A numerical procedure generates successive points (xk,yk) that satisfy y′=f(x,y), using a variable increment hk at each step. The program measures the change in slope between two consecutive points with Δfk=∣f(xk,yk)−f(xk−1,yk−1)∣. Which of the following update rules for the next increment hk+1 is most likely to reduce local error in regions where the slope varies rapidly?
a) Set hk+1=hk(Δfk/tol).
b) If Δfk exceeds a chosen tolerance, set hk+1=2hk; otherwise leave hk+1 unchanged.
c) If Δfk exceeds a chosen tolerance, set hk+1=21hk; otherwise leave hk+1 unchanged.
d) If Δfk is *below* the tolerance, set hk+1=21hk; otherwise leave hk+1 unchanged.