Answer to Question #108299 in Quantitative Methods for Garima Ahlawat

Question

Perform iterations of Newton-Raphson method to approximate a root of the equation f(x) = x^4 -x^3 +x -1 =0 , until the roots at successive iterations are closer than 10^(-5). How many iterations do you need for this much accuracy.

Answer 1

Solution:

Since the sum of the coefficients of the original equation is 0, one of the roots is the number 1. We plot the function to determine the left boundary of the interval in which the next root of the equation will be located.

$$f^/(x)=4x^3-3x^2+1$$

$$x_{n+1}=x_n-\frac {f(x_n)}{f^{/}(x_n)}$$

Answer: 5 interations