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In Secant method - $$x_{n} = x_{n-1} - f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}$$ I have to choose 2 initial - $x_0$ and $x_1$ . How could I know whether those 2 $x$'s going to converge to the zero of $f(x)$ with no calculate the above iteration ? |
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Wikipedia says: If the initial values are not close enough to the root, then there is no guarantee that the secant method converges. There is no general definition of "close enough", but the criterion has to do with how "wiggly" the function is on the interval $[x_0, x_1]$. For example, if is differentiable on that interval and there is a point where $f'=0$ on the interval, then the algorithm may not converge. The relevant page is here. If you want a method that is guaranteed to converge to a root, then you should look into bisection or regula falsi. They require initial values that "bracket" the root you're seeking, though. For some fascinating (and unusual) discussions of root-finding, I recommend papers by Bill Kahan, especially this one. |
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