Specifications

University of Pretoria etd – Combrinck, M (2006)
() ()
()
()()
() () ()
()()
() ()
()
()()
332211
2313
1321
2
1
2
3
2
1
21
3
1
3
3
3
1
3
3212
1331
2
1
2
3
2
1
31
3
1
3
3
3
1
2
3121
1332
2
1
2
3
2
1
32
3
1
3
3
3
1
1
)()()(
)(
)(
)(
)(
)(
)(
WxfWxfWxf
xxxx
xxxxxxxxxx
xf
xxxx
xxxxxxxxxx
xf
xxxx
xxxxxxxxxx
xf
++=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−
−+−+−−
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−
−+−+−−
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−
−+−+−−
=
Derivation of Simpson’s rule for unequally spaced data values (end 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 intervals
3221
xxxxxx
<
<
<< and respectively.
()()
()()
()()
()()
()()
()()
() () ()
()()
() () ()
()()
() () ()
()()
332211
2313
1221
2
1
2
2
2
1
21
3
1
3
2
3
1
3
3212
1231
2
1
2
2
2
1
31
3
1
3
2
3
1
2
3121
1232
2
1
2
2
2
1
32
3
1
3
2
3
1
1
2313
21
3
3212
31
2
3121
32
1
)()()(
)(
)(
)(
)(
)(
)(
)()()(
2
1
2
1
WxfWxfWxf
xxxx
xxxxxxxxxx
xf
xxxx
xxxxxxxxxx
xf
xxxx
xxxxxxxxxx
xf
dx
xxxx
xxxx
xf
xxxx
xxxx
xf
xxxx
xxxx
xff(x)dx
x
x
x
x
++=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−
−+−+−−
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−
−+−+−−
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−
−+−+−−
=
−−
−−
+
−−
−−
+
−−
−−
≈
∫∫
(B.3)
Now,
i
i