Partial inverse heuristic: witness problems and new algorithms

``WITNESS'' PROBLEMS AND NEW ALGORITHMS
These are problems which are difficult to solve and reveal shortcomings of most of the current methods and systems.

$x^{1.001}-x \ln x = 0$

For positive x, this equation has two solutions, one small, the other one huge in magnitude: x= 2.721005..., 0.794138... x10³⁹⁶⁰.

$\frac{\sin^{-1} x - \tan x}{x^4} = 0$

This problem and the next were proposed by Kahan. For positive x, this equation has one solution, x=0.999906... This value is too close to 1, the limit where the function can be evaluated without going into the complex plane.

$x^2+5+ \ln ( \vert x - \pi \vert ) = 0$

For positive x, this equation has two roots very close to $\pi$ , x=3.1415923... and x=3.1415930.... The convergence interval for such roots is extremely narrow, and most methods, even when started close to the roots, will diverge away from them.

sin(x) = x/100

This equation has 63 real roots in the interval -96.1<x<96.1. Any good iterator should easily converge to any root given a suitable starting guess.