Numerical recipes

4.1 Classical Formulas for Equally Spaced Abscissas 135
Extended Formulas (Open and Semi-open)
We can construct open and semi-open extended formulas by adding the closed
formulas (4.1.11)–(4.1.14), evaluated for the second and subsequent steps, to the
extrapolative open formulas for the first step, (4.1.7)–(4.1.10). As discussed
immediately above, it is consistent to use an end step that is of one order lower
than the (repeated) interior step. The resulting formulas for an interval open at
both ends are as follows:
Equations (4.1.7) and (4.1.11) give
�
3
f(x)dx = h
x1
2 f2 +f3 +f4 +···+fN−2 + 3
2 fN−1
�
Equations (4.1.8) and (4.1.12) give
� � xN
23
f(x)dx = h
x1
12 f2 + 7
12 f3 + f4 + f5+
···+ fN−3 + 7
12 fN−2 + 23
�
1
+ O
N 3
�
� xN
12 fN−1
Equations (4.1.9) and (4.1.13) give
� � xN
27
f(x)dx = h
x1
12 f2 +0+ 13
12 f4 + 4
3 f5+
···+ 4
3 fN−4 + 13
12 fN−3 +0+ 27
�
1
+ O
N 4
�
�
1
+O
N 2
�
�
12 fN−1
�
(4.1.15)
(4.1.16)
(4.1.17)
The interior points alternate 4/3 and 2/3. If we want to avoid this alternation,
we can combine equations (4.1.9) and (4.1.14), giving
� � xN
55
f(x)dx = h
x1
24 f2 − 1
6 f3 + 11
8 f4 + f5 + f6 + f7+
···+ fN−5 + fN−4 + 11
8 fN−3 − 1
6 fN−2 + 55
24 fN−1
�
�
1
+ O
N 4
�
(4.1.18)
We should mention in passing another extended open formula, for use where
the limits of integration are located halfway between tabulated abscissas. This one is
known as the extended midpoint rule, and is accurate to the same order as (4.1.15):
� xN
x1
f(x)dx = h[f 3/2 + f 5/2 + f 7/2+
�
1
···+ fN−3/2 + fN−1/2] +O
N 2
� (4.1.19)
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