Answer to Question #110504 in Quantitative Methods for Anju Jayachandran

Question

Taking the endpoints of the last interval obtained in part a) above as the initial
approximations, perform two iterations of the secant method to approximate the
root.

Answer 1

As follows from question 110503, we were given an equation $f(x)=x^3-5x^2+1$ and we established that its smallest positive root belongs to the interval

$$[x_0,x_1]=[x_{i-1},x_i]=[0.469,0.5].$$

Use the secant method to approximate the root:

$$x_{i+1}=x_i-\frac{f(x_i)\cdot(x_i-x_{i-1})}{f(x_i)-f(x_{i-1})}=\\ \space\\ =0.5-\frac{-0.125\cdot(0.5-0.469)}{(-0.125)-0.0034}=0.4698.$$

Use this root for the second iteration, where $x_i=0.4698,x_{i-1}=0.469$:

$$x_{i+1}=0.4698-\frac{1.303\cdot10^{-4}\cdot(0.4698-0.469)}{1.303\cdot10^{-4}-0.0034}=0.46983.$$

Check the value of the function:

$$f(0.46983)=9.23\cdot10^{-6}.$$