Let us first define the local truncation error at of the general explicit one-step method defined by

(3.31)

Definition: The local truncation error at of the one step method (3.31) is defined to be where                    (3.32)

and is the true solution of the initial value problem.

If we assume that no previous errors have been made, viz. , then from (3.32) and (3.11) , it follows that

and the truncation error defined by (3.32) is local.

The global truncation error of the one step method (3.11), denoted by , and is defined by where it is no longer assumed that no previous truncation errors have been made.

Definition: The local truncation error for the non-linear method (3.11) of order p is

(3.33)

where the function is called the principal error function , and is called the principal local truncation error . In other words, one can also define that the method (3.11) is of order p, if its local truncation error is of .