Experimental estimation of errors
A common maxim for the ranking of numerical methods is: the higher the or der of convergence p the better the numerical method. Therefore, for a particular numerical method, numerical analysts work hard to establish a sharp estimate of p, while the error constant C is normally left unspecified. It is clear that without an estimate of the error constant C, an error bound of this form gives information only about the rate of convergence of the numerical solutions. In other words, it tells us that if we use more mesh points then the error decreases like N ~ p. However, if the value of the error constant C is unknown, we cannot compute the magnitude of the right hand side of (8 .1) and, consequently, we have no estimate of the magnitude of the errors in our numerical approximations.