1D Newton iteration count issue(s)

[ryanelandt]

The documentation for newton_raphson_iterate says:

uintmax_t& max_iter
    An optional maximum number of iterations to perform.
    On exit, this is updated to the actual number of iterations performed. 

There are two issues relating to max_iter:

1. max_iter is copied to a variable count. After each Newton iteration, one is subtracted from count. Because of the do {...} while (count && ...) structure of the Newton iteration, if max_iter = 0, rollover occurs, making max_iter functionally infinite.
2. If 0 < max_iter, running out of iterations exits the Newton loop as if the tolerance was met.

These two observations are demonstrated in the code below.

const auto fn = [](double x) { return std::make_pair(x * x - 3, 2 * x); };
std::uintmax_t iters;
std::cout << std::setprecision(16);

for (int count = 0; count < 10; count++) {
   iters = count;
   const double x0 = boost::math::tools::newton_raphson_iterate(fn, 10.0, 0.0, 100.0, 52, iters);
   std::cout << "max_iters: " << count << " -- used_iters: " << iters << " -- x: " << x0 << std::endl;
}

max_iters: 0 -- used_iters: 8 -- x: 1.732050807568877
max_iters: 1 -- used_iters: 1 -- x: 5.15
max_iters: 2 -- used_iters: 2 -- x: 2.86626213592233
max_iters: 3 -- used_iters: 3 -- x: 1.956460731776899
max_iters: 4 -- used_iters: 4 -- x: 1.74492093914502
max_iters: 5 -- used_iters: 5 -- x: 1.732098271119538
max_iters: 6 -- used_iters: 6 -- x: 1.732050808219183
max_iters: 7 -- used_iters: 7 -- x: 1.732050807568877
max_iters: 8 -- used_iters: 8 -- x: 1.732050807568877
max_iters: 9 -- used_iters: 8 -- x: 1.732050807568877

[ryanelandt]

Because of the issue described above, the unit test https://github.com/boostorg/math/blob/50ef83a47b13c02bf9d6c0e86f9066c431ac42cb/test/test_roots.cpp#L657-L659 passes, even though the answer in produces in inaccurate due to running out of iterations.