## Copyright (C) 2002 Etienne Grossmann. All rights reserved. ## ## This program is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by the ## Free Software Foundation; either version 2, or (at your option) any ## later version. ## ## This is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License ## for more details. ## ex = poly_2_ex (l, f) - Extremum of a 1-var deg-2 polynomial ## ## l : 3 : Values of variable at which polynomial is known. ## f : 3 : f(i) = Value of the degree-2 polynomial at l(i). ## ## ex : 1 : Value for which f reaches its extremum ## ## Assuming that f(i) = a*l(i)^2 + b*l(i) + c = P(l(i)) for some a, b, c, ## ex is the extremum of the polynome P. ## function ex = poly_2_ex (l, f) ### This somewhat helps if solution is very close to one of the points. [f,i] = sort (f); l = l(i); m = (l(2) - l(1))/(l(3) - l(1)); d = (2*(f(1)*(m-1)+f(2)-f(3)*m)); if abs (d) < eps, printf ("poly_2_ex : divisor is small (solution at infinity)\n"); printf ("%8.3e %8.3e %8.3e, %8.3e %8.3e\n",\ f(1), diff (f), diff (sort (l))); ex = (2*(l(1)>l(2))-1)*inf; ## keyboard else ex = ((l(3) - l(1))*((f(1)*(m^2-1) + f(2) - f(3)*m^2))) / d ; ## Not an improvement # n = ((l(2)+l(3))*(l(2)-l(3)) + 2*(l(3)-l(2))*l(1)) / (l(3)-l(1))^2 ; # ex = ((l(3) - l(1))*((f(1)*n + f(2) - f(3)*m^2))) / \ # (2*(f(1)*(m-1)+f(2)-f(3)*m)); # if ex != ex0, # ex - ex0 # end ex = l(1) + ex; end

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