My question is simple, see example:
std::array<int,6> a = {{0,1,2,3,4,5}}; // -- given container.
auto F = []( int i ) { return  i*i; }; // -- given function.
std::vector<int> v;  // need create
// my solution:
v.reserve( a.size () );
for( std::size_t i = 0; i < a.size(); ++i )
    v.push_back( F(a[i]) ); 
// but I need something like
   std::vector<int>v( a.begin(), a.end(), <|applying each element to F|> );
Can I create container something like above not calling reserve explicitly and any reallocation?
EDIT:
The standard algorithm std::transform does exactly this!
std::vector<int> v(a.size());
std::transform(
   std::begin(a), std::end(a),
   std::begin(v),
   F
);
You can start with an empty vector and use std::back_inserter, if you like:
std::vector<int> v;
std::transform(
   std::begin(a), std::end(a),
   std::back_inserter(v),
   F
);
But you're subjecting yourself to needless re-allocations if you do that (unless you reserve first, as in your original attempt). You can decide for yourself what your priority is.
Use std::transform:
#include <algorithm> // std::transform
#include <iterator>  // std::back_inserter
....
transform(a.begin(), a.end(), back_inserter(v), F);
You may want to call v.reserve(asize()) first to avoid re-allocations.
Another solution is to use boost::transform_iterator. The benefit is that you can pass iterators to the container constructor. That avoids memory reallocations compared to when using std::back_inserter or having to call reserve or resize on the destination. All in one statement:
std::vector<int> result(
      boost::make_transform_iterator(std::begin(a), F)
    , boost::make_transform_iterator(std::end(a), F)
    );
You can achieve terser syntax though, like this:
std::vector<int> result(transform_range(a, F));
transform_range implementation:
template<class Iterator>
struct AutoSequence
{
    Iterator const beg_, end_;
    template<class T>
    operator std::vector<T>() const {
        return {beg_, end_};
    }
};
template<class Function, class InSeq>
auto transform_range(InSeq const& in) -> AutoSequence<decltype(boost::make_transform_iterator<Function>(in.begin()))> {
    return {
          boost::make_transform_iterator<Function>(std::begin(in))
        , boost::make_transform_iterator<Function>(std::end(in))
        };
}
template<class Function, class InSeq>
auto transform_range(InSeq const& in, Function&& f) -> AutoSequence<decltype(boost::make_transform_iterator(in.begin(), f))> {
    return {
          boost::make_transform_iterator(std::begin(in), f)
        , boost::make_transform_iterator(std::end(in), f)
        };
}
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