Page 21 - Numerical Analysis Using MATLAB and Excel

Chapter 1 Introduction to MATLAB

3
2
5
dp dx⁄ = 12x – 32x + 4x + 8x + 10
5

1.5 Rational Polynomials

Rational Polynomials are those which can be expressed in ratio form, that is, as

n
1
2
n –
n –
x
+
b x +
b
b x +
… +
x
b
+
b
2
1
n
n –
Rx() = Num x() ------------------------------------------------------------------------------------------------------------------------- (1.8)
0
n –
--------------------- =
1
Den x() a x + a m – 1 x m – 1 + a m – 2 x m – 2 + … + a x + a 0
m
m
1
where some of the terms in the numerator and/or denominator may be zero. We can find the roots
of the numerator and denominator with the roots(p) function as before.
Example 1.9
Let
5
4
2
p num x – 3x + 5x + 7x + 9
Rx() = ------------ = ----------------------------------------------------------------------- (1.9)
p den 2x – 8x + 4x + 10x + 12
2
4
6
Express the numerator and denominator in factored form, using the roots(p) function.
Solution:
num=[1 −3 0 5 7 9]; den=[2 0 −8 0 4 10 12];% Do not display num and den coefficients
roots_num=roots(num), roots_den=roots(den) % Display num and den roots
roots_num =
2.4186 + 1.0712i 2.4186 - 1.0712i -1.1633
-0.3370 + 0.9961i -0.3370 - 0.9961i
roots_den =
1.6760 + 0.4922i 1.6760 - 0.4922i -1.9304
-0.2108 + 0.9870i -0.2108 - 0.9870i -1.0000
As expected, the complex roots occur in complex conjugate pairs.
For the numerator, we have the factored form
p num ( x 2.4186 j1.0712 ⋅ = – – ) ( x 2.4186 + j1.0712 ⋅ ) ( x + 1.1633 ⋅ )
–
)
( x + 0.3370 j0.9961 ⋅ ( x + 0.3370 + j0.9961 )
–
and for the denominator, we have
p den ( x 1.6760 – j0.4922 ⋅ = ) ( x 1.6760 + j0.4922 ⋅ ) ( x + 1.9304 ⋅ )
–
–
( x + 0.2108 j0.9870 ⋅ – ) ( x + 0.2108 + j0.9870 ⋅ ) ( x + 1.0000 )
1−8 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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