Page 286 - Numerical Analysis Using MATLAB and Excel

Chapter 7 Finite Differences and Interpolation

Continuing with the above procedure, we obtain a new quotient whose degree is one less than
preceding quotient and therefore, the process of finding new quotients and remainders terminates
after n +( 1 ) steps.
The general form of a factorial polynomial is

p x() = r + r x() 1 () + r x() 2 () + … + r n – 1 x () ( n – 1 ) + r x() n () (7.28)
2
0
1
n
n
and from (7.16) and (7.22),
j
Δ p 0()
n
r = a = ------------------- (7.29)
j
j
j!
or
j
Δ p 0() = j!r j (7.30)
n
Example 7.3
Express the algebraic polynomial
3
4
px() = x – 5x + 3x + 4 (7.31)
as a factorial polynomial. Then, construct the difference table with h = . 1
Solution:

Since the highest power of the given polynomial px() is , we must evaluate the remainders
4
r r r r,,, 3 and ; then, we will use (7.28) to determine p x() . We can compute the remainders
r
4
0
n
2
1
by long division, but for convenience, we will use the MATLAB deconv(p,q) function which
divides the polynomial p by q.
The MATLAB script is as follows:
px=[1 −5 0 3 4]; % Coefficients of given polynomial
d0=[1 0]; % Coefficients of first divisor, i.e, x
[q0,r0]=deconv(px,d0) % Computation of first quotient and remainder
d1=[1 −1]; % Coefficients of second divisor, i.e, x−1
[q1,r1]=deconv(q0,d1) % Computation of second quotient and remainder
d2=[1 −2]; % Coefficients of third divisor, i.e, x−2
[q2,r2]=deconv(q1,d2) % Computation of third quotient and remainder
d3=[1 −3]; % Coefficients of fourth divisor, i.e, x−3
[q3,r3]=deconv(q2,d3) % Computation of fourth quotient and remainder
d4=[1 −4]; % Coefficients of fifth (last) divisor, i.e, x−4
[q4,r4]=deconv(q3,d4) % Computation of fifth (last) quotient and remainder
q0 =

7−8 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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