roots_

roots_(c) works with poorly scaled polynomials and does not return zero when c(end) is nonzero.

This fixes the MATLAB roots function to work with poorly scaled polynomials. For example,

c = 10.^(20:-1:0);

x = roots(c); max(abs(polyval(c,x))) % 7.647251195518823e-08

x = roots_(c); max(abs(polyval(c,x))) % 7.257804400984254e-14

This also fixes the MATLAB roots function to work with small roots. For example, roots([1,1,1,1e-32]) incorrectly gives a root of exact zero. roots_([1,1,1,1e-32]) gives the correct root of -1e-32.

(Reported as Case Number 04335183 on 30 Jun 2020 and Case Number 07198299 on 05 Aug 2024.)