Worse than Bisection???!!!!

[Namir]

Richard A Davis suggests in his new book Practical Numerical Methods for Chemical Engineers on page 187 the following remedies (which I am paraphrasing):

1) If a remains unchanged after two iterations use:

x = b - f(b)(a - b)/(f(b) - 0.5 f(a))

Otherwise, if b remains unchanged after two iterations use:

x = b - 0.5 f(b)(a-b)/(0.5 f(b) - f(a))

Where [a, b] is the root-bracketing interval for f(x) = 0.