# Satisfying a specified accuracy in a numerical method

In an iterative root finding numerical method, with an assumption of convergence, with every iteration, we attain more and more correct digits of the true root. If I obtain three correct decimal figures, then my estimation is said to be correct to within $$10^{-3}$$. This means

$$|p_n-r|\lt 10^{-3}$$ where $$p_n$$ is the nth approximation and r is the actual root.

Below, I find one of the roots of polynomial $$x^3+3x^2-1$$ between -3 and -2 (who are $$p_0$$ and $$p_1$$ respectively) using secant method. At sixth iteration I have three correct digits and I can state that sixth estimation is correct to within $$10^{-3}$$ However, the actual distance between the real root and the sixth iteration shows that the sixth iteration is accurate to fourth decimal place. What is going on here? Where am I making a mistake?

Iteration table https://imgur.com/a/h6oBaEo

Subtraction https://imgur.com/a/ulRhXUh

• There's no mistake. If it is accurate to the fourth decimal place, then it is also accurate to the third decimal place. What's wrong with that? – Franklin Pezzuti Dyer Oct 6 '18 at 23:41
• So, based only on the correct digits, we should say it is AT LEAST accurate to three decimal place instead of saying it is exactly accurate to three decimal place, shouldn't we? – Ali Kıral Oct 6 '18 at 23:58