A little later than planned, here is my review of the polynomial library: Documentation and Design ~~~~~~~~~~~~~~~~~~~~~~~~ As previously noted by others the Introduction doesn't adequately describe the library - some sort of simple tutorial at this point would be useful too. The "Background" section doesn't really describe the "background" as such either. This constructor: template<typename U> polynomial(const U& v); had me a little confused at first - is there any advantage to making this a template rather than simply accepting a const FieldType& as the single argument? This constructor: polynomial(std::vector<FieldType>& c); should *not* modify the original vector IMO, a construct-with-move should be a separate constructor - I'm sure we have a move/rvalue reference emulation library available somewhere? Likewise for the assignment operator polynomial<FieldType>& operator=(std::vector<FieldType>& c); It's not clear from the documentation what this constructor does: polynomial(InputIterator first1, InputIterator last1, InputIterator first2); It needs to clearly state the degree of the resulting polynomial and whether it passes through all/any of the points - in fact maybe the constructor should have a final parameter for the degree of the polynomial and use least-squares fitting when required? Likewise for the member function template<typename InputIterator> void interpolate(InputIterator first1, InputIterator last1, InputIterator first2); Operators: the / and % operators need good descriptions of what they do, given that no exact division is possible. As someone else has already noted, a function to calculate divide and remainder in one step would be useful too. GCD: presumably this is restricted to polynomials with integer coefficients? If so it should say so. Evaluation: The docs should say what method is used for the evaluate_faithfully method and give a reference. Renaming as someone else suggested may be better too, but personally I'm easy either way on that. I'm not sure whether there should be an evaluate_by_preconditioning method, shouldn't the polynomial "know" that it has been preconditioned and react accordingly. I haven't looked/checked but do the usual arithmetic operators still work if the polynomial has been preconditioned? I assume probably not, but if that's the case there should be a stern warning to that effect, *and* checks in code to prevent us from doing something stupid :-) BTW, preconditioning can be applied to polynomials of any degree I believe it's just that it gets hard to implement in the generic case. In addition I'd like to see the evaluation functions templated, so that the type being evaluated can differ from FieldType - it would surely be quite common for example to create and manipulate polynomials with integer coefficients, but then want to evaluate on a floating point type. There is machinery in Boost.Math BTW to handle the mixed argument-promotion and calculation of the result type, let me know if you need help in using this. Special Forms: the functions provided do *not* generate polynomials in the alternative forms, but rather generate the named polynomial of order N, which is rather less useful, but at the very least the docs need updating to reflect this. Implementation ~~~~~~~~~~~~~~ I haven't spent a great deal of time on this, but I note that: test_arithmetics.cpp fails on cygwin: or at least it did once I added a using std::pow; to start of operator*= to get things compiling. None of the code builds with msvc: there are a couple of issues here: friend polynomial<FieldType> boost::math::tools::gcd<>(polynomial<FieldType> u, polynomial<FieldType> v); Needs to be changed to: friend polynomial<FieldType> boost::math::tools::gcd<FieldType>(polynomial<FieldType> u, polynomial<FieldType> v); to get msvc to Grok the code. After that there are a bunch of failures due to the use of log2 which msvc doesn't support (and since it's not part of the std other compilers are likely to complain too). I haven't looked too hard at the implementation, except to echo the comments posted previously that the Conceptual requirements for FieldType need documenting and *testing*. There are concept checking classes and archetypes in Boost.Math which may be of use here. Again, shout if you need help in figuring out how to use these. I did notice that operator*= for example uses std::complex<double> internally which effectively limits the precision that can be achieved - to be useful to me - and to replace the existing "implementation detail" polynomial class in Boost.Math - I would need the polynomial class to be usable with arbitrary precision types such as NTL::RR or mpfr_class. Also as previously noted, multiplication of small polynomials may well be more efficient with the "naive" algorithm, so some performance tests and experimenting is required here. Related to this, use of "round" should be restricted to types that are known to be integers? * What is your evaluation of the potential usefulness of the library? As noted by others it's a niche library but potentially quite useful within that niche. * Do you think the library should be accepted as a Boost library? Be sure to say this explicitly so that your other comments don't obscure your overall opinion. I think that the library could be accepted into Boost, but not it's it's current form, so I would vote no for now: please note this is my personal review of the library and should not be confused with my role as review manager for this library. Regards, John Maddock.