Create symmetric positive definite Toeplitz matrix type

[rtrangucci]

Summary:

Introduce SPD Toeplitz matrix type to math library.

Description:

Implementation details:

#include <stan/math/prim/scal/err/check_less_or_equal.hpp>

template <typename T>
class spd_toeplitz {
  const Eigen::Matrix<T, Eigen::Dynamic, 1> band_vals_;
  const int non_zero_dim_;
  const int dim_;

  spd_toeplitz(Eigen::Matrix<T, Eigen::Dynamic, 1> band_vals, int dim)
  : {
       check_less_or_equal("Size of band values", 
                           "Toeplitz row and column dimensions", 
                           band_vals.size(), dim); 
       dim_(dim);
       non_zero_dim_(band_vals.size());
       for (int i = 0; i < non_zero_dim_; ++i)
         band_vals_.coeffRef(i) = band_vals.coeffRef(i);

       // Illustrative purposes only
       // We won't need to store the zeros explicitly,
       // but we will need the size of the non-zero 
       // entries
       for (int i = non_zero_dim_; i < dim_; ++i)
         band_vals_.coeffRef(i) = 0;
    }
  var& operator()(int i) {
        check_bounded("Row index", "non-zero entries", i, 0, non_zero_dim_);
        return band_vals_.coeffRef(ind);
    }
};

When vals_.size() < dim, only fill bands corresponding to implied non-zero entries in the matrix.

Current Version:

v2.11.0

[rtrangucci]

@vdorie we still need add you to the dev list but does this sound like a valid description of what we talked about yesterday?

[vdorie]

Yeah, it looks about right.

On Wed, Aug 17, 2016 at 2:30 PM, Rob Trangucci notifications@github.com
wrote:

@vdorie https://github.com/vdorie we still need add you to the dev list
but does this sound like a valid description of what we talked about
yesterday?

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[bob-carpenter]

I updated the issue because I would prefer more descriptive names or placeholders instead of meaningless long names like data; also, we want space between template and <. And I thought the member variables should be constant---being constant helps a lot with reasoning about objects, because they can't change so you know if they fulfill some criterion at construction time, they will always fullfill it.

I didn't understand the only-fill-bands bit. What are the size requirements on what I renamed to band_vals? And how do you map them to the dense matrix (even if we never construct that). What are requirements to keep it positive definite?

Also, do we really need this data type as opposed to just using the underlying Eigen vector? Specifically, where will this get built and how will it be used? Are there methods (non-static member functions) in the class, say exposing a dense and/or sparse matrix as a view? And what functions will be defined for it?

[rtrangucci]

@bob-carpenter thanks for updating the issue.

These objects would be useful in order to specialize decompositions, matrix and coefficient-wise algebra that can take advantage of the Toeplitz structure. As far as I'm concerned, these are just useful ways for storing vectors that can be used down the line as matrices with special structure.

The class should also implement operator()(int i). I'm not sure if we should allow people access to indices that are off the first column of the matrix, which is all we need to specify a Toeplitz matrix.