Use Newton's method to find all roots of the equation correct to six decimal places. (Enter your answers as a comma-separated list.) (x − 9)2 = ln(x)

Question

Use Newton's method to find all roots of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

(x − 9)² = ln(x)

Answer 1

The Newton's method for root finding is based on x_n+1 = x_n - (f(x_n)/f'(x_n)) where x_n is the initial value for the algorithm. Setting f(x)=(x-9)² - ln(x), we obtain f'(x) = 2(x-9) - 1/x. Let's set the initial value x₀= 3, then x₁= 5.829842, x₂= 7.102432, and so on. The first root is found approximately as x₆= 7.576937. Similarly, by starting another initial value like x₀= 10 we would obtain the another root as x4= 10.534488.