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ROOT FINDING: BISECTION METHODDr. Marco A. Arocha

Aug, 2014

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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.

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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.

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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.

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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:

a) Same sign, no rootsb) Different sign, one rootc) Same sign, two rootsd) Different sign, three roots

who is doing this one?

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GRAPHICAL METHODSSome 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.

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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.

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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.

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BISECTION METHOD

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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 εa rather than with its sign.

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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 εa rather than with its sign.

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BISECTION PSEUDOCODEINPUT xl, xu, es, imax

xr = (xl + xu) / 2iter = 0

DOxrold = xrxr = (xl + xu) / 2iter = iter + 1

IF xr ≠ 0 THEN % to avoid division by zeroea = ABS((xr - xrold) / xr) * 100

END IFtest = f(xl) * f(xr)IF test < 0 THEN

xu = xrELSE IF test > 0 THEN

xl = xrELSE

ea = 0END IFIF ea < es OR iter ≥ imax EXIT

END DOBisectResult = xrEND PROGRAM

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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.

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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 as

which can be solved for xr

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FALSE POSITION METHOD The value of xr 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 xu 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).

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PITFALLS

for some cases false-position method may show slow convergence