Add sparse matrix (csr format) by dense vector multiplication

[sakrejda]

I let this go stale, I'm going to fix it up so I can get it in now, the background is in pull request 1405. It needs to be two separate pull requests now, one for stan-dev/stan and one for stan-dev/math .

Fixes

• Make just one multiplication function---named "csr_matrix_times_vector"
• csr_matrix_times_vector needs to be hardened. Right now, it is unsafe.
  • v: check that each element is between 1 and n
  • u: check that it's ordered, each element is greater than or equal to 1, that every element is less than or equal to length of w
  • z: check that each number is greater than or equal to 0. Sum is equal to length of w. NOT doing this as the Eigen implementation I'm mirroring allows for unused entries of w. Instead:
    • 'z' implicitly check z by making sure calculations never run off the end of 'w' (check on "end" in code).
    • 'b' implicitly checked by making sure v is checked against the number of columns.
• change explicit 1-index to stan::error_index::value -- this is #defined as 1, but can be changed to 0
• Add a constructor of a dense matrix using the sparse format--named csr_to_dense_matrix
• Rename extractors to csr_extract_* this makes all functions in the group named as csr_*
• Add extractors for w/v/u/z that work from dense matrices, wrap around the sparse ones (work by going from Eigen dense to sparse then calling the usual extractors, less indexing code to maintain. These would usually be called only in transformed data block anyway. These will be:
  • "csr_extract_w"
  • "csr_extract_v"
  • "csr_extract_u"
  • "csr_extract_z"
• Follow Daniel's suggestion to split out each extractor function into a separate file with its own corresponding test file.

[sakrejda]

More Fixes

• function parameters are const
• function parameters passed by reference.
• Include stan/math/prim/mat/fun/Eigen.hpp in extractors.
• Usage of <T, R, C> versus <T, Dynamic, Dynamic>
  • In template function declaration.
  • In within-function variable declarations.
• use row/nze notation everywhere rather than j/p.
• Tests:
  • Move tests for csr_matrix_times_vector.hpp
  • csr_matrix_times_vector.hpp tests should include missing row/ missing column patterns.
  • Move tests for csr_extract_w.hpp
  • Move tests for csr_extract_v.hpp
  • Move tests for csr_extract_u.hpp
  • Move tests for csr_extract_z.hpp
  • Write tests for effect of makeCompressed() on extractors.
  • Fixed matrix test for csr_to_dense_matrix.hpp
  • Round-trip test for csr_to_dense_matrix.hpp and csr_extract_*.hpp
  • Write tests for effect of makeCompressed() on round-trip through csr_to_dense_matrix.hpp
  • Pass all sparse tests
  • Pass all tests in test/unit/math/prim/mat/fun/

[bgoodri]

jenkins, test this please