Apologies for the delay in putting this together... First off I'd like to thank the author for submitting this for review - I wish all our SOC students were similarly diligent! :-) We received a number of reviews of this library, but none were in favor of acceptance in its current form, and most thought that there was still a fair bit of work to do to get the library into shape. However, most thought that the library could be accepted into Boost given sufficient changes/enhancements. Therefore the library is not accepted into Boost at this time, but I would like to encourage the author to continue to work on the library and resubmit at a future time. In no particular order the main review comments are summarized below: Principal comments: * Documentation, especially the background is inadequate and needs a good proofreading a) From the examples are nothing that a competent programmer couldn't figure out from the declarations. Some more interesting or useful examples would be nice, particularly for things like the special forms. It's not exactly clear what I would do with those functions. b) There is no documentation or references to the various algorithms used. Those, too, would be nice. c) Doc.html appear to have been created 'the Hard Way'. Would be much more useful and look nicer if produced with the Quickbook, Doxygen... toolchain. And make it maintainable by other people. * Use of constructors and assignment operators that modify their arguments isn't acceptable - best to find another interface for this. * Several reviewers noted that the code didn't build for them, in particular log2 isn't available on all platforms/compilers. * evaluate_polynomial() and evaluate_polynomial_faithfully() take their coefficient arguments in the opposite order of all the other functions/constructors in the library, without providing any reason for that choice. That's seems somewhat surprising. I'd suggest fixing that, or at the very least providing an argument in the documentation why this makes some sense. * Rename evaluate() to horner() and provide evaluate() as an inline call to horner(). * Rename evaluate_faithfully() to compensated_horner(). * Things that say "Be careful about this" generally need fixing. * It would be nice to have operator* and operator/ as well as *= and /=. * Considering they share the same core algorithm, it might be nice to have a function that could do division and return the remainder at the same time. Something like poly::divide(P &divisor, P "ient, P &remainer) * FFTs are good for multiplying polynomials above a certain size, but they are rather heavy for small polynomials and can lead to unexpected crosstalk between terms. There should be an option for * that doesn't use the FFT, and it should be the default for small polynomials. * operator/= has a hardcoded "std::vector<double> new_coefficients" that later gets rounded?? This should be fixed to work with any type that fulfills the necessary conceptual requirements (and tested and documented what those requirements are - it would be nice if they were the same as the rest of Boost.Math). * The name FieldType is mathematically misleading since the instantiating type is very likely not to be a field (e.g. int or a multiple precision integer type). * the Polynomial class provides a fairly minimalist set of supported functions. This may well be deliberate (if so, apologies), but I would like to see a richer interface including such things as - content (gcd of coefficients, where that makes sense) - polynomial discriminant - resultant of 2 polynomials - evaluation of homogeneous polynomial ( f(x) = sum c_i x^i,f(a,b) = sum c_i a^i b^(d-i) ) - evaluation at a polynomial (f(g(x))) - Pseudo-division of polynomials - finding roots over C - finding roots over F_p - factorisation over Z - factorisation over F_p * We had one feature request for extending support to finite fields. * 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? * 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. * 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? Regards, John Maddock Review Manager for Polynomial Library.