autodiff::fvar doesn't work with Eigen

[wgledbetter]

Currently, the templated copy constructor for autodiff::fvar interferes with Eigen's lazy evaluation, and, in my experience, produces two types of errors:

1. Ambiguous operators

• Example error: ambiguous overload for ‘operator*’ (operand types are ‘Eigen::Transpose<Eigen::Matrix<boost::math::differentiation::autodiff_v1::detail::fvar<double, 1>, 4, 1> >’ and ‘Eigen::Matrix<boost::math::differentiation::autodiff_v1::detail::fvar<double, 1>, 4, 1>’)

2. Failed attempts to static_cast a matrix into an fvar

• Example error: error: invalid ‘static_cast’ from type ‘const Eigen::CwiseUnaryOp<Eigen::internal::scalar_cast_op<double, boost::math::differentiation::autodiff_v1::detail::fvar<double, 1> >, const Eigen::Matrix<double, 4, 4> >’ to type ‘double’

A solution would be to only define the constructor for types that are convertible to the fvar's RealType. A colleague and I worked out a potential fix using SFINAE that I'm currently testing against the existing test_autodiff files. Tests 1-3 have succeeded with the existing travis configuration, but I'm having issues in test_autodiff_4 with boost::has_right_shift as called from boost::lexical_cast.

Is there any interest in merging a solution to this problem? I can post the work I've done so far.

[NAThompson]

@wgledbetter : Yes, we're interested.

@pulver : Can your review the PR when it shows up?

[pulver]

The build errors @wgledbetter forewarned of are on https://travis-ci.org/github/boostorg/math/builds/721216811

../../../boost/static_assert.hpp:31:45: error: static assertion failed: Target type is neither std::istream`able nor std::wistream`able
 #     define BOOST_STATIC_ASSERT_MSG( ... ) static_assert(__VA_ARGS__)
                                             ^
../../../boost/lexical_cast/detail/converter_lexical.hpp:243:13: note: in expansion of macro ‘BOOST_STATIC_ASSERT_MSG’
             BOOST_STATIC_ASSERT_MSG((result_t::value || boost::has_right_shift<std::basic_istream<wchar_t>, T >::value),
             ^~~~~~~~~~~~~~~~~~~~~~~

In case this helps someone diagnose this, the larger context in boost/lexical_cast/detail/converter_lexical.hpp where this is called is:

    namespace detail  // deduce_target_char_impl<T>
    {
        // If type T is `deduce_character_type_later` type, then tries to deduce
        // character type using boost::has_right_shift<T> metafunction.
        // Otherwise supplied type T is a character type, that must be normalized
        // using normalize_single_byte_char<Char>.
        // Executed at Stage 2  (See deduce_source_char<T> and deduce_target_char<T>)
        template < class Char >
        struct deduce_target_char_impl
        {
            typedef BOOST_DEDUCED_TYPENAME normalize_single_byte_char< Char >::type type;
        };

        template < class T >
        struct deduce_target_char_impl< deduce_character_type_later<T> >
        {
            typedef boost::has_right_shift<std::basic_istream<char>, T > result_t;

#if defined(BOOST_LCAST_NO_WCHAR_T)
            BOOST_STATIC_ASSERT_MSG((result_t::value),
                "Target type is not std::istream`able and std::wistream`s are not supported by your STL implementation");
            typedef char type;
#else
            typedef BOOST_DEDUCED_TYPENAME boost::conditional<
                result_t::value, char, wchar_t
            >::type type;

            BOOST_STATIC_ASSERT_MSG((result_t::value || boost::has_right_shift<std::basic_istream<wchar_t>, T >::value), 
                "Target type is neither std::istream`able nor std::wistream`able");
#endif
        };
    }

This appears to be called when the RealType is a multiprecision:
../../../boost/lexical_cast/detail/converter_lexical.hpp: In instantiation of ‘struct boost::detail::deduce_target_char_impl<boost::detail::deduce_character_type_later<boost::math::differentiation::autodiff_v1::detail::fvar<boost::multiprecision::number<boost::multiprecision::backends::cpp_bin_float<50u> >, 5ul> > >’

[wgledbetter]

I was able to solve this problem by creating an operator>> for fvar that behaves like operator=. Now I'm getting all autodiff tests to pass.

Just a side note, I got this same has_right_shift error with RealType = double on my personal machine with clang-10 and gcc-10 and c++-17, but the new update on #425 fixes it, too.