How to calculate the point on the globe with maximum magnitude with given Besselian elements and given time for a partial solar eclipse?
Jean Meeus, Morsels 2, chapter 69 talks about the Nelder-Mead simplex algorithm for minimising a function of two variables - in this instance magnitude as a function of latitude and longitude. Maximising the magnitude equates to minimising the negative of the magnitude. Apparently the method first appeared in magazine Byte in the 1960s, there are various references on-line.
Maximum magnitude always occurs near the polar circle, so try starting with three points equally spaced round the 60th parallel. They would have to be converted to Cartesian X Y with centre at the pole (giving an equilateral triangle), and the result converted back to latitude/longitude at the end. The sign of the element Y will make it obvious which polar circle to use.
Because the shadow is conic, not cylindrical, the resulting point will not (in general) be exactly on the horizon.