Is $10^{-3}$ error or tolerance? .Find a solution accurate within $10^{-3}$ for $() = ^{3} + 3 − 5$ on$ [1, 2].$

Question

According to the algorithm,$f(c)=0$ 0r $(a-b)/2<TOL$ is stop procedure.
My problem is in this question, Is $10^{-3}$ error or tolerance. If it is an error, should I check $|aₙ-bₙ|<10^{-3}$ to stop the iteration? and if it is TOL should I check $(a-b)/2<TOL$ to stop iteration? I have confusion about the differences between tolerance and error and the relationship between them.

Answer 1

Tolerance is usually synonymous for error. As stated in the procedure, you want to stop when $(b-a)/2<10^{-3}$. Note that $|a-b|=b-a>0$ by the initial condition and that the distance the actual root is from $c$ is at most $(b-a)/2$ when the root is at $a$ or $b$ (the farthest points from $c$, the midpoint).