Specifications

University of Pretoria etd – Combrinck, M (2006)
APPENDIX A
Derivation of Simpson’s rule for unequally spaced data values (centre points).
A Lagrange polynomial
p(x) is obtained such that p(x
i
)=f(x
i
) for i=1,2,3 and integrated
over the interval
.
31
xxx <
<
()()
()()
(
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(
)
()()
(
)
(
)
()()
()()
()()
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()()
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()()
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[]
()()
[]
()()
[]
()()
() () () []
()()
() () () []
()()
() ()
() []
1321
2
1
2
3
2
1
21
3
1
3
3
3
1
2313
3
1331
2
1
2
3
2
1
31
3
1
3
3
3
1
3212
2
1332
2
1
2
3
2
1
32
3
1
3
3
3
1
3121
1
2121
2
2313
3
3131
2
3212
2
3232
2
3121
1
21
2313
3
31
3212
2
32
3121
1
2313
21
3
3212
31
2
3121
32
1
2313
21
3
3212
31
2
3121
32
1
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3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
xxxxxxxxxx
xxxx
xf
xxxxxxxxxx
xxxx
xf
xxxxxxxxxx
xxxx
xf
dxxxxxxx
xxxx
xf
dxxxxxxx
xxxx
xf
dxxxxxxx
xxxx
xf
dxxxxx
xxxx
xf
dxxx
xx
xxxx
xf
dxxxxx
xxxx
xf
dx
xxxx
xxxx
xf
xxxx
xxxx
xf
xxxx
xxxx
xff(x)dx
xxxx
xxxx
xf
xxxx
xxxx
xf
xxxx
xxxx
xfxp
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
−+−+−−
−−
+
−+−+−−
−−
+
−+−+−−
−−
=
++−
−−
+
++−
−−
+
++−
−−
=
−−
−−
+
−−
−−
+
−−
−−
=
−−
−−
+
−−
−−
+
−−
−−
≈
−−
−−
+
−−
−−
+
−−
−−
=
∫
∫
∫
∫
∫
∫
∫∫
(B.2)
So
(B.1)
i