Cleanup mdivide_* and tests

[mcol]

Summary

This cleans up the mdivide functions, mainly to use return_type_t instead of promote_common. There are a couple of unused header removed and a correction to the return type declared for the fwd version of mdivide_left_tri_low. Fixes #1714.

Tests

Reworked and expanded so that the left and right versions contain the same tests. Some boundary condition tests will be completed in #1689.

Side Effects

None.

Checklist

• Math issue remove uses of promote_common from mdivide_* functions and improve tests #1714

• Copyright holder: Marco Colombo

  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)

• 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	4.93	4.99	0.99	-1.3% slower
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	1.0	0.44% faster
eight_schools/eight_schools.stan	0.09	0.09	1.01	1.36% faster
gp_regr/gp_regr.stan	0.22	0.22	1.02	1.83% faster
irt_2pl/irt_2pl.stan	6.09	6.07	1.0	0.18% faster
performance.compilation	87.97	87.07	1.01	1.02% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	7.5	7.43	1.01	0.99% faster
pkpd/one_comp_mm_elim_abs.stan	20.82	22.02	0.95	-5.74% slower
sir/sir.stan	97.45	96.72	1.01	0.75% faster
gp_regr/gen_gp_data.stan	0.05	0.05	1.04	4.04% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.99	3.01	0.99	-0.58% slower
pkpd/sim_one_comp_mm_elim_abs.stan	0.32	0.34	0.94	-5.93% slower
arK/arK.stan	1.74	1.74	1.0	0.24% faster
arma/arma.stan	0.79	0.81	0.97	-3.17% slower
garch/garch.stan	0.58	0.59	1.0	-0.39% slower
Mean result: 0.996499834573

---

Jenkins Console Log
Blue Ocean
Commit hash: ed296c8

---

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

[mcol]

I have two more PRs I'm preparing that depend on this. @rok-cesnovar Would you mind review this?

[rok-cesnovar]

Sure. I can do this in a few hours when I will be at my computer.

[rok-cesnovar]

Nice catch!

[rok-cesnovar]

Looks great! Much nicer to read. Thanks for adding the zero size tests.

[rok-cesnovar]

Much nicer!

[mcol]

Thank you!