There are two fundamental sources of error in solving an initial value problem
numerically. Let us first assume that our computer is such that we can carry
out all computations with complete accuracy; that is, we can retain an
infinite number of decimal places. The difference 
between the solution
of the initial value problem (2) and its numerical
approximation is given by
and is known as the global truncation error. It is obtained by taking
the max of all 
's. It arises from two causes:
(approximating the curve locally by the tangent).
is not equal to 
.
, then the only error in going one step
is due to the use of an approximate formula. This error is know as the
local truncation error 
.
Note that this definition of local truncation error is different from that
in Golub/Ortega's book. They implicitly divide the term that we derive by
h.
The second fundamental source of error is that we carry out the computations
in arithmetic with only a finite number of digits (like IEEE arithmetic).
This leads to a round-off error 
defined by
where 
is the value actually computed from the given numerical
method.
The absolute value of the total error in computing 
is given by
Making use of the triangle inequality, 
, we obtain
Thus the total error is bounded by the sum of the absolute values of the truncation and round-off errors. For the numerical procedures discussed in this course it is possible to obtain useful estimates of the truncation error. However, we limit our discussion primarily to the local truncation error, which is somewhat simpler. The round-off error is clearly more random in nature. It depends on the sequence in which the computations are carried out.
