function R = approot (X,epsilon)% Input - f is object function saved as an M-file named f.m%       - X is the vector of abscissas%       - epsilon is the tolerance % Output - R is the vector of approximate locations for roots%  NUMERICAL METHODS: Matlab Programs% (c) 2004 by John H. Mathews and Kurtis D. Fink%  Complementary Software to accompany the textbook:%  NUMERICAL METHODS: Using Matlab, Fourth Edition%  ISBN: 0-13-065248-2%  Prentice-Hall Pub. Inc.%  One Lake Street%  Upper Saddle River, NJ 07458Y=f(X);yrange = max(Y)-min(Y);epsilon2 = yrange*epsilon;n=length(X);m=0;X(n+1)=X(n);Y(n+1)=Y(n);for k=2:n,	if Y(k-1)*Y(k) <= 0,		m=m+1;		R(m)=(X(k-1)+X(k))/2;	end	s=(Y(k)-Y(k-1))*(Y(k+1)-Y(k));	if (abs(Y(k)) < epsilon2) & (s <= 0),		m=m+1;		R(m)=X(k);	endend