Re: [boost] Math tools polynomial enhancements

Subject: Re: [boost] Math tools polynomial enhancements
From: Robert Ramey (ramey_at_[hidden])
Date: 2015-10-29 16:26:51

On 10/29/15 11:35 AM, John Maddock wrote:
>
>
> On 29/10/2015 15:04, Robert Ramey wrote:
>> On 10/29/15 3:22 AM, John Maddock wrote:
>>>
>>> I think there is a place for polynomial manipulation within Boost, but
>>> I'm not sure that this class is the best basis.  As we say in the docs,
>>> it's a braindead implementation that's good enough for what I needed at
>>> the time to implement Boost.Math, but not really suitable for heavy duty
>>> polynomial manipulation.
>>
>> While you're at it, How about a constexpr/tmp version which would do
>> polynomial manipulation at compile time?
>>
>
> Interesting idea, what kind of manipulation did you have in mind -
> arithmetic?  Might be possible in C++14 but certainly not in C++11.
> Probably a whole different library actually,

Right

since the order would have
> to be a compile time parameter?

I'm thinking it would be a variadic tuple.

here's what I had in mind:

// define a polynomial with coefficients of type T (int, float, etc)

template<typename T>
using polynomial = std::tuple<T ...>

// now define some operators on these polynomials

template<typename T>j
// polynomial addition - easy to implement
operator+(polynomial<T>lhs, polynomial<T>rhs);
// other ops  ... easy to implement
// polynomial division - aaaa - more compilicated to implement
operator/(polynomial<T>lhs, polynomial<T>rhs);

// very cool functions
template<typename T>
constexpr polynomial<T> derivative(polynomial<T> p);

template<typename T>
constexpr polynomial<T> integrate(polynomial<T> p);

// and of course we could evaluate any polynomial at
// specific point at either compile and/or run time
constexpr T evaluate(const polynomial<T> & p);

// a taylor series has the last two terms reserved for a rational
// representation of an error term
template<typename T>
using taylor_series = polynomial<T>;

// evaluate
T value(taylor_series<T>, T x);

// get error value
T error(taylor_series<T>, T x);

// given a sequence of derivatives of F at point x
// calculate a taylor series
template<T, F>
constexpr taylor_series::taylor_series(x = 0);

// now I can replace my clunky quadrature of function
// F with

// return value and error
template<class T, class F>
const expr std::pair<T, T> fast_integral(
   const T & start,
   const T & finish
){
   using f = integrate(taylor_series<F>(3.233));
   T value = evaluate(f, finish) - evaluate(f, start);
   T error_value = abs(error(f, finish)) + abs(error(f, start));
   return std::pair<value, error_value>
};

This would be pretty useful as it stands.  Of course it
brings to mind a few more ideas

a) T above is presumed to be a scalar variable - but it
doesn't have to be.  What I really need is to permit T
to be a tuble itself so I could handle complex and
and n dimensional space situations.

b) The creating of F is problematic and tedious.  For this we need
TMP expression template driven automatic differentiation.

I'm thinking you could put this together in a couple of days

Robert Ramey