Roots of equations (Theory & problems)

1. Numerical Analysis
Roots of equations (Theory & problems)Roots of equations (Theory & problems)

2. Years ago , we learn to use the
cuadratic formula to solve F(x).
The values calculated with this
equation are called the ‘’roots’’.
They represents the values of x
that make f(x) equal to zero.
Thus we can define the roots of
an equation as the value of x
that makes f(x)=0. For this
reason , roots are sometimes
called the zeros of the equation

3. • A root or solution of equation f(x)=0 are
the values of x for which the equation
holds true. Sometimes roots of equations
are called the zeros of the equation.
• Numerical methods for finding roots of
equations can often be easily
programmed.
ROOTS OF EQUATIONS

4. Classification
 1. GRAPHICAL METHOD
It is a simple method to obtain an
approximation to the equation root f(x) =0.
It consists of to plot the function and
determine where it crosses the x-axis. At
this point, which represents the x value where
f(x) =0, offer an initial approximation 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. Likewise, with this method we
can indentify if the function has several roots.

5. 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
domain 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:

6. The bisection method, which is alternatively called binary chopping,
interval halving, or Bolzano's method, is one type of incremental search
method in which the interval is always divided in half. If a function changes
sign over interval, the function value at the midpoint is evaluated. The
location of the root is then determinate as lying at the midpoint of the
subinternal within which the sign change occurs .The process is repeated
to obtain refined estimates.

7. A simple algorithm for the calculation is shown in the figure ( CHAPRA,
source information)

8. THE FALSE-POSITION METHOD
 “An alternative method that exploits tis graphical insight is to
join f(xi) and f(xu) by a straight line. The intersection of this
line with the x axis represents an improved estimate of the
root. The fact that the replacement of the curve by a straight
line gives a “false position”. Source: CHAPRA, numerical methods for
engineers.
The intersection of the line with the x axis can be estimated
as
Source: Internet

10. Although bisection is a perfectly valid technique for determining roots, its
‘’brute-force’’ approach is relatively inefficient . False position is an
alternative based on a graphical insight.
A shortcoming of the bisection methods is that , in dividing the interval
from xl to xa into equal halves, no account is taken of the magnitudes of
f(xl) and F(xa).
For example f(xl) is much closer to zero than f(xa), it is likely that the root
is closer to xl than to xa.
An alternative method that exploits this graphical insight is to join f(xl) and
F(xa) by a straight line
The fact that the replacement of the curve by a straight line gives a false
position of the root is the origin of the name, method of false position, it is
also called the liner interpolation method

11. • Another option to find this roots is to
incorporate an incremental search at the
beginning of the computer program. This
method consist in taking one end of the
region of interest and then evaluate the
function at small increments across the
region. The point is: When the function
changes the sign this mean that in that
point there is a root.
INCREMENTAL SEARCHES

12. A simple method for obtaining an
estimate of the root of the equation
f(x)=0 is to make a plot of the function
and observe where it crosses the x
axis. this point which represents the x
value for which f(x)=0, provides a rough
approximation of the root
Graphical techniques are of limited
practical value because they are not
precise. However, graphical methods
can be utilized to obtain rough
estimates of roots.

13. • Solution: You have to build a chart with values of x and
f(x), trying to get a crossing with the x-axis.
GRAPHICAL METHODS
Root Root
So we can see that there are two roots, one is
approximately -2 and the other is approximately 6.

14. Cases where roots could be missed because the increment length of the search
procedure is too large. note that the last root on the right is multiple and would be
missed regardless of increment length

15. INCREMENTAL SEARCHES AND DETERMINING
INITINAL GUESSE
 Another option to determinate all possible root is incorporate
an incremental search at the beginning of the computer
program. This consists of starting at one end of the region of
interest and then making function evaluations at small
increments across the region. When the function changes sign,
it is assumed that the root falls within the increment.
 A potential problem with an incremental search is the choice
of the incremental length If the length is too small, the search
can be very time consuming. On the other hand , if the length is
too great , there is a possibility that closely space roots might
be missed . The problem is compounded by the possible
existence of multiple roots. A partial remedy for such cases is
to computed the first derivate changes sign, it suggests that a
minimum or maximum may have occurred and the interval
should be examined more closely for the existence of a
possible root

16. OPEN METHODS

17. • To rearrange the function f(x)=0
• x=g(x)
SIMPLE FIXED-POINT ITERATION

18. THE NEWTON RAPHSON METHOD
Perhaps the most widely use of all root- locating formulas is
the Newton – Raphson equation . if the initial guess at the
root is xi , a tangent can be extended from the point ( xi, f(xi)).
the point where this tangent crosses the x axis usually
represents an improved estimate of the root

19. • If the initial guess at the root is xi, a
tangent can be extended from the point
{xi,x(xi)}. The point where this tangent
crosses the x axis usaually represents an
improved estimate of the root.
• The Newton-Raphson formula is:
THE NEWTON-RAPHSON
METHOD

20. The Newton-Raphson method
can be derived on the basis
on the basis of this
geometrical interpretation.,
the first derivative at x is
equivalent to slope:
Which cab be
rearranged to yield
Newton – Raphson
formula

21. Graphical depiction of the secant method. This technique is similar to the
Newton-Raphson .technique in the sense that an estimate of the root is
predicted by extrapolating a tangent of the function to the x axis . However ,
The secant method uses a difference rather than a derivative to estimate the
slope

22. • In the Secant method the derivative is approximated by
a backward finite divided difference.
• This approximation can be substituted with the following
iterative equation:
THE SECANT METHOD
The difference between the secant method and the false
position method is how one of the initial values is replaced
by the new estimate

23. A multiple root corresponds to a point where a function is
tangent to the x axis. For example, a double root results from
The equation has a double root because one value of x
makes two terms in 1st
equation equal to zero. Graphically,
this corresponds to the curve touching the x axis tangentially
at the double root. Examine Fig 6.10a at x=1. Notice that the
function touches the axis but does not cross it at the root.
A triple root corresponds to the case where one x value
makes three terms in a equation equal to zero as in
Notice that the graphical depiction ( fig6.10B) again indicates
that the function is tangent to the axis at the root , but that for
this case the axis is crossed. In general, odd multiplied roots
cross the axis, whereas even ones do not. For example, the
quadruple root in fig 6.10c does not cross the axis

24. PROBLEM
step
xi-1 xi
f(xi-1) f(xi)
xi+1
1 2 3 -1 16 2.05882
2 3 2.05882 16 -0.3908 2.08126
3 2.05882 2.08126 -0.3908 -0.1472 2.09482
2.09482 is correct up to three decimal points

25. PROBLEM
step
xi-1 xi
f(xi-1) f(xi)
xi+1
1 0 1 -1 0.6321 0.6127
2 1 0.6127 0.632 0.0708 0.5638
3 0.6127 0.5638 0.0708 -0.0052 0.5672
0.5672 is correct up to three decimal points

26. PROBLEM
step i=0,1,2….. xi xi+1
1 x0 1 0.6839397
2 x1 0.6839397 0.5774545
3 x2 0.5774545 0.5672297
4 x3 0.5672297 0.5671433
0.5671433 is correct up to five decimal points

27.  Find a real root of the equation:
 Xsinx+cosx=0
 f(x)=Xsinx+cosx
 f’(x)= xcosx+sinx-sinx = xcosx
 xi+1 =xi-{(Xisinxi+cosxi)/ xcosxi}
PROBLEM

28. solution
 Using NR method:
step i=0,1,2….. xi xi+1
1 x0 3.1416 2.8233
2 x1 2.8233 2.7986
3 x2 2.7986 2.7984
4 x3 2.7984 2.7984
2.7984 is the root of the equation

29. PROBLEM

30.  Using Bisection method:
solution
0.5672 is correct up to three decimal points

31. PROBLEM

32. solution
1.3203125 is correct up to two decimal points

33. PROBLEM

34. solution
step xi xu
Xr= xu-{f(xu)(xi-
xu)/f(xi)-f(xu)}
Sign of
f(xi)
Sign of
f(xr)
Sign of
f(xi) f(xr)
1 2 1 1.33
+ - -
2 2 1.33 1.399
+ - -
3 2 1.399 1.412 + - -
1.412 is correct up to two decimal points

35.  Introduction Method of Numerical Analysis
(S.S Sastry) Page: 22-72
 Numerical methods for engineers ( Steven c.
Chapra, Raymond p. canale) page: 124-54
Reference

36. Reference
http://www.sosmath.com/algebra/quadraticeq
/root/root.html
http://www.algebra.com/algebra/homework/Average/A
verage.faq.question.1129.html
http://www.thefreedictionary.com/Root
+of+an+equation

37. THE END
THANK YOU FOR BEING WITH
US