Page 319 - Numerical Analysis Using MATLAB and Excel

Summary

2. We divide q x() by x – 1 ) , to obtain a quotient q x() and a remainder which turns out
(
r
1
1
0
to be the constant term a 1 . Then,
)
q x() = r + ( x1 q x()
–
1
1
0
and by substitution we obtain
)
[
)
]
( +
–
p x() = r + xr + ( x1 q x() = r + r x() 1 () xx – 1 q x()
0
1
1
1
1
0
n
(
3. We divide q x() by x – 2 ) , to obtain a quotient q x() and a remainder which turns out
r
2
1
2
to be the constant term a 2 , and thus
)
q x() = r + ( x2 q x()
–
1
2
2
and by substitution we obtain
)
)
[
( +
p x() = r + r x() 1 () xx – 1 r + ( x2 q x() ]
–
0
n
2
1
2
)
)
(
( +
= r + r x() 1 () + r x() 2 () xx – 1 x – 2 q x()
1
0
2
2
and in general,
p x() = r + r x() 1 () + r x() 2 () + … + r n – 1 x () ( n – 1 ) + r x() n ()
2
n
n
1
0
where
j
Δ p 0()
n
r = a = -------------------
j
j
j!
• The antidifference of a factorial polynomial is analogous to integration in elementary calculus.
1
–
It is denoted as Δ p x() , and it is computed from
n
(
n +
1
x ()
–
1
Δ () n () = -------------------- )
x
( n + 1 )
• Antidifferences are very useful in finding sums of series.
• The definite sum of p x() for the interval a ≤ x ≤ a + ( n 1 h is
)
–
n
)
α + ( n – 1 h
∑ p x() = p α() + p α + h + p α + 2h + … + p α + ( n – 1 h ]
[
)
)
(
(
)
n
n
n
n
n
x = α
• In analogy with the fundamental theorem of integral calculus which states that
b
∫ fx() x = fb() – fa()
d
a
Numerical Analysis Using MATLAB® and Excel®, Third Edition 7−41
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