Add backward induction and beta =1 #172

[tokuma09]

This pull request is related with #172.

I try the followings.

• add backward_induction method and explanation.
• allows beta = 1 for DiscreteDP`
• write test codes for them.

close #172.

[codecov-io]

Codecov Report

Merging #228 into master will decrease coverage by 39.02%.
The diff coverage is 70%.

@@             Coverage Diff             @@
##           master     #228       +/-   ##
===========================================
- Coverage   95.85%   56.83%   -39.03%     
===========================================
  Files          25       25               
  Lines        1061     1786      +725     
===========================================
- Hits         1017     1015        -2     
- Misses         44      771      +727

Impacted Files	Coverage Δ
src/markov/ddp.jl	66.66% <70%> (-31.98%)	⬇️
src/quad.jl	8.98% <0%> (-84.96%)	⬇️
src/optimization.jl	53.06% <0%> (-46.94%)	⬇️
src/util.jl	58.88% <0%> (-39.26%)	⬇️
src/zeros.jl	42.37% <0%> (-37%)	⬇️
src/filter.jl	63.63% <0%> (-36.37%)	⬇️
src/matrix_eqn.jl	58.2% <0%> (-34.65%)	⬇️
src/markov/markov_approx.jl	64.51% <0%> (-33.86%)	⬇️
src/sampler.jl	62.96% <0%> (-31.49%)	⬇️
... and 13 more

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[oyamad]

Thanks for working on this!

[oyamad]

@sglyon What is the status of issue #136 (documentation style)? Isn't it more efficient to start following the official guidance in the new code?

[oyamad]

Will you also add a test with Rational inputs?

(Define the R and Q arrays with Rational first, then run the test with it; in the second test, convert them into Float64.)

[sglyon]

@oyamad I'd be happy to follow something more official.

I believe @jlperla and his team were working on a style guide. Perhaps they included a discussion of docstrings conventions to follow?

[oyamad]

A few more comments.

[tokuma09]

Thanks. Fix them.

[tokuma09]

Fix them.

[oyamad]

Thanks. I think this is ready to merge.

[oyamad]

@sglyon Can I merge this PR?

[sglyon]

Looks great to me, thank you for following up.

Sorry I forgot to merge this after it was approved