Page 291 - Numerical Analysis Using MATLAB and Excel

Antidifferences

b
∫ fx() x = fb() fa() (7.39)
–
d
a
we have the fundamental theorem of sum calculus which states that

)
α + ( n – 1 h – α +
1
∑ p x() = Δ p x() nh (7.40)
n
n
x = α α
Example 7.5
Derive a simple expression, in closed form, that computes the sum of the cubes of the first odd
n
integers.

Solution:

An odd number can be expressed as 2m 1– , and thus its cube is 2m 1–( ) 3 . To use (7.40), we
must express this term as a factorial polynomial. Recalling from (7.12) that

n ()
x () xx 1 x2 … – ) ( x – n + 1 ) (7.41)
(
)
( =
–
and using the MATLAB expand(f) function where f is a symbolic expression, we execute
syms m; f = (2*m−1)^3; expand(f)
and we obtain

ans =
8*m^3-12*m^2+6*m-1
Thus

2
3
–
pm()= ( 2m – 1 ) 3 = 8m – 12m + 6m 1 (7.42)
Following the procedure of Example 7.3, we find p m() with MATLAB as
n
pm=[8 −12 6 −1];
d0=[1 0];
[q0,r0]=deconv(pm,d0)
d1=[1 −1];
[q1,r1]=deconv(q0,d1)
d2=[1 −2];
[q2,r2]=deconv(q1,d2)
d3=[1 −3];
[q3,r3]=deconv(q2,d3)

q0 =
8 -12 6

Numerical Analysis Using MATLAB® and Excel®, Third Edition 7−13
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