Dr Marco A Arocha Aug 2014 1 Roots Roots problems occur when some function f can be written in terms of one or more dependent variables x where the solutions to fx0 yields the solution to the problem ID: 174209
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Slide1
Root Finding: Bisection Method
Dr. Marco A. ArochaAug, 2014
1Slide2
Roots
“Roots” problems occur when some function
f
can be written in terms of one or more dependent variables
x, where the solutions to f(x)=0 yields the solution to the problem.Examples:These problems often occur when a design problem presents an implicit equation (as oppose to an explicit equation) for a required parameter.
2Slide3
Roots or “Zeros”
Finding the roots of these equations is equivalent to finding the values of x for which f(x) and g(x) are “zero”
For this reason the procedure of solving equations (3) and (4) is frequently referred to as finding the “zeros” of the equations.
3Slide4
Bracketing methods
Methods that exploit the fact that a
function typically
changes sign in the vicinity of a
root are called bracketing methods.Two initial guesses for the root are required. Guesses must “bracket” or be on either side of the root. The particular methods employ different strategies to systematically reduce the width of the bracket and, hence, home in on the correct answer.
4Slide5
Graphical Methods
A simple method for obtaining the estimate of the root of the equation
f(x)
=0 is to make a plot of the function
as f(x) vs x and observe where it crosses the x-axis.Graphing the function can also indicate where roots may be and where some root-finding methods may fail:Same sign, no roots
Different sign, one root
Same sign, two roots
Different sign, three roots
5
who is doing this one?Slide6
Graphical methods
Some difficult cases:
Multiple
roots
that occurs when the function is tangential to the x axis. For this case, although the end points are of opposite signs, there are an even number of axis intersections for the interval. Discontinuous function where end points of opposite sign bracket an even number of roots. Special strategies are required for determining the roots for these cases.
6Slide7
GRAPHICAL METHODS
Graphical techniques alone are
of limited practical value because they are not precise.
However, graphical
methods can be utilized to obtain rough estimates of roots. These estimates can be employed as starting guesses for numerical methods.Graphical interpretations are important tools for understanding the properties of the functions and anticipating the pitfalls of the numerical methods.
7Slide8
Bisection
The
bisection method
is a
search method in which the interval is progresively divided in half.If a function changes sign over an interval, the function value at the midpoint is evaluated.The location of the root is then determined as lying within the subinterval where the sign change occurs.The absolute error is reduced by a factor of 2 for each iteration.
8Slide9
bisection method
9Slide10
Termination criteria
A
simple termination
criteria can be used:(1) An approximate error εa can be calculated:
where
is
the root
lower bound and
is
the root
upper bound from
the
present
iteration.
The absolute value is used because we are usually concerned with the magnitude
of
ε
a
rather than with its sign.
10Slide11
Termination criteria
Another
forms of termination criteria can be used:
(2) An approximate percent relative error εa can be calculated:
100%
where
is
the root for the present iteration and
is
the root from the previous iteration.
The absolute value is used because we are usually concerned with the magnitude
of
ε
a
rather than with its sign.
11Slide12
BISECTION
PSEUDOCODE
INPUT xl
,
xu, es, imaxxr = (xl + xu) / 2iter = 0
DO
xrold
=
xr
xr
=
(xl
+
xu
) /
2
iter = iter
+ 1
IF
xr
≠
0
THEN % to avoid division by zero
ea
=
ABS((
xr
-
xrold
) /
xr
) * 100
END
IF
test = f(xl) * f(xr) IF test < 0 THEN xu = xr
ELSE IF test > 0 THEN
xl =
xr
ELSE ea = 0 END IF IF ea < es OR iter
≥ imax EXITEND DO
BisectResult
=
xrEND PROGRAM
12Slide13
false position method
An alternative method to bisection,
sometimes
faster.
Exploits a graphical perception, join f(xl) 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” of the root is the origin of the name, method of false position, or in Latin,
regula falsi.
It is also
called the linear interpolation method.
13Slide14
false position method
Can derive the method’s formula by using similar
triangles.
The intersection of the straight line with the x axis can be estimated aswhich can be solved for xr
14Slide15
false position method
The
value of
x
r so computed then replaces whichever of the two initial guesses, xl or xu, yields a function value with the same sign as f(xr). By
this way, the values of xl and x
u
always bracket the true root.
The
process
is repeated
until the root is
estimated.
The
algorithm is identical to the one for
bisection with
the exception that
the above eq. is used for step 2 (slide 9). The same stopping criteria are used to terminate the computation (slides 10 and 11).
15Slide16
pitfalls
16
for some cases false-position method may show slow convergence