revert revert 2401 algebra solver newton

[SteveBronder]

Summary

Revert revert of #2401 with the same make_chainable_ptr() trick used in algebra_solver_powell for algebra_solver_newton

Tests

No new tests

Side Effects

Nope!

Release notes

Checklist

• Math issue Algebraic Solver Differentiation Speedup #2401

• Copyright holder: Johann Gaebler / Ben Bales / 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)

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

• the new changes are tested

[rok-cesnovar]

Thanks Steve!

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_pois_regr/gp_pois_regr.stan	3.5	3.43	1.02	2.12% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	0.97	-2.85% slower
eight_schools/eight_schools.stan	0.09	0.09	0.97	-3.28% slower
gp_regr/gp_regr.stan	0.14	0.14	1.03	2.55% faster
irt_2pl/irt_2pl.stan	5.1	5.11	1.0	-0.28% slower
performance.compilation	91.32	89.05	1.03	2.49% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.28	8.23	1.01	0.58% faster
pkpd/one_comp_mm_elim_abs.stan	29.54	29.51	1.0	0.1% faster
sir/sir.stan	120.61	127.03	0.95	-5.32% slower
gp_regr/gen_gp_data.stan	0.03	0.03	1.01	1.21% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	3.02	3.05	0.99	-1.13% slower
pkpd/sim_one_comp_mm_elim_abs.stan	0.41	0.38	1.1	9.0% faster
arK/arK.stan	2.06	2.05	1.01	0.53% faster
arma/arma.stan	0.25	0.25	1.01	0.58% faster
garch/garch.stan	0.61	0.6	1.01	0.75% faster
Mean result: 1.00574242707

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Jenkins Console Log
Blue Ocean
Commit hash: f6bc018

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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