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ROOTS OF EQUATIONS
Author
Lizeth Paola Barrero Riaño
Industrial University of Santander
2010
A root or solution of equation f(x)=0 are
the values of x for which the equation
holds true. The numerical methods are
used for finding roots of equations, some
of them are:
1. GRAPHICAL METHOD
It is a simple method to obtain an ap-
proximation to the equation root f(x) =0.
It consists of to plot the function and de-
termine where it crosses the x-axis. At
this point, which represents the x value
where f(x) =0, offer an initial approxima-
tion of the root.
The graphical method is necessary to use
any method to find roots, due to it allows
us to have a value or a domain values in
which the function will be evaluated, due
to these will be next to the root. Like-
wise, with this method we can indentify if
the function has several roots.
2. CLOSED METHODS
These are called closed methods because
are necessary two initial values to the
root, which should “enclose” or to be to
the both root sides. The key feature of
these methods is that we evaluate a do-
main or range in which values are close to
the function root; these methods are
known as convergent. Within the closed
methods are the following methods:
2.1 BISECTION (Also called Bol-zano method)
The method feature lie in look for an in-
terval where the function changes its sign
when is analyzed. The location of the sign
change gets more accurately by dividing
the interval in a defined amount of sub-
intervals. Each of this sub intervals are
evaluated to find the sign change. The
approximation to the root improves ac-
cording to the sub-intervals are getting
smaller.
The following is the procedure:
Step 1: Choose lower, xl, and upper, xu,
values, which enclose the root, so that the
function changes sign in the interval. This
is verified by checking that:
0l uf x f x
2
Step 2: An approximation of the xr root,
is determined by:
Step 3: Realize the following evaluations
to determine in what subinterval the root
is:
If , then the root is
within the lower or left subinterval,
so, do xu=xr and return to step 2.
If , then the root is
within the top or right subinterval,
so, do xl=xr and return to step 2.
If , the root is equal
to xr; the calculations ends.
The maximum number of iterations to ob-
tain the root value is given by the follow-
ing equation:
TOL = Tolerance
2.2 METHOD OF FALSE POSITION
Although the bisection method is techni-cally valid to determine roots its focus is
relatively inefficient. Therefore this method is an improved alternative based on an idea for a more efficient approach to the root. This method raises draw a straight line joining the two interval points (x, y) and (x1, y1), the cut generated by the x-axis allows greater approximation to the root. Using similar triangles, the intersection can be calculated as follows:
The final equation for False position
method is:
The calcula-
tion of the root xr requires replacing one
of the other two values so that they al-
ways have opposite signs, what leads
these two points always enclose the root.
Sometimes, depending on the function,
this method works poorly, while the bi-
section method leads better approxima-
tions. section method leads better ap-
0l rf x f x
0l rf x f x
0l rf x f x
2
l ur
x xx
3
section method leads better approxima-
tions.
3. OPEN METHODS
These methods are based on formulas
that require a single initial value x, or a
couple of them but do not necessarily that
contain the root. Because of this feature,
sometimes these methods diverge or
move away from the root, according to
grows the number of iterations. It is im-
portant to know that when the open
methods converge, these are more effi-
cient than methods that use intervals.
Open methods are:
3.1 SIMPLE FIXED POINT ITE-
RATION
Open methods employ a formula that pre-
dicts the root, this formula can be devel-
oped for a single iteration of a fixed point (also called point iteration or successive
substitution) to change the equation f (x)
= 0 so that is:
x=g(x)
This formula is employ to predict a new x
value in function of the previous x valor,
through:
xi+1=g(xi)
Subsequently, these iterations are used to
calculate the approximate error so that
the least error indicates the root of the
function in matter.
For instance:
Use simple iteration of a fixed point to lo-
cate the root of f(x) = e-x – x
Solution: Like f(x)=0 ó e-x – x=0
Expressing of the form x=g(x) result us:
x= e-x
Beginning with an initial value of xo=0, we
can apply the iterative equation xi+1=g(xi)
and calculate:
Thus, each iteration bring near increas-
ingly to the estimated value with the true
value of the root, that is to say
0.56714329
3.2 NEWTON RAPHSON METHOD
The most widely used formula to find
roots, is the Newton Raphson, argues that
if it indicates the initial value of x1 as the
value of the root, then it is possible to ex-
tend a tangent line from the point (x1,f
(x1)). Where this straight line crosses the
x-axis will be the point of improve ap-
proach.
=
1
1
100i ia
i
x xx
x
i xi Ea(%) Et(%)
0 0 100.0
1 1.000000 100.0 76.3
2 0.367879 171.8 35.1
3 0.692201 46.9 22.1
4 0.500473 38.3 11.8
5 0.606244 17.4 6.89
6 0.545396 11.2 3.83
7 0.579612 5.90 2.20
8 0.560115 3.48 1.24
9 0.571143 1.93 0.705
10 0.564879 1.11 0.399
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This method could derived in a graphically
or using Taylor’s series.
Newton Raphson formula
From its reorder is obtained the value of
the desired root.
Note: The Newton Raphson method has a
strong issue for its implementation, this is
due to the derivative, as in some functions
is extremely difficult to evaluate the de-
rivative.
3.3 Secant Method
The secant method allows an approxima-
tion of the derivative by means of a di-
vided difference; this method avoids fal-
ling into the Newton Raphson problem, as
it applies for all functions regardless of
whether they have difficulty in evaluating
its derivative.
The approximation of the derivative is ob-
tained as follows:
Substituting this equation in the Newton
Raphson formula we get:
The above equation is the secant formula.
4. MULTIPLES ROOTS
A multiple root corresponds to a point
where a function is tangential to the x axis
For instance, double root results of:
f(x)=(x – 3)(x – 1)(x – 1) ó f(x)=x3 – 5x2 – 7x - 3
Because a value of x makes two terms in
the previous equation are zero. Graphi-
cally this means that the curve tangen-
tially touches the x axis in the double root.
The function touches the axis but not
cross in the root.
A triple root corresponds to the case
where a value of x makes three terms in
an equation equal to zero, as:
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an equation equal to zero, as:
f(x)=(x – 3)(x – 1)(x – 1)(x – 1)
ó f(x)=x4 – 6x3 + 12x2– 10x + 3
In this case the function is tangent to the
axis at the root and crosses the axis.
In general, odd multiplicity of roots
crosses the axis, while the pair multiplicity
does not cross.
The multiple roots offer some difficulties
to numerical methods:
1. The fact that the function does not
change sign on multiple pairs roots pre-
vents the use of reliable methods that
use intervals.
2. Not only f(x) but also f'(x) approach
to zero. These problems affect the New-
ton-Raphson and the secant methods,
which contain derivatives in the denomi-
nator of their respective formulas.
Some modifications have been proposed
to alleviate these problems, Ralston and
Rabinowitz (1978) proposed the following
formulas:
a) Root Multiplicity
Where m is the root multiplicity (ie, m=2
for a double root, m=3 for a triple root,
etc.). This formulation may be unsatisfac-
tory because it presumes the root multi-
plicity knowledge.
b) New function definition
The new function is
replaced in the Newton-Raphson’s method
equation, so you get an alternative:
c) Secant Method Modification
Substituting the new function (explained
previously) in the equation of the secant
method, we have
5. POLYNOMIALS ROOTS
Below are described the methods to find
polynomial equations roots of the general
form:
Where n is the polynomial order and the a
are constant coefficients. The roots of
such polynomials have the following rules:
a. For equation of order n, there are n
real or complex roots. It should be
noted that these roots are not necessar-
ily different.
b. If n is odd, there is at least one real
root.
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c. If the roots are complex, there is a
conjugate pair (i.e., λ + µi y λ - µi) where
A predecessor of Muller method is the se-
cant method, which obtain root, estimating
a projection of a straight line on the x-axis
through two function values (Shape
1). Muller's method takes a similar view, but
projected a parabola through three points
(Shape 2).
The method consist in to obtain the coeffi-
cients of the three points, replace them in
the quadratic formula and get the point
where the parabola intersects the x-axis The
approach is easy to write, as appropriate
this would be:
Shape 1
Shape 2
Thus, this parable is used to intersect the three points [x0, f(x0)], [x1, f(x1)] and [x2, f(x2)]. The coefficients of the previous equa-tion are evaluated by substituting one of these three points to make:
The last equation generated that , in this way, we can have a system of two equations with two unknowns: Defining this way:
Substituting in the system: The coefficients are:
Finding the root, the conventional solution is implemented, but due to rounding error po-tential, we use an alternative formulation:
cxxbxxaxf )()()( 2
2
22
cxf )( 2
2
0 0 2 0 2( ) ( ) ( )f x a x x b x x c
2
1 1 2 1 2( ) ( ) ( )f x a x x b x x c
2
2 2 2 2 2( ) ( ) ( )f x a x x b x x c
2
0 2 0 2 0 2( ) ( ) ( ) ( )f x f x a x x b x x
)()()()( 21
2
2121 xxbxxaxfxf
010 xxh 1 2 1h x x
12
121
)()(
xx
xfxf
01
210
)()(
xx
xfxf
1100
2
1010 )()( hhahhbhh
11
2
11 hahbh
01
01
hha
11 ahb
)( 2xfc
acbb
cxx
4
2
223
7
Solving: The great advantage of this method is that we find real and imaginary roots. Finding the error this will be: To be an approximation method, this is per-formed sequentially and iteratively, where x1, x2, x3replace the points x0, x1, x2 carrying the error to a value close to zero. 5. 1 Bairstow’s Method Bairstow's method is an iterative process related approximately with Muller and New-ton-Raphson methods. The mathematical process depends of dividing the polynomial between a factor. Given a polynomial fn(x) find two factors, a quadratic polynomial f2(x) = x2 – rx – s y fn-2
(x). The general procedure for the Bairstow method is: 1. Given fn(x) y r0 y s0 2. Using Newton-Raphson method calcu-
late f2(x) = x2 – r0x – s0 y fn-2(x), such as, the residue of fn(x)/ f2(x) be equal to zero.
3. The roots f2(x) are determined, using the general formula.
4. Calculate fn-2(x)= fn(x)/ f2(x). 5. Do fn(x)= fn-2(x) 6. If the polynomial degree is greater
than three, back to step 2 7. If not, we finish
The main difference of this method in rela-tion to others, allows calculating all the poly-nomial roots (real and imaginaries). To calculate the polynomials division, we
use the synthetic division. So given fn(x) = anx
n + an-1xn-1 + … + a2x
2 + a1x + a0 By dividing between f2(x) = x2 – rx – s, we have as a result the following polynomial
fn-2(x) = bnx
n-2 + bn-1xn-3 + … + b3x + b2
with a residue R = b1(x-r) + b0, the residue will be zero only if b1 and b0 are. The terms b and c, are calculated using the following recurrence relation:
bn = an bn-1 = an-1 + rbn
bi = ai + rbi+1 + sbi+2
cn = bn cn-1 = bn-1 + rcn
ci = bi + rci+1 + sci+2
Finally, the approximate error in r and s can
be estimated as:
When both estimated errors failed, the roots
values can be determined as:
BIBLIOGRAPHY CHAPRA, Steven C. y CANALE, Raymond P.:
Métodos Numéricos para Ingenieros.
McGraw Hill 2007. 5th edition.
http://numerical-methods.com/roots.htm
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