Question

Taking the endpoints of the last interval obtained in f(x) = x^3−5x^2 +1 = 0 as the initial
approximations, perform two iterations of the secant method to approximate the
root.

Answer 1

According to the secant method, the root can be found by the following equation:

$$x_n=\frac{x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}$$

Take the last interval of $[x_{n-2};x_{n-1}]=[0.469;0.5]$. The first iteration gives

$$x_n=\frac{0.469f(0.5)-0.5f(0.469)}{f(0.5)-f(0.469)}=0.495.$$

Choose points for the second iteration. The signs of the function at the interval ends must be different:

$$f(0.495)f(0.5)>0,\\ f(0.469)f(0.495)<0.$$

Therefore, we choose [0.469;0.495].

The second iteration gives

$$x_n=\frac{0.469f(0.495)-0.495f(0.469)}{f(0.495)-f(0.469)}=0.470.$$