## Errors of numerical integration

Numerical integration is much more reliable process compared to numerical
differentiation. The round-off error in computing the sum of values
*I*_{k}, where *k = 0,1,...,n*, is always constant
which does not depend on the rule of numerical integration. This constant
is bounded by the product of the integration interval *T*
and the maximal round-off error *e*_{r} in
computer's representation of numbers. Thus, if the truncation error
of the numerical integration rule can be reduced by a recursive algorithm
(see Lecture 3.5),
the resulting numerical approximation represents the exact value of the integral
accurately subject to a constant total round-off error.

The truncation error can be reduced by two different ways: by reducing
the step size *h* and by using the higher-order integration formula
of the order of *O(h*^{2}), *O(h*^{4}), and so on.
If the step size *h* between two adjacent values *I*_{k}
becomes smaller, the truncation error of the numerical integration rule decays.
For example, if the step size is reduced by half, the global
truncation error of the composite trapezoidal rule is reduced by four.
The figure below presents the results from the use of two composite
trapezoidal rules on the current *I = I(t)* (view the
data values for the current).
The approximations are obtained with step size *h = 10* (green pluses)
and with step size * h = 5* (blue dots), versus the exact
integral *S*_{T}[I(t)] (red solid curve). The error of
the composite trapezoidal rule clearly reduces with smaller step size *h*
(blue dots are closer to the exact red curve compared to the green pluses).

The figure also shows that the truncation error for the integral grows with
the length of the interval. It is the global truncation error of numerical
integration over the interval *t = 0* and *t = T*. The global
truncation error is distinguished from the local truncation error, the latter
error occurs when the integral between two adjacent points is replaced by a trapezoid.

In many cases, the data samples are given with a fixed step size *h* that
can not be controled. If this is the case, the numerical approximation for
the integral can be improved by using a higher-order integration rule,
such as the Simpson's rule. Romberg integration algorithm (see
Lecture 3.5)
allows to construct a sequence of higher-order integration rules starting with
few computations of the composite trapezoidal rule. The figure below presents
comparison of the composite trapezoidal rule (green pluses) and the
composite Simpson's rule (blue dots) for the integral of the current *I = I(t)*.
The step size *h* is the same for both the numerical integrations: *h = 10*.
The exact integral *S*_{T}[I(t)] is shown by red solid curve.
The composite Simpson's rule is clearly much more accurate than the composite
trapezoidal rule.