Page 323 - Numerical Analysis Using MATLAB and Excel

Solutions to End−of−Chapter Exercises

7.11 Solutions to End−of−Chapter Exercises

1.
3
4
5
fx() = x – 2x + 4x – x + 6
The highest power of the given polynomial fx() is , we must evaluate the remainders
5
r r r r r,,,, 4 and ; then, we will use (7.28), repeated below, to determine p x() .
r
0
1
5
n
2
3
p x() = r + r x() 1 () + r x() 2 () + … + r n – 1 x () ( n – 1 ) + r x() n ()
n
n
1
2
0
We can compute the remainders by long division but, for convenience, we will use the MAT-
LAB deconv(p,q) function which divides the polynomial p by q.
The MATLAB script is as follows:
px=[1 −2 4 0 −1 6]; % 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 divisor, i.e, x−4
[q4,r4]=deconv(q3,d4) % Computation of fifth quotient and remainder
d5=[1 −5]; % Coefficients of sixth (last) divisor, i.e, x−5
[q5,r5]=deconv(q4,d5) % Computation of sixth (last) quotient and remainder

q0 =
1 -2 4 0 -1

r0 =
0 0 0 0 0 6
q1 =
1 -1 3 3
r1 =
0 0 0 0 2
q2 =
1 1 5
r2 =

0 0 0 13

Numerical Analysis Using MATLAB® and Excel®, Third Edition 7−45
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