Solving Equations

Forward and backward error

backward error is on the input side . It is the amount we would need to change the function $f$ to make the equation balance with the output approximation $x_a$. This amount is $|f(x_a)|$

Forward error is the error on the output side. It is the amount we would need to change the approximate solution to make it correct ,which is $|r-x_a|$

example

function $f(x) = \sin x - x$ has a triple root at $r = 0$.

the forward and backward error of the approximate root $x_c = 0.001$

the forward error is $|r-x_a| = 10^{-3}$
the backward error is $|f(x_a)| = |\sin(0.001)-0.001|\approx 1.6667 \times 10^{-10}$