getPolyRoots(C,tBnd​)

Find all roots of polynomial and derivative on a bounded interval

% getPolyRoots(C,tBnd)
%
% Given a vector of polynomial coefficients in C, and a bounded interval
% tBnd, getPolyRoots() finds the roots of this polynomial and all of its
% derivatives on the given interval.
%
% INPUTS:
% C = vector of polynomial coefficients, see >>help polyval for format
% tBnd = [tLow, tUpp] = time interval of interest
%
% OUTPUTS:
% R = cell array
% R{1} = vector of roots of the polynomial C, including tBnd
% R{2} = vector of roots of the derivative of C, including tBnd
% ...
% R{n} = vector of roots of the (n-1)th derivative of C
%
% NOTES:
%
% 1) Why this function? There is clearly the matlab command roots().
%
% --> The implementation of this function is efficient for medium-order
% polynomials that have many roots which are not inside of the interval.
% In particular, it performs one root-solve per root found in R, not
% counting the boundary points. Thus, if there are no roots, then there
% is no root solve. Matlab's roots() command computes eigen-values to
% find the roots, and then you need to trim any roots not in the
% interval. This is the method that I use to validate the result of this
% function.
%
% 2) Matlab's roots() command should still be faster because they have an
% awesome vectorized implementation of eig().
%
% --> True. In many cases I suspect that the roots() method will actually
% be faster, since this code is not vectorized. My motivation is that I'm
% using this code to prototype the implementation, which I will later be
% writing up in C++ for other applications. In C++, the for loops in this
% implementation will run quickly.
%