better handling of size 0 inputs

[mcol]

Summary

The following is done:

• matrix_exp_multiply, scale_matrix_exp_multiply and matrix_exp_pade: move check_square at the beginning, so that it throws if A is not square (currently it returns an empty matrix if any dimension is 0)
• scale_matrix_exp_multiply: it returns a rectangular matrix with 0 rows if A is empty and B has zero rows (currently it returns a square empty matrix)
• mdivide_left_ldlt: it returns a rectangular matrix with 0 rows if A is empty and b has zero rows (currently it returns a square empty matrix)
• mdivide_right_ldlt: it returns a rectangular matrix with 0 columns if A is empty and b has zero columns (currently it returns a square empty matrix)
• trace_inv_quad_form_ldlt: use B.rows() instead of B.size(), so we ensure that matrices are multipliable before returning 0

This also:

• avoids extra work in check_symmetric by checking if the size of the square matrix is 0
• same for check_corr_matrix, so that for size 0 matrix it returns instead of throwing

Fixes #1515.

Tests

I've added a bunch of tests that check for all of the above. It's still worth checking for more cases that I might have missed. I'm also not entirely convinced that my mental mapping of prim/rev/mix is complete or correct, so it's possible I've missed something.

Side Effects

None.

Checklist

• Math issue Improve handling of size 0 inputs #1515

• 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)
  • 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]

(stat_comp_benchmarks/benchmarks/gp_pois_regr/gp_pois_regr.stan, 0.99)
(stat_comp_benchmarks/benchmarks/low_dim_corr_gauss/low_dim_corr_gauss.stan, 0.96)
(stat_comp_benchmarks/benchmarks/irt_2pl/irt_2pl.stan, 1.01)
(stat_comp_benchmarks/benchmarks/pkpd/one_comp_mm_elim_abs.stan, 1.01)
(stat_comp_benchmarks/benchmarks/eight_schools/eight_schools.stan, 1.05)
(stat_comp_benchmarks/benchmarks/gp_regr/gp_regr.stan, 1.0)
(stat_comp_benchmarks/benchmarks/arK/arK.stan, 1.01)
(performance.compilation, 1.01)
(stat_comp_benchmarks/benchmarks/low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan, 0.98)
(stat_comp_benchmarks/benchmarks/low_dim_gauss_mix/low_dim_gauss_mix.stan, 0.99)
(stat_comp_benchmarks/benchmarks/sir/sir.stan, 1.0)
(stat_comp_benchmarks/benchmarks/pkpd/sim_one_comp_mm_elim_abs.stan, 1.0)
(stat_comp_benchmarks/benchmarks/garch/garch.stan, 1.01)
(stat_comp_benchmarks/benchmarks/gp_regr/gen_gp_data.stan, 0.99)
(stat_comp_benchmarks/benchmarks/arma/arma.stan, 1.0)
Result: 1.00046914435
Commit hash: 117e21e

[rok-cesnovar]

Thanks for cleaning these up! Two minor return cleanups then its good.

[rok-cesnovar]

Great! Thanks for the quick fixes.

[stan-buildbot]

(stat_comp_benchmarks/benchmarks/gp_pois_regr/gp_pois_regr.stan, 0.99)
(stat_comp_benchmarks/benchmarks/low_dim_corr_gauss/low_dim_corr_gauss.stan, 0.96)
(stat_comp_benchmarks/benchmarks/irt_2pl/irt_2pl.stan, 1.0)
(stat_comp_benchmarks/benchmarks/pkpd/one_comp_mm_elim_abs.stan, 1.01)
(stat_comp_benchmarks/benchmarks/eight_schools/eight_schools.stan, 0.99)
(stat_comp_benchmarks/benchmarks/gp_regr/gp_regr.stan, 1.0)
(stat_comp_benchmarks/benchmarks/arK/arK.stan, 1.0)
(performance.compilation, 1.01)
(stat_comp_benchmarks/benchmarks/low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan, 1.0)
(stat_comp_benchmarks/benchmarks/low_dim_gauss_mix/low_dim_gauss_mix.stan, 0.99)
(stat_comp_benchmarks/benchmarks/sir/sir.stan, 1.01)
(stat_comp_benchmarks/benchmarks/pkpd/sim_one_comp_mm_elim_abs.stan, 0.97)
(stat_comp_benchmarks/benchmarks/garch/garch.stan, 1.0)
(stat_comp_benchmarks/benchmarks/gp_regr/gen_gp_data.stan, 1.02)
(stat_comp_benchmarks/benchmarks/arma/arma.stan, 1.03)
Result: 0.99904817297
Commit hash: 45afd6e