1. CHAPTER 3: ROOT FINDING-b
– Roots of Polynomials:
– Müller’s Method
– Bairstow’s Method
– Packages and Libraries

2. Roots of Polynomials:
An order-n polynomial
f n ( x)
a0
a1 x a2 x 2 ... an x n
The roots of the polynomial follow these rules:
> For nth-order polynomial there are n real/complex roots.
> The roots are not necessarily distinct.
> If n is odd, there is at least one real root.
> If complex roots exist they are in conjugate pairs (a±bi).
Computing with Polynomials:
Consider a 3rd order polynomial:
f 3 ( x)
a3 x 3
a2 x 2
f 3 ( x)
(( a3 x a2 ) x a1 ) x a0 (Nested form; 3 multiplications + 3 additions)
a1 x a0
(Open form; 6 multiplications + 3 additions)
Nested form is preferred as it requires fewer operations:
> Lower cost of computation.
> Lower round-off errors.

3. Polynomial deflation:
 Once you computed a root of a polynomial, you need to remove
this root from the polynomial before proceeding to the next root
(so as not to trap at the same root again).
 This process is done by a process called polynomial deflation.
 Consider a third order polynomial. This polynomail can always be
written as
f 3 ( x)
( x r1 )( x r2 )( x r3 )
(factored form)
where r1, r2 and r3 are the roots. To remove the root x=r1, divide
the polynomail by (x-r1). The remaining polynomial will be a
second order polynomial:
f 2 ( x)
( x r2 )( x r3 )
The remainder of this divison will be zero (R=0) since you divide by
its root. If it was not a root, R=consant (a 0th order polynomial).

4.  In general, if an n-th order polyomial (dividend) is divided by an
m-th order polynomail (divisor), the resulting polynomial
(quotient) is an (n-m)th order polynomail. The reminder will be
an (m-1)th order polynomial.
dividend
Reminder (R)
x
2
2 x 24
x 4
x 6
0
divisor
quotient
 In computer the polynomial deflation is done by a process called
synthetic divison.
Synthetic division by a linear factor (x-t):
bn
an
bi
ai
an .. a0 : coefficients of the dividend
bi 1t
for i n 1 to 0
bn .. b1 : coefficients of the quotient
b0 : coefficient of the reminder
For example above: a1=1, a2=2, a3=-24, t=4 > b2=1, b1=6, b0=0

5. dividend
divisor
x 4 2 x3 2 x 24
x2 2x 3
x2 3
Reminder (R)
quotient
6x 9
Synthetic division by a quadratic factor (x2-rx-s):
bn
bn
bi
an
an
1
ai
an .. a0 : coefficients of the dividend
1
rbn
rbi
1
sbi
2
for i n 2 to 0
bn .. b2 : coefficients of the quotient
b1 ,b0 : coefficients of the reminder
 Synthetic division is subject to round-off errors. To minimize this
effect some strategies are used:
> root polishing: use root estimates as initial guess for successive
estimates.
> start from the root with smallest absolute value.

6. Conventional methods for finding roots of polynomials:
 Generally speaking, previously mentioned bracketing and open
methods can be used for finding the real roots of polynomials.
However,
> Bracketing methods are difficult for finding good initial guesses.
> Open methods are problematic due to divergence problems.
 Special methods have been developed to locate both real and
complex roots of polynomials. Two conventional methods:
- Müller’s method: Fitting a parabola to the function
- Bairstow’s method: Dividing the polynomial by a guess factor
 There are other methods too,e.g.:
- Jenkins-Traub method
- Laguerre’s method
- ...
 Plenty of software library is available for locating roots of
polynomials.

7. Müller’s Method
 This method is similar to Secant method. The difference is that
instead of drawing a straight line for two initial guesses, a
parabola is drawn with three initial guesses.
Secant
Müller
xr
xr x
x0
x1
two initial guesses
x1
2
x0
three initial guesses
 Parabolic fit function facilitates finding both real and complex roots.
 Define the parabola as
f 2 ( x)
a( x x2 ) 2 b( x x2 ) c
 We need to find the coefficients (a,b,c) such that the parabola
passes from three initial guesses: [x0, f(x0)] , [x1, f(x1)] , [x2, f(x2)] .

8. We get these three equations with there unknowns (a,b,c):
f ( x0 )
a ( x0
x2 ) 2 b( x0
x2 ) c
f ( x1 )
a( x1
x2 ) 2 b( x1
x2 ) c
f ( x2 )
a ( x2
x2 ) 2 b ( x2
x2 ) c
From the third equation => f ( x2 ) c
Then, we get two equations with two unknowns:
f ( x0 )
f ( x2 )
a ( x0
x2 ) 2 b( x0
x2 )
f ( x1 )
f ( x2 )
a( x1
x2 ) 2 b( x1
x2 )
Define
h0
x1
x0
h1
Then
(h0
h1 )b (h0
x2
x1
h1 ) 2 a
2
h1b h1 a
h1
0
h0
1
0
f ( x1 ) f ( x0 )
x1 x0
h1
1
1
f ( x2 ) f ( x1 )
x2 x1

9.  The coefficients a,b,c are determined as
a
1
0
h0
h1 h0
b
ah1
c
x1
h1
x0
f ( x1 ) f ( x0 )
x1 x0
x2
x1
f ( x2 ) f ( x1 )
x2 x1
f ( x2 )
1
0
1
 To calculate the root, calculate the point where the parabola
intersects the x-axis, i.e. f (x) 0 . Then, the roots (xr) are
xr
xr
x2
x2
2c
b2
b
4ac
2c
b
b 2 4ac
(alternative formula for
roots of quadratic polyn.)
(both real and complex
roots are facilitated.)

10.  Root is estimated iteratively (i.e., xr x3)
 Error can be defined as the difference between the current (x3)
and the previous (x2) solution:
x3
a
x2
x3
2c
100%
b
b2
1
100%
4ac x3
 A problem arises as which of the two estimated roots (xr) will be
used for the next iteration.
> choose the one that makes the error smaller.
 Once x3 is determined the process is repeated. For the three
points of the next iteration, suggested strategy:
> choose other two points that are nearest to the estimated root
(If estimated roots are real).
> just follow the method sequentially, i.e., drop x0, and take x1 ,
x2 , x3 (if a complex root is located).
 Once a root is located, it is removed from the polynomial, and
the same procedure is applied to locate other roots.

11. EX: Use Muller’s method to determine a root of the equation
f ( x)
x 3 13 x 12
With initial guesses x0=4.5 , x1=5.5 , x2=5.0.
First iteration:
f (5) 48
f (4.5) 20.625 f (5.5) 82.875
h0
0
5.5 4.5
h1
82.875 20.625
5.5 4.5
5 5.5
62.25
1
48 82.875
5 5.5
69.75
Then,
a
69.75 62.25
15 b 15( 0.5) 69.75 62.25
0.5 1
c 48

12. To determine the estimated root:
b 2 4ac
since
62.252 4(15)48 31.54
62.25 31.54
62.25 31.54
The estimated root is:
2(48)
62.25 31.54
3.976
Error:
a
3.976 5
100 %
3.976
25 .74 %
For the next iteration, assign new
guesses (sequential replacement):
x0=5.5
x1=5
x2=3.976
Xr
0
5
1
5
i
3.976
25.74
2
x3
4.001
0.6139
3
4
0.0262
4
4
< 10-5
a(%)
method rapidly converges
to the root xr=4.

13. Bairstow’s Method
 We know that dividing a polynomial by its root result in a
reminder of zero. If the divisor is not a root then the reminder
will not be zero.
 The basic approach of Bairstow’s method is starting a guess value
for root and dividing the polynomial by this value. Then the
estimated root is changed until the reminder of the division get
sufficiently small.
Consider the polynomial:
f n ( x)
a0
a1 x a2 x 2 ... an x n
If you divide the polynomial by its factor (x-t), you get a new
polynomial of the form
f n 1 ( x) b1 b2 x b3 x 2 ... bn x n
1
and a reminder R b0

14. This process can be done by using synthetic division:
bn
an
bi
ai
bi 1t
for i n 1 to 0
 In order to facilitate the complex root Bairstow’s method divides
the polynomial by a quadratic factor: (x2-rx-s). Then the result
will be in the form of
f n 2 ( x) b2 b3 x ... bn 1 x n
with a reminder
R
3
bn x n
2
b1 ( x r ) b0
The process can be done by synthetic division by a quadratic factor:
bn
bn
an
an
1
1
rbn
for i n 2 to 0
Then, the problem is to find (r,s) values that make the reminder
term zero (i.e. b1=b0=0).
bi
ai
rbi
1
sbi
2

15.  We need a systematic way to modify (r,s) values to reach the
correct root. In order to this, we use an approach that is similar
to Newton-Raphson method.
Consider ing b0 and b1 are both functions of (r,s), and expand them
using Taylor series:
b1 (r
b0 (r
r, s
r, s
s) b1
b1
r
r
s) b0
b0
r
r
b1
s
s
b0
s
s
(no higher order terms)
 We force b0 and b1 to be zero to reach the root; then we get
b1
r
r
b1
s
s
b0
r
r
b0
s
s
b1
b0
Need to solve these two
equations for two
unknowns: r and s

16. Define
b0
r
c1
b1
r
c2
b0
s
c3
b1
s
Then, we have the following simultaneous equations:
c2 r c3 s
b1
c1 r c2 s
b0
 We previously applied synthetic division to derive values of “b”
from values of ”a”. Bairstow showed that we can evaluate values
of “c” from values of “b” in a way similar to synthetic division:
cn
cn
ci
bn
bn
1
bi
1
rc n
rci
1
sci
2
for
i
n 2 to 1
Once r and s are calculated using the simultaneous equations,
Then, for the next iteration (r,s) values are improved as:
r
r
r
s
s
s

17.  Error at each iteration can be calculated for r and s seperately:
a ,r
r
100 %
r
and
s
100 %
s
a ,s
 When both values fall below a stopping criterion, the roots can
be determined as the roots of x 2 rx s :
x1, 2
r
r2
2
4s
 Both real and complex roots are obtained.
 Once the roots are located, these roots are removed, and the
same procedure is applied to locate other roots.
EX: Use Bairstow’s method to determine the roots of the equation
f 5 ( x)
x 5 3.5 x 4
2.75 x 3
2.125 x 2 3.875 x 1.25
using initial guess of r=s=-1, and stopping criterion s=1%

18. First iteration:
Apply synthetic division to obtain b values:
b5=1 ; b4=-4.5 ; b3=6.25 ; b2=0.375 ; b1=-10.5 ; b0=11.375
Apply synthetic division to obtain c values:
c5=1 ; c4=-5.5 ; c3=10.75 ; c2=-4.875 ; c1=-16.375
Solve the simultaneous equations:
r=0.3558 and s=1.1381.
Revise (r, s) values: r+ r=-0.6442 and s+ s=0.1381
Error:
a,r=55.23% and a,s=824.1%
Second iteration:
b5=1 ; b4=-4.1442 ; b3=5.5578 ; b2=-2.0276 ; b1=-1.8013 ; b0=2.1304
c5=1 ; c4=-4.7884 ; c3=8.7806 ; c2=-8.3454 ; c1=4.7874
Solution of the simultaneous equations: r=0.1331 and s=0.3316.
Revise (r, s) values: r+ r=-0.5111 and s+ s=0.4697
Error : a,r=26.0% and a,s=70.6%
Third iteration:
……

19. Forth iteration:
r=-0.5 with
s=0.5 with
a,r=0.063%
a,s=0.040%
Errors satisfy the error criterion. Then, we obtain first two roots as:
r1, 2
0.5
( 0.5) 2 4(0.5)
2
0.5
1.0
For the other roots, we use the deflated polynomial:
f ( x)
x 3 4 x 2 5.25 x 2.5
Apply Bairstow’s method with starting guesses r=-0.5 and s=0.5
(solution of the previous step) .
After five iterations, we get r=2 and s=-1.249. Then, the roots are
r3, 4
2
22 4( 1.249)
2
1 0.499i
We can easily obtain the fifth root by deflating the polynomial into
the first order. The fifth root will be r5=2.

20. Libraries and Packages
Matlab
function
Method
fzero
initial guesses
fzero(function, x0)
fzero(function [x0 x1])
sign change
finds the real root of a single non-linear
equation
 A combination of bisection method + secant
(Müller’s) methods
 It first search for sign change, then applies
secant algorithm.
 If the root falls outside the interval, bisection
method is applied until an acceptable result is
obtained with fatser methods.
 Usually it starts with bisection, and as the root
is appraoched it shifts to faster methods
roots
roots(c)
vector containing coefficients
 determines all real and complex roots of
polynomials.
 uses the method of eigenvalues.

21. IMSL:
routine
method
ZREAL
Finds real zeros of real function using Müller’s method.
ZBREN
Finds a zero of a real function in a given interval with
sign change
ZANLY
Find zeros of a univariate complex function using
Müller’s method
roots of functions
ZPORC
Finds zeros of a polynomial with real coefficients using
Jenkins-Traub algorithm
ZPLRC
Finds zeros of a polynomial with real coefficients using
Laguerre method
ZPOCC
roots of polynomials
Finds zeros of polynomials with complex coefficients
using Jenkins-Traub algorithm.