 3.00GHz

G++:
Configured with: --prefix=/Applications/Xcode.app/Contents/Developer/usr --with-gxx-include-dir=/usr/include/c++/4.2.1
Apple LLVM version 7.0.2 (clang-700.1.81)
Target: x86_64-apple-darwin15.6.0
Thread model: posix

Clang:
Apple LLVM version 7.0.2 (clang-700.1.81)
Target: x86_64-apple-darwin15.6.0
Thread model: posix

[SteveBronder]

@andrjohns would you mind taking a look at this?

[andrjohns]

Sorry for taking so long to get to this! I think with these binary functions where we need overloads for the various combinations of scalar/matrix/arith/var, we're going to end up with a lot of code duplication if we're not careful.

I know you've mentioned before that you're not a huge fan of the forward_as approach because it's more abstract, but I think we're going to end up with much more code to maintain otherwise

[syclik]

@SteveBronder, I merged develop back in... let's see if the tests get kicked off again.

[andrjohns]

@SteveBronder this was something I was keen to have in, so I made some changes and updates after merging develop. Feel absolutely free to revert/change anything you disagree with!

[SteveBronder]

Thanks @andrjohns those changes look good to me!