download searchable PDF of Circuit Design book - IEEE Global ...

18 Some Fundamental Numerical Methods
least six decimal places of true value. The value for In 2 in Figure 2.5 is off in
the ninth place.
2.3.4. Simpson's Integration Rule. The order of truncation errors for repeated
linear interpolation is shown in the top row of Figure 2.5. The j = 0
column is the trapezoid rule, and the j =I column happens to be the wellknown
Simpson rule. The other columns represent increasing orders of accuracy,
but they do not coincide with other frequently used methods, such as
Weddle's rule (see Ley, 1970, p. 246). Simpson's rule is to Ge applied in
Section 4.6, where independently incremented function data will be integrated.
Therefore, it will be convenient to obtain a closed formula for Simpson's rule,
comparable to (2.19) for the trapezoid rule. Recall that the area in Figure 2.3
is an estimate of an integral of f(x) from a to b; call it To.,' From Figures 2.3
and 2.5, To.o=h(f, +fb). So, with i=O in (2.26), T,.o becomes
where
T,.o=~(f,+4fHh+fb)' (2.28)
h ~ b-a 2 . (2.29)
This is Simpson's three-point rule.
The general formula for Simpson's rule can be recognized by first finding
the five-point rule, namely, Tl,p using h, = h/2. Extending the analysis
evident in Figure 2.3, the five-point trapezoid rule is
h ( I I )
To., = 1 If,+fHh/,+fah+f'+3h/'+ lfb .
Substituting (2.30) and (2.28) into (2.26), with i= I, yields
(2.30)
h/2
T,., = -3-(f,+4f'+h/' + 2fa+h +4f'+3h/' + f b ). (2.31)
Deducing Simpson's rule from (2.29) and (2.31) and putting it into standard
form, using variable t, we obtain
r'"
where n is even and
at
J", f(t)db 3 (f o+4f,+2f,+ ... +4f n _,+f n ),
(2.32)
t _L
at= _n_'1) • (2.33)
n
Recall that errors in the trapezoid rule were proportional to (M)'. Simpson's
rule errors are proportional to (at)4.
2.3.5. Summary of Integration. Romberg integration is based on numerically
stable trapezoidal integration. The number of trapezoid sections neces-